| Step | Hyp | Ref
| Expression |
| 1 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(Base‘𝑈) =
(Base‘𝑈) |
| 2 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(.r‘𝑈) = (.r‘𝑈) |
| 3 | | evlselv.u |
. . . . . . . . . . . . . . . 16
⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) |
| 4 | | evlselv.i |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐼 ∈ 𝑉) |
| 5 | | difssd 4137 |
. . . . . . . . . . . . . . . . 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 42557 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑈 ∈ CRing) |
| 9 | 8 | crngringd 20243 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑈 ∈ Ring) |
| 10 | 9 | ad2antrr 726 |
. . . . . . . . . . . . 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 2737 |
. . . . . . . . . . . . . . . 16
⊢
(Base‘𝑇) =
(Base‘𝑇) |
| 13 | | eqid 2737 |
. . . . . . . . . . . . . . . 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, 7, 16, 17 | selvcl 42593 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ (Base‘𝑇)) |
| 19 | 11, 1, 12, 13, 18 | mplelf 22018 |
. . . . . . . . . . . . . . 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 7104 |
. . . . . . . . . . . . 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 2737 |
. . . . . . . . . . . . . 14
⊢
(mulGrp‘𝑈) =
(mulGrp‘𝑈) |
| 23 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(.g‘(mulGrp‘𝑈)) =
(.g‘(mulGrp‘𝑈)) |
| 24 | 4, 16 | ssexd 5324 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐽 ∈ V) |
| 25 | 24 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝐽 ∈ V) |
| 26 | 8 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑈 ∈ CRing) |
| 27 | | fvexd 6921 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (Base‘𝑈) ∈ V) |
| 28 | | evlselv.k |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝐾 = (Base‘𝑅) |
| 29 | 28 | fvexi 6920 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝐾 ∈ V |
| 30 | 29 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐾 ∈ V) |
| 31 | | evlselv.l |
. . . . . . . . . . . . . . . . . 18
⊢ 𝐿 = (algSc‘𝑈) |
| 32 | 7 | crngringd 20243 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑅 ∈ Ring) |
| 33 | 3, 1, 28, 31, 6, 32 | mplasclf 22089 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐿:𝐾⟶(Base‘𝑈)) |
| 34 | 27, 30, 33 | elmapdd 8881 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐿 ∈ ((Base‘𝑈) ↑m 𝐾)) |
| 35 | | evlselv.a |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) |
| 36 | 35, 16 | elmapssresd 42282 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐴 ↾ 𝐽) ∈ (𝐾 ↑m 𝐽)) |
| 37 | 34, 36 | mapcod 42284 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐿 ∘ (𝐴 ↾ 𝐽)) ∈ ((Base‘𝑈) ↑m 𝐽)) |
| 38 | 37 | ad2antrr 726 |
. . . . . . . . . . . . . 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 42571 |
. . . . . . . . . . . . 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 20257 |
. . . . . . . . . . . 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 2738 |
. . . . . . . . . . . 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 2738 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) = (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐))) |
| 44 | | fveq1 6905 |
. . . . . . . . . . . 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 7149 |
. . . . . . . . . . 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 726 |
. . . . . . . . . . . . . . . . . . . 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 2737 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(mulGrp‘𝑅) =
(mulGrp‘𝑅) |
| 48 | 47, 28 | mgpbas 20142 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝐾 =
(Base‘(mulGrp‘𝑅)) |
| 49 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(.g‘(mulGrp‘𝑅)) =
(.g‘(mulGrp‘𝑅)) |
| 50 | 47 | ringmgp 20236 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑅 ∈ Ring →
(mulGrp‘𝑅) ∈
Mnd) |
| 51 | 32, 50 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd) |
| 52 | 51 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . . 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 21938 |
. . . . . . . . . . . . . . . . . . . . . . 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 7104 |
. . . . . . . . . . . . . . . . . . . . 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 8889 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝐴 ∈ (𝐾 ↑m 𝐼) → 𝐴:𝐼⟶𝐾) |
| 57 | 35, 56 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝐴:𝐼⟶𝐾) |
| 58 | 57, 16 | fssresd 6775 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (𝐴 ↾ 𝐽):𝐽⟶𝐾) |
| 59 | 58 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . 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 7104 |
. . . . . . . . . . . . . . . . . . . . 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 19113 |
. . . . . . . . . . . . . . . . . . . 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 7152 |
. . . . . . . . . . . . . . . . . . 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 22042 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝐼 ∖ 𝐽) ∈ V ∧ 𝑅 ∈ CRing) → 𝑈 ∈ AssAlg) |
| 64 | 6, 7, 63 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝑈 ∈ AssAlg) |
| 65 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(Scalar‘𝑈) =
(Scalar‘𝑈) |
| 66 | 31, 65 | asclrhm 21910 |
. . . . . . . . . . . . . . . . . . . . . . . . . 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 22033 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → 𝑅 = (Scalar‘𝑈)) |
| 69 | 68 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (Scalar‘𝑈) = 𝑅) |
| 70 | 69 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → ((Scalar‘𝑈) RingHom 𝑈) = (𝑅 RingHom 𝑈)) |
| 71 | 67, 70 | eleqtrd 2843 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝐿 ∈ (𝑅 RingHom 𝑈)) |
| 72 | 47, 22 | rhmmhm 20479 |
. . . . . . . . . . . . . . . . . . . . . . . 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 730 |
. . . . . . . . . . . . . . . . . . . . . 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 19133 |
. . . . . . . . . . . . . . . . . . . . . 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 1373 |
. . . . . . . . . . . . . . . . . . . . 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 730 |
. . . . . . . . . . . . . . . . . . . . . . 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 7009 |
. . . . . . . . . . . . . . . . . . . . . 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 7447 |
. . . . . . . . . . . . . . . . . . . . 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 2780 |
. . . . . . . . . . . . . . . . . . . 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 5242 |
. . . . . . . . . . . . . . . . . . 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 2777 |
. . . . . . . . . . . . . . . . . 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 7447 |
. . . . . . . . . . . . . . . . 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 2737 |
. . . . . . . . . . . . . . . . . 18
⊢
(Base‘(mulGrp‘(Scalar‘𝑈))) =
(Base‘(mulGrp‘(Scalar‘𝑈))) |
| 86 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢
(0g‘(mulGrp‘(Scalar‘𝑈))) =
(0g‘(mulGrp‘(Scalar‘𝑈))) |
| 87 | 68, 7 | eqeltrrd 2842 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (Scalar‘𝑈) ∈ CRing) |
| 88 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(mulGrp‘(Scalar‘𝑈)) = (mulGrp‘(Scalar‘𝑈)) |
| 89 | 88 | crngmgp 20238 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((Scalar‘𝑈)
∈ CRing → (mulGrp‘(Scalar‘𝑈)) ∈ CMnd) |
| 90 | 87, 89 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 →
(mulGrp‘(Scalar‘𝑈)) ∈ CMnd) |
| 91 | 90 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . 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 20236 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑈 ∈ Ring →
(mulGrp‘𝑈) ∈
Mnd) |
| 93 | 9, 92 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (mulGrp‘𝑈) ∈ Mnd) |
| 94 | 93 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . 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 20479 |
. . . . . . . . . . . . . . . . . . . 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 726 |
. . . . . . . . . . . . . . . . . 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 6910 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (Base‘𝑅) =
(Base‘(Scalar‘𝑈))) |
| 99 | 28, 98 | eqtrid 2789 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑈))) |
| 100 | 99 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . . 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 2843 |
. . . . . . . . . . . . . . . . . . . 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 2737 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈)) |
| 103 | 88, 102 | mgpbas 20142 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(Base‘(Scalar‘𝑈)) =
(Base‘(mulGrp‘(Scalar‘𝑈))) |
| 104 | 101, 103 | eleqtrdi 2851 |
. . . . . . . . . . . . . . . . . . 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 7135 |
. . . . . . . . . . . . . . . . . 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 | 54 | feqmptd 6977 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑒 = (𝑗 ∈ 𝐽 ↦ (𝑒‘𝑗))) |
| 107 | 13 | psrbagfsupp 21939 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} → 𝑒 finSupp 0) |
| 108 | 107 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑒 finSupp 0) |
| 109 | 106, 108 | eqbrtrrd 5167 |
. . . . . . . . . . . . . . . . . . . . 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) |
| 110 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(0g‘(mulGrp‘𝑅)) =
(0g‘(mulGrp‘𝑅)) |
| 111 | 48, 110, 49 | mulg0 19092 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ 𝐾 →
(0(.g‘(mulGrp‘𝑅))𝑘) = (0g‘(mulGrp‘𝑅))) |
| 112 | 111 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((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))) |
| 113 | | fvexd 6921 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
(0g‘(mulGrp‘𝑅)) ∈ V) |
| 114 | 109, 112,
55, 60, 113 | fsuppssov1 9424 |
. . . . . . . . . . . . . . . . . . . 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))) |
| 115 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(1r‘𝑅) = (1r‘𝑅) |
| 116 | 47, 115 | ringidval 20180 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(1r‘𝑅) = (0g‘(mulGrp‘𝑅)) |
| 117 | 114, 116 | breqtrrdi 5185 |
. . . . . . . . . . . . . . . . . . 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)) |
| 118 | 68 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (1r‘𝑅) =
(1r‘(Scalar‘𝑈))) |
| 119 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(1r‘(Scalar‘𝑈)) =
(1r‘(Scalar‘𝑈)) |
| 120 | 88, 119 | ringidval 20180 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(1r‘(Scalar‘𝑈)) =
(0g‘(mulGrp‘(Scalar‘𝑈))) |
| 121 | 118, 120 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (1r‘𝑅) =
(0g‘(mulGrp‘(Scalar‘𝑈)))) |
| 122 | 121 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . 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)))) |
| 123 | 117, 122 | breqtrd 5169 |
. . . . . . . . . . . . . . . . . 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)))) |
| 124 | 85, 86, 91, 94, 25, 97, 105, 123 | gsummhm 19956 |
. . . . . . . . . . . . . . . . 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)))))) |
| 125 | 84, 124 | eqtr3d 2779 |
. . . . . . . . . . . . . . . 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)))))) |
| 126 | 125 | oveq2d 7447 |
. . . . . . . . . . . . . . 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))))))) |
| 127 | 64 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑈 ∈ AssAlg) |
| 128 | 101 | fmpttd 7135 |
. . . . . . . . . . . . . . . . 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))) |
| 129 | 123, 120 | breqtrrdi 5185 |
. . . . . . . . . . . . . . . . 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))) |
| 130 | 103, 120,
91, 25, 128, 129 | gsumcl 19933 |
. . . . . . . . . . . . . . . 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))) |
| 131 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢ (
·𝑠 ‘𝑈) = ( ·𝑠
‘𝑈) |
| 132 | 31, 65, 102, 1, 2, 131 | asclmul2 21907 |
. . . . . . . . . . . . . . . 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))) |
| 133 | 127, 130,
21, 132 | syl3anc 1373 |
. . . . . . . . . . . . . . 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))) |
| 134 | 126, 133 | eqtrd 2777 |
. . . . . . . . . . . . . 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))) |
| 135 | 134 | fveq1d 6908 |
. . . . . . . . . . . . 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)) |
| 136 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(.r‘𝑅) = (.r‘𝑅) |
| 137 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈
Fin} |
| 138 | 99 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝐾 =
(Base‘(Scalar‘𝑈))) |
| 139 | 130, 138 | eleqtrrd 2844 |
. . . . . . . . . . . . . 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) |
| 140 | | simplr 769 |
. . . . . . . . . . . . . 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}) |
| 141 | 3, 131, 28, 1, 136, 137, 139, 21, 140 | mplvscaval 22036 |
. . . . . . . . . . . . 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))) |
| 142 | 135, 141 | eqtrd 2777 |
. . . . . . . . . . . 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))) |
| 143 | 142 | mpteq2dva 5242 |
. . . . . . . . . . 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)))) |
| 144 | 45, 143 | eqtrd 2777 |
. . . . . . . . . 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)))) |
| 145 | 144 | oveq2d 7447 |
. . . . . . . . 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))))) |
| 146 | 69 | fveq2d 6910 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 →
(mulGrp‘(Scalar‘𝑈)) = (mulGrp‘𝑅)) |
| 147 | 146 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
(mulGrp‘(Scalar‘𝑈)) = (mulGrp‘𝑅)) |
| 148 | 147 | oveq1d 7446 |
. . . . . . . . . . . . 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))))) |
| 149 | 148 | oveq1d 7446 |
. . . . . . . . . . . 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))) |
| 150 | 7 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing) |
| 151 | 148, 139 | eqeltrrd 2842 |
. . . . . . . . . . . . 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) |
| 152 | 19 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒) ∈ (Base‘𝑈)) |
| 153 | 3, 28, 1, 137, 152 | mplelf 22018 |
. . . . . . . . . . . . . . 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) |
| 154 | 153 | ffvelcdmda 7104 |
. . . . . . . . . . . . . 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) |
| 155 | 154 | an32s 652 |
. . . . . . . . . . . . 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) |
| 156 | 28, 136, 150, 151, 155 | crngcomd 20252 |
. . . . . . . . . . . 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)))))) |
| 157 | 149, 156 | eqtrd 2777 |
. . . . . . . . . . 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)))))) |
| 158 | 157 | mpteq2dva 5242 |
. . . . . . . . . 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))))))) |
| 159 | 158 | oveq2d 7447 |
. . . . . . . . 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)))))))) |
| 160 | 145, 159 | eqtrd 2777 |
. . . . . . . 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)))))))) |
| 161 | 160 | oveq1d 7446 |
. . . . . . 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)))))) |
| 162 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) = (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) |
| 163 | | fveq1 6905 |
. . . . . . . . . 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)) |
| 164 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (𝐽 eval 𝑈) = (𝐽 eval 𝑈) |
| 165 | 164, 11, 12, 13, 1, 22, 23, 2, 24, 8, 18, 37 | evlvvval 42583 |
. . . . . . . . . . . 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)))))))) |
| 166 | 164, 11, 12, 1, 24, 8, 18, 37 | evlcl 42582 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽))) ∈ (Base‘𝑈)) |
| 167 | 165, 166 | eqeltrrd 2842 |
. . . . . . . . . . 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)) |
| 168 | 167 | 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)) |
| 169 | | fvexd 6921 |
. . . . . . . . . 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) |
| 170 | 162, 163,
168, 169 | fvmptd3 7039 |
. . . . . . . . 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)) |
| 171 | | eqid 2737 |
. . . . . . . . . 10
⊢
(0g‘𝑈) = (0g‘𝑈) |
| 172 | 9 | ringcmnd 20281 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑈 ∈ CMnd) |
| 173 | 172 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑈 ∈ CMnd) |
| 174 | 7 | crnggrpd 20244 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑅 ∈ Grp) |
| 175 | 174 | grpmndd 18964 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑅 ∈ Mnd) |
| 176 | 175 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Mnd) |
| 177 | | ovex 7464 |
. . . . . . . . . . . 12
⊢
(ℕ0 ↑m 𝐽) ∈ V |
| 178 | 177 | rabex 5339 |
. . . . . . . . . . 11
⊢ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ∈
V |
| 179 | 178 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ∈
V) |
| 180 | 6 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐼 ∖ 𝐽) ∈ V) |
| 181 | 174 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Grp) |
| 182 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈
Fin}) |
| 183 | 3, 1, 137, 162, 180, 181, 182 | mplmapghm 42566 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∈ (𝑈 GrpHom 𝑅)) |
| 184 | | ghmmhm 19244 |
. . . . . . . . . . 11
⊢ ((𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∈ (𝑈 GrpHom 𝑅) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∈ (𝑈 MndHom 𝑅)) |
| 185 | 183, 184 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑢 ∈ (Base‘𝑈) ↦ (𝑢‘𝑐)) ∈ (𝑈 MndHom 𝑅)) |
| 186 | 41 | fmpttd 7135 |
. . . . . . . . . 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)) |
| 187 | 24 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝐽 ∈ V) |
| 188 | 8 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑈 ∈ CRing) |
| 189 | 18 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ (Base‘𝑇)) |
| 190 | 37 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐿 ∘ (𝐴 ↾ 𝐽)) ∈ ((Base‘𝑈) ↑m 𝐽)) |
| 191 | 13, 11, 12, 1, 22, 23, 2, 187, 188, 189, 190 | evlvvvallem 42584 |
. . . . . . . . . 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)) |
| 192 | 1, 171, 173, 176, 179, 185, 186, 191 | gsummhm 19956 |
. . . . . . . . 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))))))))) |
| 193 | 165 | 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)))))))) |
| 194 | 193 | fveq1d 6908 |
. . . . . . . . 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)) |
| 195 | 170, 192,
194 | 3eqtr4rd 2788 |
. . . . . . . 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))))))))) |
| 196 | 195 | oveq1d 7446 |
. . . . . . 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)))))) |
| 197 | | eqid 2737 |
. . . . . . . 8
⊢
(0g‘𝑅) = (0g‘𝑅) |
| 198 | 32 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring) |
| 199 | 47 | crngmgp 20238 |
. . . . . . . . . . 11
⊢ (𝑅 ∈ CRing →
(mulGrp‘𝑅) ∈
CMnd) |
| 200 | 7, 199 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (mulGrp‘𝑅) ∈ CMnd) |
| 201 | 200 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(mulGrp‘𝑅) ∈
CMnd) |
| 202 | 51 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (mulGrp‘𝑅) ∈ Mnd) |
| 203 | 137 | psrbagf 21938 |
. . . . . . . . . . . . 13
⊢ (𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑐:(𝐼 ∖ 𝐽)⟶ℕ0) |
| 204 | 203 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑐:(𝐼 ∖ 𝐽)⟶ℕ0) |
| 205 | 204 | ffvelcdmda 7104 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (𝑐‘𝑘) ∈
ℕ0) |
| 206 | 57, 5 | fssresd 6775 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴 ↾ (𝐼 ∖ 𝐽)):(𝐼 ∖ 𝐽)⟶𝐾) |
| 207 | 206 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐴 ↾ (𝐼 ∖ 𝐽)):(𝐼 ∖ 𝐽)⟶𝐾) |
| 208 | 207 | ffvelcdmda 7104 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘) ∈ 𝐾) |
| 209 | 48, 49, 202, 205, 208 | mulgnn0cld 19113 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) ∈ 𝐾) |
| 210 | 209 | fmpttd 7135 |
. . . . . . . . 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) |
| 211 | 204 | feqmptd 6977 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑐 = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑐‘𝑘))) |
| 212 | 137 | psrbagfsupp 21939 |
. . . . . . . . . . . 12
⊢ (𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑐 finSupp 0) |
| 213 | 212 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑐 finSupp 0) |
| 214 | 211, 213 | eqbrtrrd 5167 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (𝑐‘𝑘)) finSupp 0) |
| 215 | 48, 110, 49 | mulg0 19092 |
. . . . . . . . . . 11
⊢ (𝑣 ∈ 𝐾 →
(0(.g‘(mulGrp‘𝑅))𝑣) = (0g‘(mulGrp‘𝑅))) |
| 216 | 215 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣 ∈ 𝐾) →
(0(.g‘(mulGrp‘𝑅))𝑣) = (0g‘(mulGrp‘𝑅))) |
| 217 | | fvexd 6921 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (𝑐‘𝑘) ∈ V) |
| 218 | | fvexd 6921 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(0g‘(mulGrp‘𝑅)) ∈ V) |
| 219 | 214, 216,
217, 208, 218 | fsuppssov1 9424 |
. . . . . . . . 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))) |
| 220 | 48, 110, 201, 180, 210, 219 | gsumcl 19933 |
. . . . . . . 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) |
| 221 | 32 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring) |
| 222 | 28, 136, 221, 155, 151 | ringcld 20257 |
. . . . . . . 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) |
| 223 | 178 | mptex 7243 |
. . . . . . . . . . 11
⊢ (𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) ∈ V |
| 224 | 223 | 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) |
| 225 | | fvexd 6921 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(0g‘𝑅)
∈ V) |
| 226 | | funmpt 6604 |
. . . . . . . . . . 11
⊢ Fun
(𝑒 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) |
| 227 | 226 | 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))) |
| 228 | 11, 12, 171, 18 | mplelsfi 22015 |
. . . . . . . . . . 11
⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) finSupp (0g‘𝑈)) |
| 229 | 228 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) finSupp (0g‘𝑈)) |
| 230 | | ssidd 4007 |
. . . . . . . . . . . . . 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))) |
| 231 | | fvexd 6921 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
(0g‘𝑈)
∈ V) |
| 232 | 20, 230, 179, 231 | suppssr 8220 |
. . . . . . . . . . . . 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)) |
| 233 | 232 | fveq1d 6908 |
. . . . . . . . . . . 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)) |
| 234 | 3, 137, 197, 171, 6, 174 | mpl0 22026 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (0g‘𝑈) = ({𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ×
{(0g‘𝑅)})) |
| 235 | 234 | 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)})) |
| 236 | 235 | fveq1d 6908 |
. . . . . . . . . . . . . 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)) |
| 237 | | fvex 6919 |
. . . . . . . . . . . . . . . 16
⊢
(0g‘𝑅) ∈ V |
| 238 | 237 | fvconst2 7224 |
. . . . . . . . . . . . . . 15
⊢ (𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} → (({𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ×
{(0g‘𝑅)})‘𝑐) = (0g‘𝑅)) |
| 239 | 238 | 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)) |
| 240 | 236, 239 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) →
((0g‘𝑈)‘𝑐) = (0g‘𝑅)) |
| 241 | 240 | 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)) |
| 242 | 233, 241 | eqtrd 2777 |
. . . . . . . . . . 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)) |
| 243 | 242, 179 | suppss2 8225 |
. . . . . . . . . 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))) |
| 244 | 224, 225,
227, 229, 243 | fsuppsssuppgd 9422 |
. . . . . . . . 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)) |
| 245 | 32 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣 ∈ 𝐾) → 𝑅 ∈ Ring) |
| 246 | | simpr 484 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣 ∈ 𝐾) → 𝑣 ∈ 𝐾) |
| 247 | 28, 136, 197, 245, 246 | ringlzd 20292 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑐 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑣 ∈ 𝐾) → ((0g‘𝑅)(.r‘𝑅)𝑣) = (0g‘𝑅)) |
| 248 | 244, 247,
155, 151, 225 | fsuppssov1 9424 |
. . . . . . . 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)) |
| 249 | 28, 197, 136, 198, 179, 220, 222, 248 | gsummulc1 20313 |
. . . . . . 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)))))) |
| 250 | 161, 196,
249 | 3eqtr4d 2787 |
. . . . . 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)))))))) |
| 251 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑒 → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)) |
| 252 | 251 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)) |
| 253 | | simpl 482 |
. . . . . . . . . . . . 13
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → 𝑏 = 𝑐) |
| 254 | 252, 253 | fveq12d 6913 |
. . . . . . . . . . . 12
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎)‘𝑏) = (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑒)‘𝑐)) |
| 255 | | fveq1 6905 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑒 → (𝑎‘𝑗) = (𝑒‘𝑗)) |
| 256 | 255 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → (𝑎‘𝑗) = (𝑒‘𝑗)) |
| 257 | 256 | oveq1d 7446 |
. . . . . . . . . . . . . 14
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → ((𝑎‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = ((𝑒‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) |
| 258 | 257 | mpteq2dv 5244 |
. . . . . . . . . . . . 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)))) |
| 259 | 258 | oveq2d 7447 |
. . . . . . . . . . . 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))))) |
| 260 | 254, 259 | oveq12d 7449 |
. . . . . . . . . . 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)))))) |
| 261 | | fveq1 6905 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = 𝑐 → (𝑏‘𝑘) = (𝑐‘𝑘)) |
| 262 | 261 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → (𝑏‘𝑘) = (𝑐‘𝑘)) |
| 263 | 262 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢ ((𝑏 = 𝑐 ∧ 𝑎 = 𝑒) → ((𝑏‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) = ((𝑐‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) |
| 264 | 263 | mpteq2dv 5244 |
. . . . . . . . . . . 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)))) |
| 265 | 264 | oveq2d 7447 |
. . . . . . . . . . 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))))) |
| 266 | 260, 265 | oveq12d 7449 |
. . . . . . . . . 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)))))) |
| 267 | | eqid 2737 |
. . . . . . . . . 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)))))) |
| 268 | | ovex 7464 |
. . . . . . . . . 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 |
| 269 | 266, 267,
268 | ovmpoa 7588 |
. . . . . . . . 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)))))) |
| 270 | 269 | adantll 714 |
. . . . . . . 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)))))) |
| 271 | 270 | mpteq2dva 5242 |
. . . . . . 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))))))) |
| 272 | 271 | oveq2d 7447 |
. . . . . 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)))))))) |
| 273 | 250, 272 | eqtr4d 2780 |
. . . . 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)))) |
| 274 | 273 | mpteq2dva 5242 |
. . . 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))))) |
| 275 | 274 | oveq2d 7447 |
. . 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)))))) |
| 276 | 32 | ringcmnd 20281 |
. . . . 5
⊢ (𝜑 → 𝑅 ∈ CMnd) |
| 277 | | ovex 7464 |
. . . . . . 7
⊢
(ℕ0 ↑m 𝐼) ∈ V |
| 278 | 277 | rabex 5339 |
. . . . . 6
⊢ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ∈
V |
| 279 | 278 | a1i 11 |
. . . . 5
⊢ (𝜑 → {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ∈
V) |
| 280 | 32 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring) |
| 281 | 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)) |
| 282 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} |
| 283 | 4 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑉) |
| 284 | 16 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐽 ⊆ 𝐼) |
| 285 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin}) |
| 286 | 282, 13, 283, 284, 285 | psrbagres 42556 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ↾ 𝐽) ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈
Fin}) |
| 287 | 281, 286 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽)) ∈ (Base‘𝑈)) |
| 288 | 3, 28, 1, 137, 287 | mplelf 22018 |
. . . . . . . . 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) |
| 289 | | difssd 4137 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝐼 ∖ 𝐽) ⊆ 𝐼) |
| 290 | 282, 137,
283, 289, 285 | psrbagres 42556 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ↾ (𝐼 ∖ 𝐽)) ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈
Fin}) |
| 291 | 288, 290 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽))) ∈ 𝐾) |
| 292 | 200 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
(mulGrp‘𝑅) ∈
CMnd) |
| 293 | 24 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐽 ∈ V) |
| 294 | 51 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (mulGrp‘𝑅) ∈ Mnd) |
| 295 | 282 | psrbagf 21938 |
. . . . . . . . . . . . . 14
⊢ (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑑:𝐼⟶ℕ0) |
| 296 | 295 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑:𝐼⟶ℕ0) |
| 297 | 296, 284 | fssresd 6775 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ↾ 𝐽):𝐽⟶ℕ0) |
| 298 | 297 | ffvelcdmda 7104 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝑑 ↾ 𝐽)‘𝑗) ∈
ℕ0) |
| 299 | 58 | ffvelcdmda 7104 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐽) → ((𝐴 ↾ 𝐽)‘𝑗) ∈ 𝐾) |
| 300 | 299 | adantlr 715 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝐴 ↾ 𝐽)‘𝑗) ∈ 𝐾) |
| 301 | 48, 49, 294, 298, 300 | mulgnn0cld 19113 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) ∈ 𝐾) |
| 302 | 301 | fmpttd 7135 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))):𝐽⟶𝐾) |
| 303 | 24 | mptexd 7244 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) ∈ V) |
| 304 | 303 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) ∈ V) |
| 305 | | fvexd 6921 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
(0g‘(mulGrp‘𝑅)) ∈ V) |
| 306 | | funmpt 6604 |
. . . . . . . . . . 11
⊢ Fun
(𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) |
| 307 | 306 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → Fun
(𝑗 ∈ 𝐽 ↦ (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)))) |
| 308 | 282 | psrbagfsupp 21939 |
. . . . . . . . . . . 12
⊢ (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑑 finSupp 0) |
| 309 | 308 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 finSupp 0) |
| 310 | | 0zd 12625 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 0 ∈
ℤ) |
| 311 | 309, 310 | fsuppres 9433 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ↾ 𝐽) finSupp 0) |
| 312 | | ssidd 4007 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑑 ↾ 𝐽) supp 0) ⊆ ((𝑑 ↾ 𝐽) supp 0)) |
| 313 | 297, 312,
293, 310 | suppssr 8220 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ (𝐽 ∖ ((𝑑 ↾ 𝐽) supp 0))) → ((𝑑 ↾ 𝐽)‘𝑗) = 0) |
| 314 | 313 | oveq1d 7446 |
. . . . . . . . . . . 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))) |
| 315 | | eldifi 4131 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ (𝐽 ∖ ((𝑑 ↾ 𝐽) supp 0)) → 𝑗 ∈ 𝐽) |
| 316 | 48, 110, 49 | mulg0 19092 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ↾ 𝐽)‘𝑗) ∈ 𝐾 →
(0(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅))) |
| 317 | 300, 316 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) →
(0(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = (0g‘(mulGrp‘𝑅))) |
| 318 | 315, 317 | sylan2 593 |
. . . . . . . . . . . 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))) |
| 319 | 314, 318 | eqtrd 2777 |
. . . . . . . . . . 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))) |
| 320 | 319, 293 | suppss2 8225 |
. . . . . . . . . 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)) |
| 321 | 304, 305,
307, 311, 320 | fsuppsssuppgd 9422 |
. . . . . . . . 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))) |
| 322 | 48, 110, 292, 293, 302, 321 | gsumcl 19933 |
. . . . . . . 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) |
| 323 | 28, 136, 280, 291, 322 | ringcld 20257 |
. . . . . . 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) |
| 324 | 6 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝐼 ∖ 𝐽) ∈ V) |
| 325 | 51 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → (mulGrp‘𝑅) ∈ Mnd) |
| 326 | 296, 289 | fssresd 6775 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ↾ (𝐼 ∖ 𝐽)):(𝐼 ∖ 𝐽)⟶ℕ0) |
| 327 | 326 | ffvelcdmda 7104 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘) ∈
ℕ0) |
| 328 | 206 | ffvelcdmda 7104 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘) ∈ 𝐾) |
| 329 | 328 | adantlr 715 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘) ∈ 𝐾) |
| 330 | 48, 49, 325, 327, 329 | mulgnn0cld 19113 |
. . . . . . . . 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) |
| 331 | 330 | fmpttd 7135 |
. . . . . . . 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) |
| 332 | 324 | mptexd 7244 |
. . . . . . . . 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) |
| 333 | | funmpt 6604 |
. . . . . . . . . 10
⊢ Fun
(𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘))) |
| 334 | 333 | 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)))) |
| 335 | 309, 310 | fsuppres 9433 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑑 ↾ (𝐼 ∖ 𝐽)) finSupp 0) |
| 336 | | ssidd 4007 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0) ⊆ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0)) |
| 337 | 326, 336,
324, 310 | suppssr 8220 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼 ∖ 𝐽) ∖ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0))) → ((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘) = 0) |
| 338 | 337 | oveq1d 7446 |
. . . . . . . . . . 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))) |
| 339 | | eldifi 4131 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ((𝐼 ∖ 𝐽) ∖ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0)) → 𝑘 ∈ (𝐼 ∖ 𝐽)) |
| 340 | 339, 329 | sylan2 593 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ ((𝐼 ∖ 𝐽) ∖ ((𝑑 ↾ (𝐼 ∖ 𝐽)) supp 0))) → ((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘) ∈ 𝐾) |
| 341 | 48, 110, 49 | mulg0 19092 |
. . . . . . . . . . . 12
⊢ (((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘) ∈ 𝐾 →
(0(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)) = (0g‘(mulGrp‘𝑅))) |
| 342 | 340, 341 | 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))) |
| 343 | 338, 342 | eqtrd 2777 |
. . . . . . . . . 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))) |
| 344 | 343, 324 | suppss2 8225 |
. . . . . . . . 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)) |
| 345 | 332, 305,
334, 335, 344 | fsuppsssuppgd 9422 |
. . . . . . . 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))) |
| 346 | 48, 110, 292, 324, 331, 345 | gsumcl 19933 |
. . . . . . 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) |
| 347 | 28, 136, 280, 323, 346 | ringcld 20257 |
. . . . . 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) |
| 348 | 347 | fmpttd 7135 |
. . . . 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) |
| 349 | 7 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing) |
| 350 | 17 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐹 ∈ 𝐵) |
| 351 | 282, 14, 15, 349, 284, 350, 285 | selvvvval 42595 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽))) = (𝐹‘𝑑)) |
| 352 | 351 | mpteq2dva 5242 |
. . . . . . . 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))) |
| 353 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(Base‘𝑅) =
(Base‘𝑅) |
| 354 | 14, 353, 15, 282, 17 | mplelf 22018 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:{ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
| 355 | 354 | feqmptd 6977 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 = (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘𝑑))) |
| 356 | 14, 15, 197, 17 | mplelsfi 22015 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 finSupp (0g‘𝑅)) |
| 357 | 355, 356 | eqbrtrrd 5167 |
. . . . . . . 8
⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝐹‘𝑑)) finSupp (0g‘𝑅)) |
| 358 | 352, 357 | eqbrtrd 5165 |
. . . . . . 7
⊢ (𝜑 → (𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽)))) finSupp (0g‘𝑅)) |
| 359 | 32 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑣 ∈ 𝐾) → 𝑅 ∈ Ring) |
| 360 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑣 ∈ 𝐾) → 𝑣 ∈ 𝐾) |
| 361 | 28, 136, 197, 359, 360 | ringlzd 20292 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑣 ∈ 𝐾) → ((0g‘𝑅)(.r‘𝑅)𝑣) = (0g‘𝑅)) |
| 362 | | fvexd 6921 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) →
(((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽))‘(𝑑 ↾ (𝐼 ∖ 𝐽))) ∈ V) |
| 363 | | fvexd 6921 |
. . . . . . 7
⊢ (𝜑 → (0g‘𝑅) ∈ V) |
| 364 | 358, 361,
362, 322, 363 | fsuppssov1 9424 |
. . . . . 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)) |
| 365 | | ovexd 7466 |
. . . . . 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))))) ∈ V) |
| 366 | 364, 361,
365, 346, 363 | fsuppssov1 9424 |
. . . . 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)) |
| 367 | | eqid 2737 |
. . . . . 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)) |
| 368 | 282, 13, 137, 367, 4, 16 | evlselvlem 42596 |
. . . . 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}) |
| 369 | 28, 197, 276, 279, 348, 366, 368 | gsumf1o 19934 |
. . . 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))))) |
| 370 | 137 | psrbagf 21938 |
. . . . . . . . . 10
⊢ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑏:(𝐼 ∖ 𝐽)⟶ℕ0) |
| 371 | 370 | ad2antrl 728 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑏:(𝐼 ∖ 𝐽)⟶ℕ0) |
| 372 | 13 | psrbagf 21938 |
. . . . . . . . . 10
⊢ (𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} → 𝑎:𝐽⟶ℕ0) |
| 373 | 372 | ad2antll 729 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑎:𝐽⟶ℕ0) |
| 374 | | disjdifr 4473 |
. . . . . . . . . 10
⊢ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅ |
| 375 | 374 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) |
| 376 | 371, 373,
375 | fun2d 6772 |
. . . . . . . 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) |
| 377 | | undifr 4483 |
. . . . . . . . . . 11
⊢ (𝐽 ⊆ 𝐼 ↔ ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼) |
| 378 | 16, 377 | sylib 218 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐼 ∖ 𝐽) ∪ 𝐽) = 𝐼) |
| 379 | 378 | 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) |
| 380 | 379 | feq2d 6722 |
. . . . . . . 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)) |
| 381 | 376, 380 | mpbid 232 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝑏 ∪ 𝑎):𝐼⟶ℕ0) |
| 382 | | vex 3484 |
. . . . . . . . . . 11
⊢ 𝑏 ∈ V |
| 383 | | vex 3484 |
. . . . . . . . . . 11
⊢ 𝑎 ∈ V |
| 384 | 382, 383 | unex 7764 |
. . . . . . . . . 10
⊢ (𝑏 ∪ 𝑎) ∈ V |
| 385 | 384 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝑏 ∪ 𝑎) ∈ V) |
| 386 | | 0zd 12625 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 0
∈ ℤ) |
| 387 | 381 | ffund 6740 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → Fun
(𝑏 ∪ 𝑎)) |
| 388 | 137 | psrbagfsupp 21939 |
. . . . . . . . . . 11
⊢ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑏 finSupp 0) |
| 389 | 388 | ad2antrl 728 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑏 finSupp 0) |
| 390 | 13 | psrbagfsupp 21939 |
. . . . . . . . . . 11
⊢ (𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} → 𝑎 finSupp 0) |
| 391 | 390 | ad2antll 729 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑎 finSupp 0) |
| 392 | 389, 391 | fsuppun 9427 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((𝑏 ∪ 𝑎) supp 0) ∈
Fin) |
| 393 | 385, 386,
387, 392 | isfsuppd 9406 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → (𝑏 ∪ 𝑎) finSupp 0) |
| 394 | | fcdmnn0fsuppg 12586 |
. . . . . . . . 9
⊢ (((𝑏 ∪ 𝑎) ∈ V ∧ (𝑏 ∪ 𝑎):𝐼⟶ℕ0) → ((𝑏 ∪ 𝑎) finSupp 0 ↔ (◡(𝑏 ∪ 𝑎) “ ℕ) ∈
Fin)) |
| 395 | 385, 381,
394 | syl2anc 584 |
. . . . . . . 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)) |
| 396 | 393, 395 | mpbid 232 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → (◡(𝑏 ∪ 𝑎) “ ℕ) ∈
Fin) |
| 397 | 4 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝐼 ∈ 𝑉) |
| 398 | 282 | psrbag 21937 |
. . . . . . . 8
⊢ (𝐼 ∈ 𝑉 → ((𝑏 ∪ 𝑎) ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↔ ((𝑏 ∪ 𝑎):𝐼⟶ℕ0 ∧ (◡(𝑏 ∪ 𝑎) “ ℕ) ∈
Fin))) |
| 399 | 397, 398 | 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))) |
| 400 | 381, 396,
399 | mpbir2and 713 |
. . . . . 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}) |
| 401 | | eqidd 2738 |
. . . . . 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))) |
| 402 | | eqidd 2738 |
. . . . . 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))))))) |
| 403 | | reseq1 5991 |
. . . . . . . . . . . 12
⊢ (𝑑 = (𝑏 ∪ 𝑎) → (𝑑 ↾ 𝐽) = ((𝑏 ∪ 𝑎) ↾ 𝐽)) |
| 404 | 403 | fveq2d 6910 |
. . . . . . . . . . 11
⊢ (𝑑 = (𝑏 ∪ 𝑎) → ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑑 ↾ 𝐽)) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘((𝑏 ∪ 𝑎) ↾ 𝐽))) |
| 405 | | reseq1 5991 |
. . . . . . . . . . 11
⊢ (𝑑 = (𝑏 ∪ 𝑎) → (𝑑 ↾ (𝐼 ∖ 𝐽)) = ((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))) |
| 406 | 404, 405 | fveq12d 6913 |
. . . . . . . . . 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)))) |
| 407 | 403 | fveq1d 6908 |
. . . . . . . . . . . . 13
⊢ (𝑑 = (𝑏 ∪ 𝑎) → ((𝑑 ↾ 𝐽)‘𝑗) = (((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)) |
| 408 | 407 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢ (𝑑 = (𝑏 ∪ 𝑎) → (((𝑑 ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗)) = ((((𝑏 ∪ 𝑎) ↾ 𝐽)‘𝑗)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑗))) |
| 409 | 408 | mpteq2dv 5244 |
. . . . . . . . . . 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)))) |
| 410 | 409 | oveq2d 7447 |
. . . . . . . . . 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))))) |
| 411 | 406, 410 | oveq12d 7449 |
. . . . . . . . 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)))))) |
| 412 | 405 | fveq1d 6908 |
. . . . . . . . . . . 12
⊢ (𝑑 = (𝑏 ∪ 𝑎) → ((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘) = (((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽))‘𝑘)) |
| 413 | 412 | oveq1d 7446 |
. . . . . . . . . . 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))) |
| 414 | 413 | mpteq2dv 5244 |
. . . . . . . . . 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)))) |
| 415 | 414 | oveq2d 7447 |
. . . . . . . . 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))))) |
| 416 | 411, 415 | oveq12d 7449 |
. . . . . . . 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)))))) |
| 417 | 384, 416 | csbie 3934 |
. . . . . . 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))))) |
| 418 | 370 | ffnd 6737 |
. . . . . . . . . . . . 13
⊢ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑏 Fn (𝐼 ∖ 𝐽)) |
| 419 | 418 | ad2antrl 728 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑏 Fn (𝐼 ∖ 𝐽)) |
| 420 | 373 | ffnd 6737 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑎 Fn 𝐽) |
| 421 | | fnunres2 6681 |
. . . . . . . . . . . 12
⊢ ((𝑏 Fn (𝐼 ∖ 𝐽) ∧ 𝑎 Fn 𝐽 ∧ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) → ((𝑏 ∪ 𝑎) ↾ 𝐽) = 𝑎) |
| 422 | 419, 420,
375, 421 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((𝑏 ∪ 𝑎) ↾ 𝐽) = 𝑎) |
| 423 | 422 | fveq2d 6910 |
. . . . . . . . . 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)) |
| 424 | | fnunres1 6680 |
. . . . . . . . . . 11
⊢ ((𝑏 Fn (𝐼 ∖ 𝐽) ∧ 𝑎 Fn 𝐽 ∧ ((𝐼 ∖ 𝐽) ∩ 𝐽) = ∅) → ((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽)) = 𝑏) |
| 425 | 419, 420,
375, 424 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) →
((𝑏 ∪ 𝑎) ↾ (𝐼 ∖ 𝐽)) = 𝑏) |
| 426 | 423, 425 | fveq12d 6913 |
. . . . . . . . 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)) |
| 427 | 422 | fveq1d 6908 |
. . . . . . . . . . . 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)) |
| 428 | 427 | oveq1d 7446 |
. . . . . . . . . . 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))) |
| 429 | 428 | mpteq2dv 5244 |
. . . . . . . . . 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)))) |
| 430 | 429 | oveq2d 7447 |
. . . . . . . . 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))))) |
| 431 | 426, 430 | oveq12d 7449 |
. . . . . . . 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)))))) |
| 432 | 425 | fveq1d 6908 |
. . . . . . . . . . 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)) |
| 433 | 432 | oveq1d 7446 |
. . . . . . . . . 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))) |
| 434 | 433 | mpteq2dv 5244 |
. . . . . . . . 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)))) |
| 435 | 434 | oveq2d 7447 |
. . . . . . . 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))))) |
| 436 | 431, 435 | oveq12d 7449 |
. . . . . . 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)))))) |
| 437 | 417, 436 | eqtrid 2789 |
. . . . . 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)))))) |
| 438 | 400, 401,
402, 437 | fmpocos 42275 |
. . . . 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))))))) |
| 439 | 438 | oveq2d 7447 |
. . . 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)))))))) |
| 440 | | ovex 7464 |
. . . . . . 7
⊢
(ℕ0 ↑m (𝐼 ∖ 𝐽)) ∈ V |
| 441 | 440 | rabex 5339 |
. . . . . 6
⊢ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈
V |
| 442 | 441 | a1i 11 |
. . . . 5
⊢ (𝜑 → {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈
V) |
| 443 | 178 | a1i 11 |
. . . . 5
⊢ (𝜑 → {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin} ∈
V) |
| 444 | 32 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑅 ∈ Ring) |
| 445 | 19 | ffvelcdmda 7104 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin}) →
((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘𝑎) ∈ (Base‘𝑈)) |
| 446 | 3, 28, 1, 137, 445 | mplelf 22018 |
. . . . . . . . . . . 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) |
| 447 | 446 | ffvelcdmda 7104 |
. . . . . . . . . . 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) |
| 448 | 447 | an32s 652 |
. . . . . . . . . 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) |
| 449 | 448 | 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) |
| 450 | 24 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝐽 ∈ V) |
| 451 | 7 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ {𝑓 ∈ (ℕ0
↑m (𝐼
∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑎 ∈ {𝑔 ∈ (ℕ0
↑m 𝐽)
∣ (◡𝑔 “ ℕ) ∈ Fin})) → 𝑅 ∈ CRing) |
| 452 | 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)) |
| 453 | | simprr 773 |
. . . . . . . . . 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}) |
| 454 | 13, 28, 47, 49, 450, 451, 452, 453 | evlsvvvallem 42571 |
. . . . . . . . 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) |
| 455 | 28, 136, 444, 449, 454 | ringcld 20257 |
. . . . . . . 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) |
| 456 | 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) |
| 457 | 35, 5 | elmapssresd 42282 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴 ↾ (𝐼 ∖ 𝐽)) ∈ (𝐾 ↑m (𝐼 ∖ 𝐽))) |
| 458 | 457 | 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))) |
| 459 | | simprl 771 |
. . . . . . . . 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}) |
| 460 | 137, 28, 47, 49, 456, 451, 458, 459 | evlsvvvallem 42571 |
. . . . . . . 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) |
| 461 | 28, 136, 444, 455, 460 | ringcld 20257 |
. . . . . . 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) |
| 462 | 461 | ralrimivva 3202 |
. . . . . 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) |
| 463 | 267 | fmpo 8093 |
. . . . . 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) |
| 464 | 462, 463 | sylib 218 |
. . . . 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) |
| 465 | | f1of1 6847 |
. . . . . . . 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}) |
| 466 | 368, 465 | 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}) |
| 467 | 278 | mptex 7243 |
. . . . . . . 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)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) ∈ V |
| 468 | 467 | 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)))))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ (((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘)(.g‘(mulGrp‘𝑅))((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘)))))) ∈ V) |
| 469 | 366, 466,
363, 468 | fsuppco 9442 |
. . . . . 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)) |
| 470 | 438, 469 | eqbrtrrd 5167 |
. . . . 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)) |
| 471 | 28, 197, 276, 442, 443, 464, 470 | gsumxp 19994 |
. . . 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)))))) |
| 472 | 369, 439,
471 | 3eqtrd 2781 |
. . 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)))))) |
| 473 | 28, 136, 280, 291, 322, 346 | ringassd 20254 |
. . . . . 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))))))) |
| 474 | 47, 136 | mgpplusg 20141 |
. . . . . . . . 9
⊢
(.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
| 475 | 51 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (mulGrp‘𝑅) ∈ Mnd) |
| 476 | 296 | ffvelcdmda 7104 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝑑‘𝑖) ∈
ℕ0) |
| 477 | 57 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐴:𝐼⟶𝐾) |
| 478 | 477 | ffvelcdmda 7104 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → (𝐴‘𝑖) ∈ 𝐾) |
| 479 | 48, 49, 475, 476, 478 | mulgnn0cld 19113 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑖 ∈ 𝐼) → ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)) ∈ 𝐾) |
| 480 | 479 | fmpttd 7135 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))):𝐼⟶𝐾) |
| 481 | 296 | feqmptd 6977 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑑 = (𝑖 ∈ 𝐼 ↦ (𝑑‘𝑖))) |
| 482 | 481, 309 | eqbrtrrd 5167 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑖 ∈ 𝐼 ↦ (𝑑‘𝑖)) finSupp 0) |
| 483 | 111 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ 𝐾) →
(0(.g‘(mulGrp‘𝑅))𝑘) = (0g‘(mulGrp‘𝑅))) |
| 484 | 482, 483,
476, 478, 305 | fsuppssov1 9424 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑖 ∈ 𝐼 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) finSupp
(0g‘(mulGrp‘𝑅))) |
| 485 | | disjdif 4472 |
. . . . . . . . . 10
⊢ (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅ |
| 486 | 485 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝐽 ∩ (𝐼 ∖ 𝐽)) = ∅) |
| 487 | | undif 4482 |
. . . . . . . . . . . 12
⊢ (𝐽 ⊆ 𝐼 ↔ (𝐽 ∪ (𝐼 ∖ 𝐽)) = 𝐼) |
| 488 | 16, 487 | sylib 218 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐽 ∪ (𝐼 ∖ 𝐽)) = 𝐼) |
| 489 | 488 | eqcomd 2743 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐼 = (𝐽 ∪ (𝐼 ∖ 𝐽))) |
| 490 | 489 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝐼 = (𝐽 ∪ (𝐼 ∖ 𝐽))) |
| 491 | 48, 110, 474, 292, 283, 480, 484, 486, 490 | gsumsplit 19946 |
. . . . . . . 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))))) |
| 492 | 284 | resmptd 6058 |
. . . . . . . . . . 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)))) |
| 493 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 𝑗 → (𝑑‘𝑖) = (𝑑‘𝑗)) |
| 494 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 𝑗 → (𝐴‘𝑖) = (𝐴‘𝑗)) |
| 495 | 493, 494 | oveq12d 7449 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)) = ((𝑑‘𝑗)(.g‘(mulGrp‘𝑅))(𝐴‘𝑗))) |
| 496 | 495 | cbvmptv 5255 |
. . . . . . . . . . . 12
⊢ (𝑖 ∈ 𝐽 ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) = (𝑗 ∈ 𝐽 ↦ ((𝑑‘𝑗)(.g‘(mulGrp‘𝑅))(𝐴‘𝑗))) |
| 497 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → 𝑗 ∈ 𝐽) |
| 498 | 497 | fvresd 6926 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝑑 ↾ 𝐽)‘𝑗) = (𝑑‘𝑗)) |
| 499 | 497 | fvresd 6926 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑗 ∈ 𝐽) → ((𝐴 ↾ 𝐽)‘𝑗) = (𝐴‘𝑗)) |
| 500 | 498, 499 | oveq12d 7449 |
. . . . . . . . . . . . . 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))) |
| 501 | 500 | eqcomd 2743 |
. . . . . . . . . . . . 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))) |
| 502 | 501 | mpteq2dva 5242 |
. . . . . . . . . . . 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)))) |
| 503 | 496, 502 | eqtrid 2789 |
. . . . . . . . . . 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)))) |
| 504 | 492, 503 | eqtrd 2777 |
. . . . . . . . . 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)))) |
| 505 | 504 | oveq2d 7447 |
. . . . . . . . 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))))) |
| 506 | 289 | resmptd 6058 |
. . . . . . . . . . 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)))) |
| 507 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 𝑘 → (𝑑‘𝑖) = (𝑑‘𝑘)) |
| 508 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 𝑘 → (𝐴‘𝑖) = (𝐴‘𝑘)) |
| 509 | 507, 508 | oveq12d 7449 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑘 → ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)) = ((𝑑‘𝑘)(.g‘(mulGrp‘𝑅))(𝐴‘𝑘))) |
| 510 | 509 | cbvmptv 5255 |
. . . . . . . . . . . 12
⊢ (𝑖 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑑‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) = (𝑘 ∈ (𝐼 ∖ 𝐽) ↦ ((𝑑‘𝑘)(.g‘(mulGrp‘𝑅))(𝐴‘𝑘))) |
| 511 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → 𝑘 ∈ (𝐼 ∖ 𝐽)) |
| 512 | 511 | fvresd 6926 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝑑 ↾ (𝐼 ∖ 𝐽))‘𝑘) = (𝑑‘𝑘)) |
| 513 | 511 | fvresd 6926 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑘 ∈ (𝐼 ∖ 𝐽)) → ((𝐴 ↾ (𝐼 ∖ 𝐽))‘𝑘) = (𝐴‘𝑘)) |
| 514 | 512, 513 | oveq12d 7449 |
. . . . . . . . . . . . . 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))) |
| 515 | 514 | eqcomd 2743 |
. . . . . . . . . . . . 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))) |
| 516 | 515 | mpteq2dva 5242 |
. . . . . . . . . . . 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)))) |
| 517 | 510, 516 | eqtrid 2789 |
. . . . . . . . . . 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)))) |
| 518 | 506, 517 | eqtrd 2777 |
. . . . . . . . . 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)))) |
| 519 | 518 | oveq2d 7447 |
. . . . . . . . 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))))) |
| 520 | 505, 519 | oveq12d 7449 |
. . . . . . . 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)))))) |
| 521 | 491, 520 | eqtr2d 2778 |
. . . . . . 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))))) |
| 522 | 351, 521 | oveq12d 7449 |
. . . . . 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)))))) |
| 523 | 473, 522 | eqtrd 2777 |
. . . . 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)))))) |
| 524 | 523 | mpteq2dva 5242 |
. . . 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))))))) |
| 525 | 524 | oveq2d 7447 |
. . 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)))))))) |
| 526 | 275, 472,
525 | 3eqtr2d 2783 |
. 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)))))))) |
| 527 | | eqid 2737 |
. . 3
⊢ ((𝐼 ∖ 𝐽) eval 𝑅) = ((𝐼 ∖ 𝐽) eval 𝑅) |
| 528 | 527, 3, 1, 137, 28, 47, 49, 136, 6, 7, 166, 457 | evlvvval 42583 |
. 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)))))))) |
| 529 | | eqid 2737 |
. . 3
⊢ (𝐼 eval 𝑅) = (𝐼 eval 𝑅) |
| 530 | 529, 14, 15, 282, 28, 47, 49, 136, 4, 7, 17, 35 | evlvvval 42583 |
. 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)))))))) |
| 531 | 526, 528,
530 | 3eqtr4d 2787 |
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)) |