Step | Hyp | Ref
| Expression |
1 | | eqid 2731 |
. . . . . . . . . . . . 13
⊢
(Base‘𝑈) =
(Base‘𝑈) |
2 | | eqid 2731 |
. . . . . . . . . . . . 13
⊢
(.r‘𝑈) = (.r‘𝑈) |
3 | | evlselv.u |
. . . . . . . . . . . . . . . 16
⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) |
4 | | evlselv.i |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐼 ∈ 𝑉) |
5 | | difssd 4132 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐼 ∖ 𝐽) ⊆ 𝐼) |
6 | 4, 5 | ssexd 5324 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐼 ∖ 𝐽) ∈ V) |
7 | | evlselv.r |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑅 ∈ CRing) |
8 | 3, 6, 7 | mplcrngd 41584 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑈 ∈ CRing) |
9 | 8 | crngringd 20147 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑈 ∈ Ring) |
10 | 9 | ad2antrr 723 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑈 ∈ Ring) |
11 | | evlselv.t |
. . . . . . . . . . . . . . . 16
⊢ 𝑇 = (𝐽 mPoly 𝑈) |
12 | | eqid 2731 |
. . . . . . . . . . . . . . . 16
⊢
(Base‘𝑇) =
(Base‘𝑇) |
13 | | eqid 2731 |
. . . . . . . . . . . . . . . 16
⊢ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} = {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈
Fin} |
14 | | evlselv.p |
. . . . . . . . . . . . . . . . 17
⊢ 𝑃 = (𝐼 mPoly 𝑅) |
15 | | evlselv.b |
. . . . . . . . . . . . . . . . 17
⊢ 𝐵 = (Base‘𝑃) |
16 | | evlselv.j |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐽 ⊆ 𝐼) |
17 | | evlselv.f |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐹 ∈ 𝐵) |
18 | 14, 15, 3, 11, 12, 4, 7, 16, 17 | selvcl 41621 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ (Base‘𝑇)) |
19 | 11, 1, 12, 13, 18 | mplelf 21869 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹):{𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈
Fin}⟶(Base‘𝑈)) |
20 | 19 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹):{𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈
Fin}⟶(Base‘𝑈)) |
21 | 20 | ffvelcdmda 7086 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒) ∈ (Base‘𝑈)) |
22 | | eqid 2731 |
. . . . . . . . . . . . . 14
⊢
(mulGrp‘𝑈) =
(mulGrp‘𝑈) |
23 | | eqid 2731 |
. . . . . . . . . . . . . 14
⊢
(.g‘(mulGrp‘𝑈)) =
(.g‘(mulGrp‘𝑈)) |
24 | 4, 16 | ssexd 5324 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐽 ∈ V) |
25 | 24 | ad2antrr 723 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝐽 ∈ V) |
26 | 8 | ad2antrr 723 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑈 ∈ CRing) |
27 | | fvexd 6906 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (Base‘𝑈) ∈ V) |
28 | | evlselv.k |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝐾 = (Base‘𝑅) |
29 | 28 | fvexi 6905 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝐾 ∈ V |
30 | 29 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐾 ∈ V) |
31 | | evlselv.l |
. . . . . . . . . . . . . . . . . 18
⊢ 𝐿 = (algSc‘𝑈) |
32 | 7 | crngringd 20147 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑅 ∈ Ring) |
33 | 3, 1, 28, 31, 6, 32 | mplasclf 21938 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐿:𝐾⟶(Base‘𝑈)) |
34 | 27, 30, 33 | elmapdd 8841 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐿 ∈ ((Base‘𝑈) ↑m 𝐾)) |
35 | | evlselv.a |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) |
36 | 35, 16 | elmapssresd 41536 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐴 ↾ 𝐽) ∈ (𝐾 ↑m 𝐽)) |
37 | 34, 36 | mapcod 41537 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐿 ∘ (𝐴 ↾ 𝐽)) ∈ ((Base‘𝑈) ↑m 𝐽)) |
38 | 37 | ad2antrr 723 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝐿 ∘ (𝐴 ↾ 𝐽)) ∈ ((Base‘𝑈) ↑m 𝐽)) |
39 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈
Fin}) |
40 | 13, 1, 22, 23, 25, 26, 38, 39 | evlsvvvallem 41599 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((mulGrp‘𝑈)
Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))) ∈ (Base‘𝑈)) |
41 | 1, 2, 10, 21, 40 | ringcld 20158 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))) ∈ (Base‘𝑈)) |
42 | | eqidd 2732 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))) = (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))) |
43 | | eqidd 2732 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) = (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐))) |
44 | | fveq1 6890 |
. . . . . . . . . . . 12
⊢ (𝑢 = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))) → (𝑢‘𝑐) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))‘𝑐)) |
45 | 41, 42, 43, 44 | fmptco 7129 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))) = (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))‘𝑐))) |
46 | 33 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝐿:𝐾⟶(Base‘𝑈)) |
47 | | eqid 2731 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(mulGrp‘𝑅) =
(mulGrp‘𝑅) |
48 | 47, 28 | mgpbas 20041 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝐾 =
(Base‘(mulGrp‘𝑅)) |
49 | | eqid 2731 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(.g‘(mulGrp‘𝑅)) =
(.g‘(mulGrp‘𝑅)) |
50 | 47 | ringmgp 20140 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑅 ∈ Ring →
(mulGrp‘𝑅) ∈
Mnd) |
51 | 32, 50 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd) |
52 | 51 | ad3antrrr 727 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (mulGrp‘𝑅) ∈ Mnd) |
53 | 13 | psrbagf 21782 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} → 𝑒:𝐽⟶ℕ0) |
54 | 53 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑒:𝐽⟶ℕ0) |
55 | 54 | ffvelcdmda 7086 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (𝑒‘𝑗) ∈
ℕ0) |
56 | | elmapi 8849 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝐴 ∈ (𝐾 ↑m 𝐼) → 𝐴:𝐼⟶𝐾) |
57 | 35, 56 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝐴:𝐼⟶𝐾) |
58 | 57, 16 | fssresd 6758 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (𝐴 ↾ 𝐽):𝐽⟶𝐾) |
59 | 58 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝐴 ↾ 𝐽):𝐽⟶𝐾) |
60 | 59 | ffvelcdmda 7086 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝐴 ↾ 𝐽)‘𝑗) ∈ 𝐾) |
61 | 48, 49, 52, 55, 60 | mulgnn0cld 19018 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) ∈ 𝐾) |
62 | 46, 61 | cofmpt 7132 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝐿 ∘ (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) = (𝑗 ∈ 𝐽 ↦ (𝐿‘((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) |
63 | 3 | mplassa 21893 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝐼 ∖ 𝐽) ∈ V ∧ 𝑅 ∈ CRing) → 𝑈 ∈ AssAlg) |
64 | 6, 7, 63 | syl2anc 583 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝑈 ∈ AssAlg) |
65 | | eqid 2731 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(Scalar‘𝑈) =
(Scalar‘𝑈) |
66 | 31, 65 | asclrhm 21755 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑈 ∈ AssAlg → 𝐿 ∈ ((Scalar‘𝑈) RingHom 𝑈)) |
67 | 64, 66 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → 𝐿 ∈ ((Scalar‘𝑈) RingHom 𝑈)) |
68 | 3, 6, 7 | mplsca 21884 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → 𝑅 = (Scalar‘𝑈)) |
69 | 68 | eqcomd 2737 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (Scalar‘𝑈) = 𝑅) |
70 | 69 | oveq1d 7427 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → ((Scalar‘𝑈) RingHom 𝑈) = (𝑅 RingHom 𝑈)) |
71 | 67, 70 | eleqtrd 2834 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝐿 ∈ (𝑅 RingHom 𝑈)) |
72 | 47, 22 | rhmmhm 20377 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝐿 ∈ (𝑅 RingHom 𝑈) → 𝐿 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑈))) |
73 | 71, 72 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝐿 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑈))) |
74 | 73 | ad3antrrr 727 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → 𝐿 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑈))) |
75 | 48, 49, 23 | mhmmulg 19038 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐿 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑈)) ∧ (𝑒‘𝑗) ∈ ℕ0 ∧ ((𝐴 ↾ 𝐽)‘𝑗) ∈ 𝐾) → (𝐿‘((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) = ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))(𝐿‘((𝐴 ↾ 𝐽)‘𝑗)))) |
76 | 74, 55, 60, 75 | syl3anc 1370 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (𝐿‘((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) = ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))(𝐿‘((𝐴 ↾ 𝐽)‘𝑗)))) |
77 | 58 | ad3antrrr 727 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (𝐴 ↾ 𝐽):𝐽⟶𝐾) |
78 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → 𝑗 ∈ 𝐽) |
79 | 77, 78 | fvco3d 6991 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗) = (𝐿‘((𝐴 ↾ 𝐽)‘𝑗))) |
80 | 79 | oveq2d 7428 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)) = ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))(𝐿‘((𝐴 ↾ 𝐽)‘𝑗)))) |
81 | 76, 80 | eqtr4d 2774 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (𝐿‘((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) = ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))) |
82 | 81 | mpteq2dva 5248 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ (𝐿‘((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) = (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))) |
83 | 62, 82 | eqtrd 2771 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝐿 ∘ (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) = (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))) |
84 | 83 | oveq2d 7428 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((mulGrp‘𝑈)
Σg (𝐿 ∘ (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) = ((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))) |
85 | | eqid 2731 |
. . . . . . . . . . . . . . . . . 18
⊢
(Base‘(mulGrp‘(Scalar‘𝑈))) =
(Base‘(mulGrp‘(Scalar‘𝑈))) |
86 | | eqid 2731 |
. . . . . . . . . . . . . . . . . 18
⊢
(0g‘(mulGrp‘(Scalar‘𝑈))) =
(0g‘(mulGrp‘(Scalar‘𝑈))) |
87 | 68, 7 | eqeltrrd 2833 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (Scalar‘𝑈) ∈ CRing) |
88 | | eqid 2731 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(mulGrp‘(Scalar‘𝑈)) = (mulGrp‘(Scalar‘𝑈)) |
89 | 88 | crngmgp 20142 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((Scalar‘𝑈)
∈ CRing → (mulGrp‘(Scalar‘𝑈)) ∈ CMnd) |
90 | 87, 89 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 →
(mulGrp‘(Scalar‘𝑈)) ∈ CMnd) |
91 | 90 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
(mulGrp‘(Scalar‘𝑈)) ∈ CMnd) |
92 | 22 | ringmgp 20140 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑈 ∈ Ring →
(mulGrp‘𝑈) ∈
Mnd) |
93 | 9, 92 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (mulGrp‘𝑈) ∈ Mnd) |
94 | 93 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
(mulGrp‘𝑈) ∈
Mnd) |
95 | 88, 22 | rhmmhm 20377 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐿 ∈ ((Scalar‘𝑈) RingHom 𝑈) → 𝐿 ∈ ((mulGrp‘(Scalar‘𝑈)) MndHom (mulGrp‘𝑈))) |
96 | 67, 95 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐿 ∈ ((mulGrp‘(Scalar‘𝑈)) MndHom (mulGrp‘𝑈))) |
97 | 96 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝐿 ∈
((mulGrp‘(Scalar‘𝑈)) MndHom (mulGrp‘𝑈))) |
98 | 68 | fveq2d 6895 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (Base‘𝑅) =
(Base‘(Scalar‘𝑈))) |
99 | 28, 98 | eqtrid 2783 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑈))) |
100 | 99 | ad3antrrr 727 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → 𝐾 = (Base‘(Scalar‘𝑈))) |
101 | 61, 100 | eleqtrd 2834 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) ∈ (Base‘(Scalar‘𝑈))) |
102 | | eqid 2731 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈)) |
103 | 88, 102 | mgpbas 20041 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(Base‘(Scalar‘𝑈)) =
(Base‘(mulGrp‘(Scalar‘𝑈))) |
104 | 101, 103 | eleqtrdi 2842 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) ∈
(Base‘(mulGrp‘(Scalar‘𝑈)))) |
105 | 104 | fmpttd 7116 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))):𝐽⟶(Base‘(mulGrp‘(Scalar‘𝑈)))) |
106 | 24 | mptexd 7228 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) ∈ V) |
107 | 106 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) ∈ V) |
108 | | fvexd 6906 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
(0g‘(mulGrp‘𝑅)) ∈ V) |
109 | | funmpt 6586 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ Fun
(𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) |
110 | 109 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → Fun
(𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) |
111 | 54 | feqmptd 6960 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑒 = (𝑗 ∈ 𝐽 ↦ (𝑒‘𝑗))) |
112 | 13 | psrbagfsupp 21784 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} → 𝑒 finSupp 0) |
113 | 112 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑒 finSupp 0) |
114 | 111, 113 | eqbrtrrd 5172 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ (𝑒‘𝑗)) finSupp 0) |
115 | | ssidd 4005 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((𝑗 ∈ 𝐽 ↦ (𝑒‘𝑗)) supp 0) ⊆ ((𝑗 ∈ 𝐽 ↦ (𝑒‘𝑗)) supp 0)) |
116 | | eqid 2731 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(0g‘(mulGrp‘𝑅)) =
(0g‘(mulGrp‘𝑅)) |
117 | 48, 116, 49 | mulg0 19000 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ 𝐾 →
(0(.g‘(mulGrp‘𝑅))𝑘) = (0g‘(mulGrp‘𝑅))) |
118 | 117 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ 𝐾) →
(0(.g‘(mulGrp‘𝑅))𝑘) = (0g‘(mulGrp‘𝑅))) |
119 | | 0zd 12577 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 0
∈ ℤ) |
120 | 115, 118,
55, 60, 119 | suppssov1 8188 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → ((𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) supp
(0g‘(mulGrp‘𝑅))) ⊆ ((𝑗 ∈ 𝐽 ↦ (𝑒‘𝑗)) supp 0)) |
121 | 107, 108,
110, 114, 120 | fsuppsssuppgd 41534 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) finSupp
(0g‘(mulGrp‘𝑅))) |
122 | | eqid 2731 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(1r‘𝑅) = (1r‘𝑅) |
123 | 47, 122 | ringidval 20084 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(1r‘𝑅) = (0g‘(mulGrp‘𝑅)) |
124 | 121, 123 | breqtrrdi 5190 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) finSupp (1r‘𝑅)) |
125 | 68 | fveq2d 6895 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (1r‘𝑅) =
(1r‘(Scalar‘𝑈))) |
126 | | eqid 2731 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(1r‘(Scalar‘𝑈)) =
(1r‘(Scalar‘𝑈)) |
127 | 88, 126 | ringidval 20084 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(1r‘(Scalar‘𝑈)) =
(0g‘(mulGrp‘(Scalar‘𝑈))) |
128 | 125, 127 | eqtrdi 2787 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (1r‘𝑅) =
(0g‘(mulGrp‘(Scalar‘𝑈)))) |
129 | 128 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
(1r‘𝑅) =
(0g‘(mulGrp‘(Scalar‘𝑈)))) |
130 | 124, 129 | breqtrd 5174 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) finSupp
(0g‘(mulGrp‘(Scalar‘𝑈)))) |
131 | 85, 86, 91, 94, 25, 97, 105, 130 | gsummhm 19854 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((mulGrp‘𝑈)
Σg (𝐿 ∘ (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) = (𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg
(𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) |
132 | 84, 131 | eqtr3d 2773 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((mulGrp‘𝑈)
Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))) = (𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg
(𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) |
133 | 132 | oveq2d 7428 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)(𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg
(𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))))) |
134 | 64 | ad2antrr 723 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑈 ∈ AssAlg) |
135 | 101 | fmpttd 7116 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))):𝐽⟶(Base‘(Scalar‘𝑈))) |
136 | 130, 127 | breqtrrdi 5190 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) finSupp
(1r‘(Scalar‘𝑈))) |
137 | 103, 127,
91, 25, 135, 136 | gsumcl 19831 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) ∈ (Base‘(Scalar‘𝑈))) |
138 | | eqid 2731 |
. . . . . . . . . . . . . . . . 17
⊢ (
·𝑠 ‘𝑈) = ( ·𝑠
‘𝑈) |
139 | 31, 65, 102, 1, 2, 138 | asclmul2 21752 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑈 ∈ AssAlg ∧
((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) ∈ (Base‘(Scalar‘𝑈)) ∧ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒) ∈ (Base‘𝑈)) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)(𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg
(𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) = (((mulGrp‘(Scalar‘𝑈)) Σg
(𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))( ·𝑠
‘𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))) |
140 | 134, 137,
21, 139 | syl3anc 1370 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)(𝐿‘((mulGrp‘(Scalar‘𝑈)) Σg
(𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) = (((mulGrp‘(Scalar‘𝑈)) Σg
(𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))( ·𝑠
‘𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))) |
141 | 133, 140 | eqtrd 2771 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))) = (((mulGrp‘(Scalar‘𝑈)) Σg
(𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))( ·𝑠
‘𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))) |
142 | 141 | fveq1d 6893 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))‘𝑐) = ((((mulGrp‘(Scalar‘𝑈)) Σg
(𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))( ·𝑠
‘𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))‘𝑐)) |
143 | | eqid 2731 |
. . . . . . . . . . . . . 14
⊢
(.r‘𝑅) = (.r‘𝑅) |
144 | | eqid 2731 |
. . . . . . . . . . . . . 14
⊢ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈
Fin} |
145 | 99 | ad2antrr 723 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝐾 =
(Base‘(Scalar‘𝑈))) |
146 | 137, 145 | eleqtrrd 2835 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) ∈ 𝐾) |
147 | | simplr 766 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈
Fin}) |
148 | 3, 138, 28, 1, 143, 144, 146, 21, 147 | mplvscaval 21887 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))( ·𝑠
‘𝑈)((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒))‘𝑐) = (((mulGrp‘(Scalar‘𝑈)) Σg
(𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))) |
149 | 142, 148 | eqtrd 2771 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))‘𝑐) = (((mulGrp‘(Scalar‘𝑈)) Σg
(𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))) |
150 | 149 | mpteq2dva 5248 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))‘𝑐)) = (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))) |
151 | 45, 150 | eqtrd 2771 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))) = (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))) |
152 | 151 | oveq2d 7428 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg
((𝑢 ∈
(Base‘𝑈) ↦
(𝑢‘𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))))) |
153 | 69 | fveq2d 6895 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 →
(mulGrp‘(Scalar‘𝑈)) = (mulGrp‘𝑅)) |
154 | 153 | ad2antrr 723 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
(mulGrp‘(Scalar‘𝑈)) = (mulGrp‘𝑅)) |
155 | 154 | oveq1d 7427 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) = ((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) |
156 | 155 | oveq1d 7427 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
(((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) = (((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))) |
157 | 7 | ad2antrr 723 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing) |
158 | 155, 146 | eqeltrrd 2833 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((mulGrp‘𝑅)
Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) ∈ 𝐾) |
159 | 19 | ffvelcdmda 7086 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒) ∈ (Base‘𝑈)) |
160 | 3, 28, 1, 144, 159 | mplelf 21869 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒):{𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) |
161 | 160 | ffvelcdmda 7086 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐) ∈ 𝐾) |
162 | 161 | an32s 649 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐) ∈ 𝐾) |
163 | 28, 143, 157, 158, 162 | crngcomd 41555 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
(((mulGrp‘𝑅)
Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) |
164 | 156, 163 | eqtrd 2771 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
(((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) |
165 | 164 | mpteq2dva 5248 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))) = (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))))) |
166 | 165 | oveq2d 7428 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg
(𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((mulGrp‘(Scalar‘𝑈)) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))))) |
167 | 152, 166 | eqtrd 2771 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg
((𝑢 ∈
(Base‘𝑈) ↦
(𝑢‘𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))))) |
168 | 167 | oveq1d 7427 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑅 Σg
((𝑢 ∈
(Base‘𝑈) ↦
(𝑢‘𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))))(.r‘𝑅)((mulGrp‘𝑅) Σg
(𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = ((𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) |
169 | | eqid 2731 |
. . . . . . . . . 10
⊢ (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) = (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) |
170 | | fveq1 6890 |
. . . . . . . . . 10
⊢ (𝑢 = (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))) → (𝑢‘𝑐) = ((𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))‘𝑐)) |
171 | | eqid 2731 |
. . . . . . . . . . . . 13
⊢ (𝐽 eval 𝑈) = (𝐽 eval 𝑈) |
172 | 171, 11, 12, 13, 1, 22, 23, 2, 24, 8, 18, 37 | evlvvval 41611 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽))) = (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))) |
173 | 171, 11, 12, 1, 24, 8, 18, 37 | evlcl 41610 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽))) ∈ (Base‘𝑈)) |
174 | 172, 173 | eqeltrrd 2833 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))) ∈ (Base‘𝑈)) |
175 | 174 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑈 Σg
(𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))) ∈ (Base‘𝑈)) |
176 | | fvexd 6906 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑈 Σg
(𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))‘𝑐) ∈ V) |
177 | 169, 170,
175, 176 | fvmptd3 7021 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐))‘(𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))) = ((𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))‘𝑐)) |
178 | | eqid 2731 |
. . . . . . . . . 10
⊢
(0g‘𝑈) = (0g‘𝑈) |
179 | 9 | ringcmnd 20179 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑈 ∈ CMnd) |
180 | 179 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑈 ∈ CMnd) |
181 | 7 | crnggrpd 20148 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑅 ∈ Grp) |
182 | 181 | grpmndd 18874 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑅 ∈ Mnd) |
183 | 182 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Mnd) |
184 | | ovex 7445 |
. . . . . . . . . . . 12
⊢
(ℕ0 ↑m 𝐽) ∈ V |
185 | 184 | rabex 5332 |
. . . . . . . . . . 11
⊢ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ∈
V |
186 | 185 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ∈
V) |
187 | 6 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐼 ∖ 𝐽) ∈ V) |
188 | 181 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Grp) |
189 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈
Fin}) |
190 | 3, 1, 144, 169, 187, 188, 189 | mplmapghm 41594 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∈ (𝑈 GrpHom 𝑅)) |
191 | | ghmmhm 19147 |
. . . . . . . . . . 11
⊢ ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∈ (𝑈 GrpHom 𝑅) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∈ (𝑈 MndHom 𝑅)) |
192 | 190, 191 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∈ (𝑈 MndHom 𝑅)) |
193 | 41 | fmpttd 7116 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))):{𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈
Fin}⟶(Base‘𝑈)) |
194 | 24 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝐽 ∈ V) |
195 | 8 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑈 ∈ CRing) |
196 | 18 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ (Base‘𝑇)) |
197 | 37 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐿 ∘ (𝐴 ↾ 𝐽)) ∈ ((Base‘𝑈) ↑m 𝐽)) |
198 | 13, 11, 12, 1, 22, 23, 2, 194, 195, 196, 197 | evlvvvallem 41612 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))) finSupp (0g‘𝑈)) |
199 | 1, 178, 180, 183, 186, 192, 193, 198 | gsummhm 19854 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg
((𝑢 ∈
(Base‘𝑈) ↦
(𝑢‘𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))) = ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐))‘(𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))))) |
200 | 172 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽))) = (𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))) |
201 | 200 | fveq1d 6893 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽)))‘𝑐) = ((𝑈 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗)))))))‘𝑐)) |
202 | 177, 199,
201 | 3eqtr4rd 2782 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽)))‘𝑐) = (𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))))) |
203 | 202 | oveq1d 7427 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽)))‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = ((𝑅 Σg ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∘ (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)(.r‘𝑈)((mulGrp‘𝑈) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑈))((𝐿 ∘ (𝐴 ↾ 𝐽))‘𝑗))))))))(.r‘𝑅)((mulGrp‘𝑅) Σg
(𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) |
204 | | eqid 2731 |
. . . . . . . 8
⊢
(0g‘𝑅) = (0g‘𝑅) |
205 | 32 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring) |
206 | 47 | crngmgp 20142 |
. . . . . . . . . . 11
⊢ (𝑅 ∈ CRing →
(mulGrp‘𝑅) ∈
CMnd) |
207 | 7, 206 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (mulGrp‘𝑅) ∈ CMnd) |
208 | 207 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(mulGrp‘𝑅) ∈
CMnd) |
209 | 51 | ad2antrr 723 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (mulGrp‘𝑅) ∈ Mnd) |
210 | 144 | psrbagf 21782 |
. . . . . . . . . . . . 13
⊢ (𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑐:(𝐼 ∖ 𝐽)⟶ℕ0) |
211 | 210 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑐:(𝐼 ∖ 𝐽)⟶ℕ0) |
212 | 211 | ffvelcdmda 7086 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (𝑐‘𝑘) ∈
ℕ0) |
213 | 57, 5 | fssresd 6758 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴 ↾ (𝐼 ∖ 𝐽)):(𝐼 ∖ 𝐽)⟶𝐾) |
214 | 213 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐴 ↾ (𝐼 ∖ 𝐽)):(𝐼 ∖ 𝐽)⟶𝐾) |
215 | 214 | ffvelcdmda 7086 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘) ∈ 𝐾) |
216 | 48, 49, 209, 212, 215 | mulgnn0cld 19018 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) ∈ 𝐾) |
217 | 216 | fmpttd 7116 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))):(𝐼 ∖ 𝐽)⟶𝐾) |
218 | 6 | mptexd 7228 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) ∈ V) |
219 | 218 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) ∈ V) |
220 | | fvexd 6906 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(0g‘(mulGrp‘𝑅)) ∈ V) |
221 | | funmpt 6586 |
. . . . . . . . . . 11
⊢ Fun
(𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) |
222 | 221 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → Fun
(𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) |
223 | 211 | feqmptd 6960 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑐 = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑐‘𝑘))) |
224 | 144 | psrbagfsupp 21784 |
. . . . . . . . . . . 12
⊢ (𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑐 finSupp 0) |
225 | 224 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑐 finSupp 0) |
226 | 223, 225 | eqbrtrrd 5172 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑐‘𝑘)) finSupp 0) |
227 | | ssidd 4005 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑐‘𝑘)) supp 0) ⊆ ((𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑐‘𝑘)) supp 0)) |
228 | 48, 116, 49 | mulg0 19000 |
. . . . . . . . . . . 12
⊢ (𝑣 ∈ 𝐾 →
(0(.g‘(mulGrp‘𝑅))𝑣) = (0g‘(mulGrp‘𝑅))) |
229 | 228 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣 ∈ 𝐾) →
(0(.g‘(mulGrp‘𝑅))𝑣) = (0g‘(mulGrp‘𝑅))) |
230 | | fvexd 6906 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (𝑐‘𝑘) ∈ V) |
231 | | 0zd 12577 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 0
∈ ℤ) |
232 | 227, 229,
230, 215, 231 | suppssov1 8188 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) supp
(0g‘(mulGrp‘𝑅))) ⊆ ((𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑐‘𝑘)) supp 0)) |
233 | 219, 220,
222, 226, 232 | fsuppsssuppgd 41534 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) finSupp
(0g‘(mulGrp‘𝑅))) |
234 | 48, 116, 208, 187, 217, 233 | gsumcl 19831 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
((mulGrp‘𝑅)
Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) ∈ 𝐾) |
235 | 32 | ad2antrr 723 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring) |
236 | 28, 143, 235, 162, 158 | ringcld 20158 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) ∈ 𝐾) |
237 | 185 | mptex 7227 |
. . . . . . . . . 10
⊢ (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) ∈ V |
238 | 237 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) ∈ V) |
239 | | fvexd 6906 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(0g‘𝑅)
∈ V) |
240 | | funmpt 6586 |
. . . . . . . . . 10
⊢ Fun
(𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) |
241 | 240 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → Fun
(𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))))) |
242 | 185 | mptex 7227 |
. . . . . . . . . . 11
⊢ (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) ∈ V |
243 | 242 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) ∈ V) |
244 | | funmpt 6586 |
. . . . . . . . . . 11
⊢ Fun
(𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) |
245 | 244 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → Fun
(𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐))) |
246 | 11, 12, 178, 18, 8 | mplelsfi 21866 |
. . . . . . . . . . 11
⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) finSupp (0g‘𝑈)) |
247 | 246 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) finSupp (0g‘𝑈)) |
248 | | ssidd 4005 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g‘𝑈)) ⊆ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g‘𝑈))) |
249 | | fvexd 6906 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(0g‘𝑈)
∈ V) |
250 | 20, 248, 186, 249 | suppssr 8186 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ ({𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ∖
((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g‘𝑈)))) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒) = (0g‘𝑈)) |
251 | 250 | fveq1d 6893 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ ({𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ∖
((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g‘𝑈)))) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐) = ((0g‘𝑈)‘𝑐)) |
252 | 3, 144, 204, 178, 6, 181 | mpl0 21877 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (0g‘𝑈) = ({𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ×
{(0g‘𝑅)})) |
253 | 252 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(0g‘𝑈) =
({𝑓 ∈
(ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ×
{(0g‘𝑅)})) |
254 | 253 | fveq1d 6893 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
((0g‘𝑈)‘𝑐) = (({𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ×
{(0g‘𝑅)})‘𝑐)) |
255 | | fvex 6904 |
. . . . . . . . . . . . . . . 16
⊢
(0g‘𝑅) ∈ V |
256 | 255 | fvconst2 7207 |
. . . . . . . . . . . . . . 15
⊢ (𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} → (({𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ×
{(0g‘𝑅)})‘𝑐) = (0g‘𝑅)) |
257 | 256 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(({𝑓 ∈
(ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ×
{(0g‘𝑅)})‘𝑐) = (0g‘𝑅)) |
258 | 254, 257 | eqtrd 2771 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
((0g‘𝑈)‘𝑐) = (0g‘𝑅)) |
259 | 258 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ ({𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ∖
((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g‘𝑈)))) →
((0g‘𝑈)‘𝑐) = (0g‘𝑅)) |
260 | 251, 259 | eqtrd 2771 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ ({𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ∖
((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g‘𝑈)))) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐) = (0g‘𝑅)) |
261 | 260, 186 | suppss2 8191 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) supp (0g‘𝑅)) ⊆ ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) supp (0g‘𝑈))) |
262 | 243, 239,
245, 247, 261 | fsuppsssuppgd 41534 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) finSupp (0g‘𝑅)) |
263 | | ssidd 4005 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) supp (0g‘𝑅)) ⊆ ((𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) supp (0g‘𝑅))) |
264 | 32 | ad2antrr 723 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣 ∈ 𝐾) → 𝑅 ∈ Ring) |
265 | | simpr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣 ∈ 𝐾) → 𝑣 ∈ 𝐾) |
266 | 28, 143, 204, 264, 265 | ringlzd 20190 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣 ∈ 𝐾) → ((0g‘𝑅)(.r‘𝑅)𝑣) = (0g‘𝑅)) |
267 | 263, 266,
162, 158, 239 | suppssov1 8188 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) supp (0g‘𝑅)) ⊆ ((𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) supp (0g‘𝑅))) |
268 | 238, 239,
241, 262, 267 | fsuppsssuppgd 41534 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) finSupp (0g‘𝑅)) |
269 | 28, 204, 143, 205, 186, 234, 236, 268 | gsummulc1 20211 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg
(𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) = ((𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) |
270 | 168, 203,
269 | 3eqtr4d 2781 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽)))‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))))) |
271 | | fveq2 6891 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑒 → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)) |
272 | 271 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)) |
273 | | simpl 482 |
. . . . . . . . . . . . 13
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → 𝑏 = 𝑐) |
274 | 272, 273 | fveq12d 6898 |
. . . . . . . . . . . 12
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) |
275 | | fveq1 6890 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑒 → (𝑎‘𝑗) = (𝑒‘𝑗)) |
276 | 275 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → (𝑎‘𝑗) = (𝑒‘𝑗)) |
277 | 276 | oveq1d 7427 |
. . . . . . . . . . . . . 14
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) |
278 | 277 | mpteq2dv 5250 |
. . . . . . . . . . . . 13
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) = (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) |
279 | 278 | oveq2d 7428 |
. . . . . . . . . . . 12
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → ((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) = ((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) |
280 | 274, 279 | oveq12d 7430 |
. . . . . . . . . . 11
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) |
281 | | fveq1 6890 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = 𝑐 → (𝑏‘𝑘) = (𝑐‘𝑘)) |
282 | 281 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → (𝑏‘𝑘) = (𝑐‘𝑘)) |
283 | 282 | oveq1d 7427 |
. . . . . . . . . . . . 13
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) = ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) |
284 | 283 | mpteq2dv 5250 |
. . . . . . . . . . . 12
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) |
285 | 284 | oveq2d 7428 |
. . . . . . . . . . 11
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) = ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) |
286 | 280, 285 | oveq12d 7430 |
. . . . . . . . . 10
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) |
287 | | eqid 2731 |
. . . . . . . . . 10
⊢ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) = (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) |
288 | | ovex 7445 |
. . . . . . . . . 10
⊢
(((((((𝐼 selectVars
𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) ∈ V |
289 | 286, 287,
288 | ovmpoa 7566 |
. . . . . . . . 9
⊢ ((𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))𝑒) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) |
290 | 289 | adantll 711 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))𝑒) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) |
291 | 290 | mpteq2dva 5248 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))𝑒)) = (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) |
292 | 291 | oveq2d 7428 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg
(𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))𝑒))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))))) |
293 | 270, 292 | eqtr4d 2774 |
. . . . 5
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽)))‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = (𝑅 Σg (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))𝑒)))) |
294 | 293 | mpteq2dva 5248 |
. . . 4
⊢ (𝜑 → (𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦
(((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽)))‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) = (𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg
(𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))𝑒))))) |
295 | 294 | oveq2d 7428 |
. . 3
⊢ (𝜑 → (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦
(((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽)))‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) = (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg
(𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))𝑒)))))) |
296 | 32 | ringcmnd 20179 |
. . . . 5
⊢ (𝜑 → 𝑅 ∈ CMnd) |
297 | | ovex 7445 |
. . . . . . 7
⊢
(ℕ0 ↑m 𝐼) ∈ V |
298 | 297 | rabex 5332 |
. . . . . 6
⊢ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ∈
V |
299 | 298 | a1i 11 |
. . . . 5
⊢ (𝜑 → {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ∈
V) |
300 | 32 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring) |
301 | 19 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹):{𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈
Fin}⟶(Base‘𝑈)) |
302 | | eqid 2731 |
. . . . . . . . . . . 12
⊢ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} |
303 | 4 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑉) |
304 | 16 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐽 ⊆ 𝐼) |
305 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin}) |
306 | 302, 13, 303, 304, 305 | psrbagres 41581 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ↾ 𝐽) ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈
Fin}) |
307 | 301, 306 | ffvelcdmd 7087 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽)) ∈ (Base‘𝑈)) |
308 | 3, 28, 1, 144, 307 | mplelf 21869 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽)):{𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) |
309 | | difssd 4132 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝐼 ∖ 𝐽) ⊆ 𝐼) |
310 | 302, 144,
303, 309, 305 | psrbagres 41581 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ↾ (𝐼 ∖ 𝐽)) ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈
Fin}) |
311 | 308, 310 | ffvelcdmd 7087 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽))) ∈ 𝐾) |
312 | 207 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
(mulGrp‘𝑅) ∈
CMnd) |
313 | 24 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐽 ∈ V) |
314 | 51 | ad2antrr 723 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (mulGrp‘𝑅) ∈ Mnd) |
315 | 302 | psrbagf 21782 |
. . . . . . . . . . . . . 14
⊢ (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑑:𝐼⟶ℕ0) |
316 | 315 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑:𝐼⟶ℕ0) |
317 | 316, 304 | fssresd 6758 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ↾ 𝐽):𝐽⟶ℕ0) |
318 | 317 | ffvelcdmda 7086 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝑑 ↾ 𝐽)‘𝑗) ∈
ℕ0) |
319 | 58 | ffvelcdmda 7086 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐽) → ((𝐴 ↾ 𝐽)‘𝑗) ∈ 𝐾) |
320 | 319 | adantlr 712 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝐴 ↾ 𝐽)‘𝑗) ∈ 𝐾) |
321 | 48, 49, 314, 318, 320 | mulgnn0cld 19018 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) ∈ 𝐾) |
322 | 321 | fmpttd 7116 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))):𝐽⟶𝐾) |
323 | 24 | mptexd 7228 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) ∈ V) |
324 | 323 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) ∈ V) |
325 | | fvexd 6906 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
(0g‘(mulGrp‘𝑅)) ∈ V) |
326 | | funmpt 6586 |
. . . . . . . . . . 11
⊢ Fun
(𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) |
327 | 326 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → Fun
(𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) |
328 | 302 | psrbagfsupp 21784 |
. . . . . . . . . . . 12
⊢ (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑑 finSupp 0) |
329 | 328 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 finSupp 0) |
330 | | 0zd 12577 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 0 ∈
ℤ) |
331 | 329, 330 | fsuppres 9394 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ↾ 𝐽) finSupp 0) |
332 | | ssidd 4005 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑑 ↾ 𝐽) supp 0) ⊆ ((𝑑 ↾ 𝐽) supp 0)) |
333 | 317, 332,
313, 330 | suppssr 8186 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑 ↾ 𝐽) supp 0))) → ((𝑑 ↾ 𝐽)‘𝑗) = 0) |
334 | 333 | oveq1d 7427 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑 ↾ 𝐽) supp 0))) → (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) =
(0(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) |
335 | | eldifi 4126 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ (𝐽 ∖ ((𝑑 ↾ 𝐽) supp 0)) → 𝑗 ∈ 𝐽) |
336 | 48, 116, 49 | mulg0 19000 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ↾ 𝐽)‘𝑗) ∈ 𝐾 →
(0(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅))) |
337 | 320, 336 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) →
(0(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅))) |
338 | 335, 337 | sylan2 592 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑 ↾ 𝐽) supp 0))) →
(0(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅))) |
339 | 334, 338 | eqtrd 2771 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑 ↾ 𝐽) supp 0))) → (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅))) |
340 | 339, 313 | suppss2 8191 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) supp
(0g‘(mulGrp‘𝑅))) ⊆ ((𝑑 ↾ 𝐽) supp 0)) |
341 | 324, 325,
327, 331, 340 | fsuppsssuppgd 41534 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) finSupp
(0g‘(mulGrp‘𝑅))) |
342 | 48, 116, 312, 313, 322, 341 | gsumcl 19831 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
((mulGrp‘𝑅)
Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) ∈ 𝐾) |
343 | 28, 143, 300, 311, 342 | ringcld 20158 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) ∈ 𝐾) |
344 | 6 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝐼 ∖ 𝐽) ∈ V) |
345 | 51 | ad2antrr 723 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (mulGrp‘𝑅) ∈ Mnd) |
346 | 316, 309 | fssresd 6758 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ↾ (𝐼 ∖ 𝐽)):(𝐼 ∖ 𝐽)⟶ℕ0) |
347 | 346 | ffvelcdmda 7086 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘) ∈
ℕ0) |
348 | 213 | ffvelcdmda 7086 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘) ∈ 𝐾) |
349 | 348 | adantlr 712 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘) ∈ 𝐾) |
350 | 48, 49, 345, 347, 349 | mulgnn0cld 19018 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) ∈ 𝐾) |
351 | 350 | fmpttd 7116 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))):(𝐼 ∖ 𝐽)⟶𝐾) |
352 | 344 | mptexd 7228 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) ∈ V) |
353 | | funmpt 6586 |
. . . . . . . . . 10
⊢ Fun
(𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) |
354 | 353 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → Fun
(𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) |
355 | 329, 330 | fsuppres 9394 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ↾ (𝐼 ∖ 𝐽)) finSupp 0) |
356 | | ssidd 4005 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0) ⊆ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0)) |
357 | 346, 356,
344, 330 | suppssr 8186 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼 ∖ 𝐽) ∖ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0))) → ((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘) = 0) |
358 | 357 | oveq1d 7427 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼 ∖ 𝐽) ∖ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0))) → (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) =
(0(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) |
359 | | eldifi 4126 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ((𝐼 ∖ 𝐽) ∖ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0)) → 𝑘 ∈ (𝐼 ∖ 𝐽)) |
360 | 359, 349 | sylan2 592 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼 ∖ 𝐽) ∖ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0))) → ((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘) ∈ 𝐾) |
361 | 48, 116, 49 | mulg0 19000 |
. . . . . . . . . . . 12
⊢ (((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘) ∈ 𝐾 →
(0(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) = (0g‘(mulGrp‘𝑅))) |
362 | 360, 361 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼 ∖ 𝐽) ∖ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0))) →
(0(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) = (0g‘(mulGrp‘𝑅))) |
363 | 358, 362 | eqtrd 2771 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼 ∖ 𝐽) ∖ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0))) → (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) = (0g‘(mulGrp‘𝑅))) |
364 | 363, 344 | suppss2 8191 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) supp
(0g‘(mulGrp‘𝑅))) ⊆ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0)) |
365 | 352, 325,
354, 355, 364 | fsuppsssuppgd 41534 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) finSupp
(0g‘(mulGrp‘𝑅))) |
366 | 48, 116, 312, 344, 351, 365 | gsumcl 19831 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
((mulGrp‘𝑅)
Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) ∈ 𝐾) |
367 | 28, 143, 300, 343, 366 | ringcld 20158 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) ∈ 𝐾) |
368 | 367 | fmpttd 7116 |
. . . . 5
⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))):{ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}⟶𝐾) |
369 | 298 | mptex 7227 |
. . . . . . 7
⊢ (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) ∈ V |
370 | 369 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) ∈ V) |
371 | | fvexd 6906 |
. . . . . 6
⊢ (𝜑 → (0g‘𝑅) ∈ V) |
372 | | funmpt 6586 |
. . . . . . 7
⊢ Fun
(𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) |
373 | 372 | a1i 11 |
. . . . . 6
⊢ (𝜑 → Fun (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) |
374 | 298 | mptex 7227 |
. . . . . . . 8
⊢ (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) ∈ V |
375 | 374 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) ∈ V) |
376 | | funmpt 6586 |
. . . . . . . 8
⊢ Fun
(𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) |
377 | 376 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → Fun (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))))) |
378 | 7 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing) |
379 | 17 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐹 ∈ 𝐵) |
380 | 302, 14, 15, 303, 378, 304, 379, 305 | selvvvval 41623 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽))) = (𝐹‘𝑑)) |
381 | 380 | mpteq2dva 5248 |
. . . . . . . 8
⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))) = (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘𝑑))) |
382 | | eqid 2731 |
. . . . . . . . . . 11
⊢
(Base‘𝑅) =
(Base‘𝑅) |
383 | 14, 382, 15, 302, 17 | mplelf 21869 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:{ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
384 | 383 | feqmptd 6960 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 = (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘𝑑))) |
385 | 14, 15, 204, 17, 7 | mplelsfi 21866 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 finSupp (0g‘𝑅)) |
386 | 384, 385 | eqbrtrrd 5172 |
. . . . . . . 8
⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘𝑑)) finSupp (0g‘𝑅)) |
387 | 381, 386 | eqbrtrd 5170 |
. . . . . . 7
⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))) finSupp (0g‘𝑅)) |
388 | | ssidd 4005 |
. . . . . . . 8
⊢ (𝜑 → ((𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))) supp (0g‘𝑅)) ⊆ ((𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))) supp (0g‘𝑅))) |
389 | 32 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑣 ∈ 𝐾) → 𝑅 ∈ Ring) |
390 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑣 ∈ 𝐾) → 𝑣 ∈ 𝐾) |
391 | 28, 143, 204, 389, 390 | ringlzd 20190 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑣 ∈ 𝐾) → ((0g‘𝑅)(.r‘𝑅)𝑣) = (0g‘𝑅)) |
392 | | fvexd 6906 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽))) ∈ V) |
393 | 388, 391,
392, 342, 371 | suppssov1 8188 |
. . . . . . 7
⊢ (𝜑 → ((𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) supp (0g‘𝑅)) ⊆ ((𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))) supp (0g‘𝑅))) |
394 | 375, 371,
377, 387, 393 | fsuppsssuppgd 41534 |
. . . . . 6
⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) finSupp (0g‘𝑅)) |
395 | | ssidd 4005 |
. . . . . . 7
⊢ (𝜑 → ((𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) supp (0g‘𝑅)) ⊆ ((𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) supp (0g‘𝑅))) |
396 | | ovexd 7447 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) ∈ V) |
397 | 395, 391,
396, 366, 371 | suppssov1 8188 |
. . . . . 6
⊢ (𝜑 → ((𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) supp (0g‘𝑅)) ⊆ ((𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) supp (0g‘𝑅))) |
398 | 370, 371,
373, 394, 397 | fsuppsssuppgd 41534 |
. . . . 5
⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) finSupp (0g‘𝑅)) |
399 | | eqid 2731 |
. . . . . 6
⊢ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎)) = (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎)) |
400 | 302, 13, 144, 399, 4, 16 | evlselvlem 41624 |
. . . . 5
⊢ (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎)):({𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})–1-1-onto→{ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin}) |
401 | 28, 204, 296, 299, 368, 398, 400 | gsumf1o 19832 |
. . . 4
⊢ (𝜑 → (𝑅 Σg (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) = (𝑅 Σg ((𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) ∘ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎))))) |
402 | 144 | psrbagf 21782 |
. . . . . . . . . 10
⊢ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑏:(𝐼 ∖ 𝐽)⟶ℕ0) |
403 | 402 | ad2antrl 725 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑏:(𝐼 ∖ 𝐽)⟶ℕ0) |
404 | 13 | psrbagf 21782 |
. . . . . . . . . 10
⊢ (𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} → 𝑎:𝐽⟶ℕ0) |
405 | 404 | ad2antll 726 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑎:𝐽⟶ℕ0) |
406 | | disjdifr 4472 |
. . . . . . . . . 10
⊢ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅ |
407 | 406 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) |
408 | 403, 405,
407 | fun2d 6755 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝑏 ∪ 𝑎):((𝐼 ∖ 𝐽) ∪ 𝐽)⟶ℕ0) |
409 | | undifr 4482 |
. . . . . . . . . . 11
⊢ (𝐽 ⊆ 𝐼 ↔ ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼) |
410 | 16, 409 | sylib 217 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼) |
411 | 410 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼) |
412 | 411 | feq2d 6703 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((𝑏 ∪ 𝑎):((𝐼 ∖ 𝐽) ∪ 𝐽)⟶ℕ0 ↔ (𝑏 ∪ 𝑎):𝐼⟶ℕ0)) |
413 | 408, 412 | mpbid 231 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝑏 ∪ 𝑎):𝐼⟶ℕ0) |
414 | | vex 3477 |
. . . . . . . . . . 11
⊢ 𝑏 ∈ V |
415 | | vex 3477 |
. . . . . . . . . . 11
⊢ 𝑎 ∈ V |
416 | 414, 415 | unex 7737 |
. . . . . . . . . 10
⊢ (𝑏 ∪ 𝑎) ∈ V |
417 | 416 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝑏 ∪ 𝑎) ∈ V) |
418 | | 0zd 12577 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 0
∈ ℤ) |
419 | 413 | ffund 6721 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → Fun
(𝑏 ∪ 𝑎)) |
420 | 144 | psrbagfsupp 21784 |
. . . . . . . . . . 11
⊢ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑏 finSupp 0) |
421 | 420 | ad2antrl 725 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑏 finSupp 0) |
422 | 13 | psrbagfsupp 21784 |
. . . . . . . . . . 11
⊢ (𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} → 𝑎 finSupp 0) |
423 | 422 | ad2antll 726 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑎 finSupp 0) |
424 | 421, 423 | fsuppun 9388 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((𝑏 ∪ 𝑎) supp 0) ∈
Fin) |
425 | 417, 418,
419, 424 | isfsuppd 9372 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝑏 ∪ 𝑎) finSupp 0) |
426 | | fcdmnn0fsuppg 12538 |
. . . . . . . . 9
⊢ (((𝑏 ∪ 𝑎) ∈ V ∧ (𝑏 ∪ 𝑎):𝐼⟶ℕ0) → ((𝑏 ∪ 𝑎) finSupp 0 ↔ (◡(𝑏 ∪ 𝑎) “ ℕ) ∈
Fin)) |
427 | 417, 413,
426 | syl2anc 583 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((𝑏 ∪ 𝑎) finSupp 0 ↔ (◡(𝑏 ∪ 𝑎) “ ℕ) ∈
Fin)) |
428 | 425, 427 | mpbid 231 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → (◡(𝑏 ∪ 𝑎) “ ℕ) ∈
Fin) |
429 | 4 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝐼 ∈ 𝑉) |
430 | 302 | psrbag 21781 |
. . . . . . . 8
⊢ (𝐼 ∈ 𝑉 → ((𝑏 ∪ 𝑎) ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↔ ((𝑏 ∪ 𝑎):𝐼⟶ℕ0 ∧ (◡(𝑏 ∪ 𝑎) “ ℕ) ∈
Fin))) |
431 | 429, 430 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((𝑏 ∪ 𝑎) ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↔ ((𝑏 ∪ 𝑎):𝐼⟶ℕ0 ∧ (◡(𝑏 ∪ 𝑎) “ ℕ) ∈
Fin))) |
432 | 413, 428,
431 | mpbir2and 710 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝑏 ∪ 𝑎) ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin}) |
433 | | eqidd 2732 |
. . . . . 6
⊢ (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎)) = (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎))) |
434 | | eqidd 2732 |
. . . . . 6
⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) = (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) |
435 | | reseq1 5975 |
. . . . . . . . . . . 12
⊢ (𝑑 = (𝑏 ∪ 𝑎) → (𝑑 ↾ 𝐽) = ((𝑏 ∪ 𝑎) ↾ 𝐽)) |
436 | 435 | fveq2d 6895 |
. . . . . . . . . . 11
⊢ (𝑑 = (𝑏 ∪ 𝑎) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽)) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽))) |
437 | | reseq1 5975 |
. . . . . . . . . . 11
⊢ (𝑑 = (𝑏 ∪ 𝑎) → (𝑑 ↾ (𝐼 ∖ 𝐽)) = ((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))) |
438 | 436, 437 | fveq12d 6898 |
. . . . . . . . . 10
⊢ (𝑑 = (𝑏 ∪ 𝑎) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽))) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽))‘((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽)))) |
439 | 435 | fveq1d 6893 |
. . . . . . . . . . . . 13
⊢ (𝑑 = (𝑏 ∪ 𝑎) → ((𝑑 ↾ 𝐽)‘𝑗) = (((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)) |
440 | 439 | oveq1d 7427 |
. . . . . . . . . . . 12
⊢ (𝑑 = (𝑏 ∪ 𝑎) → (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) |
441 | 440 | mpteq2dv 5250 |
. . . . . . . . . . 11
⊢ (𝑑 = (𝑏 ∪ 𝑎) → (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) = (𝑗 ∈ 𝐽 ↦ ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) |
442 | 441 | oveq2d 7428 |
. . . . . . . . . 10
⊢ (𝑑 = (𝑏 ∪ 𝑎) → ((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) = ((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) |
443 | 438, 442 | oveq12d 7430 |
. . . . . . . . 9
⊢ (𝑑 = (𝑏 ∪ 𝑎) → ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽))‘((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) |
444 | 437 | fveq1d 6893 |
. . . . . . . . . . . 12
⊢ (𝑑 = (𝑏 ∪ 𝑎) → ((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘) = (((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)) |
445 | 444 | oveq1d 7427 |
. . . . . . . . . . 11
⊢ (𝑑 = (𝑏 ∪ 𝑎) → (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) = ((((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) |
446 | 445 | mpteq2dv 5250 |
. . . . . . . . . 10
⊢ (𝑑 = (𝑏 ∪ 𝑎) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) |
447 | 446 | oveq2d 7428 |
. . . . . . . . 9
⊢ (𝑑 = (𝑏 ∪ 𝑎) → ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) = ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) |
448 | 443, 447 | oveq12d 7430 |
. . . . . . . 8
⊢ (𝑑 = (𝑏 ∪ 𝑎) → (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽))‘((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) |
449 | 416, 448 | csbie 3929 |
. . . . . . 7
⊢
⦋(𝑏
∪ 𝑎) / 𝑑⦌(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽))‘((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) |
450 | 402 | ffnd 6718 |
. . . . . . . . . . . . 13
⊢ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑏 Fn (𝐼 ∖ 𝐽)) |
451 | 450 | ad2antrl 725 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑏 Fn (𝐼 ∖ 𝐽)) |
452 | 405 | ffnd 6718 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑎 Fn 𝐽) |
453 | | fnunres2 6662 |
. . . . . . . . . . . 12
⊢ ((𝑏 Fn (𝐼 ∖ 𝐽) ∧ 𝑎 Fn 𝐽 ∧ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) → ((𝑏 ∪ 𝑎) ↾ 𝐽) = 𝑎) |
454 | 451, 452,
407, 453 | syl3anc 1370 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((𝑏 ∪ 𝑎) ↾ 𝐽) = 𝑎) |
455 | 454 | fveq2d 6895 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽)) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)) |
456 | | fnunres1 6661 |
. . . . . . . . . . 11
⊢ ((𝑏 Fn (𝐼 ∖ 𝐽) ∧ 𝑎 Fn 𝐽 ∧ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) → ((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽)) = 𝑏) |
457 | 451, 452,
407, 456 | syl3anc 1370 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽)) = 𝑏) |
458 | 455, 457 | fveq12d 6898 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽))‘((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)) |
459 | 454 | fveq1d 6893 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
(((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗) = (𝑎‘𝑗)) |
460 | 459 | oveq1d 7427 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) |
461 | 460 | mpteq2dv 5250 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝑗 ∈ 𝐽 ↦ ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) = (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) |
462 | 461 | oveq2d 7428 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((mulGrp‘𝑅)
Σg (𝑗 ∈ 𝐽 ↦ ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) = ((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) |
463 | 458, 462 | oveq12d 7430 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽))‘((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))) |
464 | 457 | fveq1d 6893 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
(((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘) = (𝑏‘𝑘)) |
465 | 464 | oveq1d 7427 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) = ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) |
466 | 465 | mpteq2dv 5250 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) |
467 | 466 | oveq2d 7428 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((mulGrp‘𝑅)
Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) = ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) |
468 | 463, 467 | oveq12d 7430 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽))‘((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) |
469 | 449, 468 | eqtrid 2783 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
⦋(𝑏 ∪
𝑎) / 𝑑⦌(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) |
470 | 432, 433,
434, 469 | fmpocos 41526 |
. . . . 5
⊢ (𝜑 → ((𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) ∘ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎))) = (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) |
471 | 470 | oveq2d 7428 |
. . . 4
⊢ (𝜑 → (𝑅 Σg ((𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) ∘ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎)))) = (𝑅 Σg (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))))) |
472 | | ovex 7445 |
. . . . . . 7
⊢
(ℕ0 ↑m (𝐼 ∖ 𝐽)) ∈ V |
473 | 472 | rabex 5332 |
. . . . . 6
⊢ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈
V |
474 | 473 | a1i 11 |
. . . . 5
⊢ (𝜑 → {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈
V) |
475 | 185 | a1i 11 |
. . . . 5
⊢ (𝜑 → {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ∈
V) |
476 | 32 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑅 ∈ Ring) |
477 | 19 | ffvelcdmda 7086 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎) ∈ (Base‘𝑈)) |
478 | 3, 28, 1, 144, 477 | mplelf 21869 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎):{𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶𝐾) |
479 | 478 | ffvelcdmda 7086 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) ∧ 𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) ∈ 𝐾) |
480 | 479 | an32s 649 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) ∈ 𝐾) |
481 | 480 | anasss 466 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) ∈ 𝐾) |
482 | 24 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝐽 ∈ V) |
483 | 7 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑅 ∈ CRing) |
484 | 36 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝐴 ↾ 𝐽) ∈ (𝐾 ↑m 𝐽)) |
485 | | simprr 770 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈
Fin}) |
486 | 13, 28, 47, 49, 482, 483, 484, 485 | evlsvvvallem 41599 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((mulGrp‘𝑅)
Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) ∈ 𝐾) |
487 | 28, 143, 476, 481, 486 | ringcld 20158 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) ∈ 𝐾) |
488 | 6 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝐼 ∖ 𝐽) ∈ V) |
489 | 35, 5 | elmapssresd 41536 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴 ↾ (𝐼 ∖ 𝐽)) ∈ (𝐾 ↑m (𝐼 ∖ 𝐽))) |
490 | 489 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝐴 ↾ (𝐼 ∖ 𝐽)) ∈ (𝐾 ↑m (𝐼 ∖ 𝐽))) |
491 | | simprl 768 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈
Fin}) |
492 | 144, 28, 47, 49, 488, 483, 490, 491 | evlsvvvallem 41599 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((mulGrp‘𝑅)
Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) ∈ 𝐾) |
493 | 28, 143, 476, 487, 492 | ringcld 20158 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) ∈ 𝐾) |
494 | 493 | ralrimivva 3199 |
. . . . . 6
⊢ (𝜑 → ∀𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}∀𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) ∈ 𝐾) |
495 | 287 | fmpo 8058 |
. . . . . 6
⊢
(∀𝑏 ∈
{𝑓 ∈
(ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}∀𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} (((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) ∈ 𝐾 ↔ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))):({𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})⟶𝐾) |
496 | 494, 495 | sylib 217 |
. . . . 5
⊢ (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))):({𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})⟶𝐾) |
497 | | f1of1 6832 |
. . . . . . . 8
⊢ ((𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎)):({𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})–1-1-onto→{ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} → (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎)):({𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})–1-1→{ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin}) |
498 | 400, 497 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎)):({𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})–1-1→{ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin}) |
499 | 398, 498,
371, 370 | fsuppco 9403 |
. . . . . 6
⊢ (𝜑 → ((𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) ∘ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑏 ∪ 𝑎))) finSupp (0g‘𝑅)) |
500 | 470, 499 | eqbrtrrd 5172 |
. . . . 5
⊢ (𝜑 → (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) finSupp (0g‘𝑅)) |
501 | 28, 204, 296, 474, 475, 496, 500 | gsumxp 19892 |
. . . 4
⊢ (𝜑 → (𝑅 Σg (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) = (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg
(𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))𝑒)))))) |
502 | 401, 471,
501 | 3eqtrd 2775 |
. . 3
⊢ (𝜑 → (𝑅 Σg (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) = (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg
(𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦ (𝑐(𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}, 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))𝑒)))))) |
503 | 28, 143, 300, 311, 342, 366 | ringassd 20157 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = ((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)(((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) |
504 | 47, 143 | mgpplusg 20039 |
. . . . . . . . 9
⊢
(.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
505 | 51 | ad2antrr 723 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (mulGrp‘𝑅) ∈ Mnd) |
506 | 316 | ffvelcdmda 7086 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈
ℕ0) |
507 | 57 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐴:𝐼⟶𝐾) |
508 | 507 | ffvelcdmda 7086 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝐴‘𝑖) ∈ 𝐾) |
509 | 48, 49, 505, 506, 508 | mulgnn0cld 19018 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)) ∈ 𝐾) |
510 | 509 | fmpttd 7116 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))):𝐼⟶𝐾) |
511 | 4 | mptexd 7228 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ∈ V) |
512 | 511 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ∈ V) |
513 | | funmpt 6586 |
. . . . . . . . . . 11
⊢ Fun
(𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) |
514 | 513 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → Fun
(𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))) |
515 | 316 | feqmptd 6960 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 = (𝑖 ∈ 𝐼 ↦ (𝑑‘𝑖))) |
516 | 515, 329 | eqbrtrrd 5172 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑖 ∈ 𝐼 ↦ (𝑑‘𝑖)) finSupp 0) |
517 | | ssidd 4005 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑖 ∈ 𝐼 ↦ (𝑑‘𝑖)) supp 0) ⊆ ((𝑖 ∈ 𝐼 ↦ (𝑑‘𝑖)) supp 0)) |
518 | 117 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ 𝐾) →
(0(.g‘(mulGrp‘𝑅))𝑘) = (0g‘(mulGrp‘𝑅))) |
519 | 517, 518,
506, 508, 330 | suppssov1 8188 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) supp
(0g‘(mulGrp‘𝑅))) ⊆ ((𝑖 ∈ 𝐼 ↦ (𝑑‘𝑖)) supp 0)) |
520 | 512, 325,
514, 516, 519 | fsuppsssuppgd 41534 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) finSupp
(0g‘(mulGrp‘𝑅))) |
521 | | disjdif 4471 |
. . . . . . . . . 10
⊢ (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅ |
522 | 521 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅) |
523 | | undif 4481 |
. . . . . . . . . . . 12
⊢ (𝐽 ⊆ 𝐼 ↔ (𝐽 ∪ (𝐼 ∖ 𝐽)) = 𝐼) |
524 | 16, 523 | sylib 217 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐽 ∪ (𝐼 ∖ 𝐽)) = 𝐼) |
525 | 524 | eqcomd 2737 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐼 = (𝐽 ∪ (𝐼 ∖ 𝐽))) |
526 | 525 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 = (𝐽 ∪ (𝐼 ∖ 𝐽))) |
527 | 48, 116, 504, 312, 303, 510, 520, 522, 526 | gsumsplit 19844 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
((mulGrp‘𝑅)
Σg (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))) = (((mulGrp‘𝑅) Σg ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ 𝐽))(.r‘𝑅)((mulGrp‘𝑅) Σg ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ (𝐼 ∖ 𝐽))))) |
528 | 304 | resmptd 6040 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ 𝐽) = (𝑖 ∈ 𝐽 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))) |
529 | | fveq2 6891 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 𝑗 → (𝑑‘𝑖) = (𝑑‘𝑗)) |
530 | | fveq2 6891 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 𝑗 → (𝐴‘𝑖) = (𝐴‘𝑗)) |
531 | 529, 530 | oveq12d 7430 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)) = ((𝑑‘𝑗)(.g‘(mulGrp‘𝑅))(𝐴‘𝑗))) |
532 | 531 | cbvmptv 5261 |
. . . . . . . . . . . 12
⊢ (𝑖 ∈ 𝐽 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) = (𝑗 ∈ 𝐽 ↦ ((𝑑‘𝑗)(.g‘(mulGrp‘𝑅))(𝐴‘𝑗))) |
533 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → 𝑗 ∈ 𝐽) |
534 | 533 | fvresd 6911 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝑑 ↾ 𝐽)‘𝑗) = (𝑑‘𝑗)) |
535 | 533 | fvresd 6911 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝐴 ↾ 𝐽)‘𝑗) = (𝐴‘𝑗)) |
536 | 534, 535 | oveq12d 7430 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = ((𝑑‘𝑗)(.g‘(mulGrp‘𝑅))(𝐴‘𝑗))) |
537 | 536 | eqcomd 2737 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝑑‘𝑗)(.g‘(mulGrp‘𝑅))(𝐴‘𝑗)) = (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) |
538 | 537 | mpteq2dva 5248 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ ((𝑑‘𝑗)(.g‘(mulGrp‘𝑅))(𝐴‘𝑗))) = (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) |
539 | 532, 538 | eqtrid 2783 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑖 ∈ 𝐽 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) = (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) |
540 | 528, 539 | eqtrd 2771 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ 𝐽) = (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) |
541 | 540 | oveq2d 7428 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
((mulGrp‘𝑅)
Σg ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ 𝐽)) = ((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))) |
542 | 309 | resmptd 6040 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ (𝐼 ∖ 𝐽)) = (𝑖 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))) |
543 | | fveq2 6891 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 𝑘 → (𝑑‘𝑖) = (𝑑‘𝑘)) |
544 | | fveq2 6891 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 𝑘 → (𝐴‘𝑖) = (𝐴‘𝑘)) |
545 | 543, 544 | oveq12d 7430 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑘 → ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)) = ((𝑑‘𝑘)(.g‘(mulGrp‘𝑅))(𝐴‘𝑘))) |
546 | 545 | cbvmptv 5261 |
. . . . . . . . . . . 12
⊢ (𝑖 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑑‘𝑘)(.g‘(mulGrp‘𝑅))(𝐴‘𝑘))) |
547 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → 𝑘 ∈ (𝐼 ∖ 𝐽)) |
548 | 547 | fvresd 6911 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘) = (𝑑‘𝑘)) |
549 | 547 | fvresd 6911 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘) = (𝐴‘𝑘)) |
550 | 548, 549 | oveq12d 7430 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) = ((𝑑‘𝑘)(.g‘(mulGrp‘𝑅))(𝐴‘𝑘))) |
551 | 550 | eqcomd 2737 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑑‘𝑘)(.g‘(mulGrp‘𝑅))(𝐴‘𝑘)) = (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) |
552 | 551 | mpteq2dva 5248 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑑‘𝑘)(.g‘(mulGrp‘𝑅))(𝐴‘𝑘))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) |
553 | 546, 552 | eqtrid 2783 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑖 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) |
554 | 542, 553 | eqtrd 2771 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ (𝐼 ∖ 𝐽)) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))) |
555 | 554 | oveq2d 7428 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
((mulGrp‘𝑅)
Σg ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ (𝐼 ∖ 𝐽))) = ((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) |
556 | 541, 555 | oveq12d 7430 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
(((mulGrp‘𝑅)
Σg ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ 𝐽))(.r‘𝑅)((mulGrp‘𝑅) Σg ((𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) ↾ (𝐼 ∖ 𝐽)))) = (((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) |
557 | 527, 556 | eqtr2d 2772 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
(((mulGrp‘𝑅)
Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = ((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))) |
558 | 380, 557 | oveq12d 7430 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)(((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) = ((𝐹‘𝑑)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))))) |
559 | 503, 558 | eqtrd 2771 |
. . . . 5
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))) = ((𝐹‘𝑑)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))))) |
560 | 559 | mpteq2dva 5248 |
. . . 4
⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) = (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ ((𝐹‘𝑑)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))))) |
561 | 560 | oveq2d 7428 |
. . 3
⊢ (𝜑 → (𝑅 Σg (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) = (𝑅 Σg (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ ((𝐹‘𝑑)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))))))) |
562 | 295, 502,
561 | 3eqtr2d 2777 |
. 2
⊢ (𝜑 → (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦
(((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽)))‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))))))) = (𝑅 Σg (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ ((𝐹‘𝑑)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))))))) |
563 | | eqid 2731 |
. . 3
⊢ ((𝐼 ∖ 𝐽) eval 𝑅) = ((𝐼 ∖ 𝐽) eval 𝑅) |
564 | 563, 3, 1, 144, 28, 47, 49, 143, 6, 7, 173, 489 | evlvvval 41611 |
. 2
⊢ (𝜑 → ((((𝐼 ∖ 𝐽) eval 𝑅)‘(((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽))))‘(𝐴 ↾ (𝐼 ∖ 𝐽))) = (𝑅 Σg (𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦
(((((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽)))‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))))) |
565 | | eqid 2731 |
. . 3
⊢ (𝐼 eval 𝑅) = (𝐼 eval 𝑅) |
566 | 565, 14, 15, 302, 28, 47, 49, 143, 4, 7, 17, 35 | evlvvval 41611 |
. 2
⊢ (𝜑 → (((𝐼 eval 𝑅)‘𝐹)‘𝐴) = (𝑅 Σg (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ ((𝐹‘𝑑)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))))))) |
567 | 562, 564,
566 | 3eqtr4d 2781 |
1
⊢ (𝜑 → ((((𝐼 ∖ 𝐽) eval 𝑅)‘(((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽))))‘(𝐴 ↾ (𝐼 ∖ 𝐽))) = (((𝐼 eval 𝑅)‘𝐹)‘𝐴)) |