Step | Hyp | Ref
| Expression |
1 | | vex 3403 |
. . . . . 6
⊢ 𝑥 ∈ V |
2 | | eqid 2739 |
. . . . . . 7
⊢ (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) = (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) |
3 | 2 | elrnmpt 5800 |
. . . . . 6
⊢ (𝑥 ∈ V → (𝑥 ∈ ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ↔ ∃𝑘 ∈ 𝑆 𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) |
4 | 1, 3 | mp1i 13 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ↔ ∃𝑘 ∈ 𝑆 𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) |
5 | | vex 3403 |
. . . . . . . 8
⊢ 𝑦 ∈ V |
6 | 2 | elrnmpt 5800 |
. . . . . . . 8
⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ↔ ∃𝑘 ∈ 𝑆 𝑦 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) |
7 | 5, 6 | mp1i 13 |
. . . . . . 7
⊢ (𝜑 → (𝑦 ∈ ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ↔ ∃𝑘 ∈ 𝑆 𝑦 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) |
8 | | fveq2 6677 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑙 → (𝑌‘𝑘) = (𝑌‘𝑙)) |
9 | | fveq2 6677 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑙 → (𝑉‘𝑘) = (𝑉‘𝑙)) |
10 | 8, 9 | oveq12d 7191 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑙 → ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) = ((𝑌‘𝑙) ↑ (𝑉‘𝑙))) |
11 | 10 | eqeq2d 2750 |
. . . . . . . . 9
⊢ (𝑘 = 𝑙 → (𝑦 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ↔ 𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙)))) |
12 | 11 | cbvrexvw 3351 |
. . . . . . . 8
⊢
(∃𝑘 ∈
𝑆 𝑦 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ↔ ∃𝑙 ∈ 𝑆 𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙))) |
13 | | eqid 2739 |
. . . . . . . . . . . . . 14
⊢
(Base‘𝑃) =
(Base‘𝑃) |
14 | | mplcoe2.g |
. . . . . . . . . . . . . . . 16
⊢ 𝐺 = (mulGrp‘𝑃) |
15 | | eqid 2739 |
. . . . . . . . . . . . . . . 16
⊢
(.r‘𝑃) = (.r‘𝑃) |
16 | 14, 15 | mgpplusg 19365 |
. . . . . . . . . . . . . . 15
⊢
(.r‘𝑃) = (+g‘𝐺) |
17 | 16 | eqcomi 2748 |
. . . . . . . . . . . . . 14
⊢
(+g‘𝐺) = (.r‘𝑃) |
18 | | mplcoe2.m |
. . . . . . . . . . . . . 14
⊢ ↑ =
(.g‘𝐺) |
19 | | mplcoe1.i |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐼 ∈ 𝑊) |
20 | | mplcoe5.r |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑅 ∈ Ring) |
21 | | mplcoe1.p |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑃 = (𝐼 mPoly 𝑅) |
22 | 21 | mplring 20837 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring) |
23 | 19, 20, 22 | syl2anc 587 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑃 ∈ Ring) |
24 | | ringsrg 19464 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑃 ∈ Ring → 𝑃 ∈ SRing) |
25 | 23, 24 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑃 ∈ SRing) |
26 | 25 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → 𝑃 ∈ SRing) |
27 | 26 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → 𝑃 ∈ SRing) |
28 | 14 | ringmgp 19425 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑃 ∈ Ring → 𝐺 ∈ Mnd) |
29 | 23, 28 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐺 ∈ Mnd) |
30 | 29 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → 𝐺 ∈ Mnd) |
31 | | mplcoe5.s |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑆 ⊆ 𝐼) |
32 | 31 | sseld 3877 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑙 ∈ 𝑆 → 𝑙 ∈ 𝐼)) |
33 | 32 | imdistani 572 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → (𝜑 ∧ 𝑙 ∈ 𝐼)) |
34 | | mplcoe5.y |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑌 ∈ 𝐷) |
35 | | mplcoe1.d |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝐷 = {𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈
Fin} |
36 | 35 | psrbag 20733 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐼 ∈ 𝑊 → (𝑌 ∈ 𝐷 ↔ (𝑌:𝐼⟶ℕ0 ∧ (◡𝑌 “ ℕ) ∈
Fin))) |
37 | 19, 36 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑌 ∈ 𝐷 ↔ (𝑌:𝐼⟶ℕ0 ∧ (◡𝑌 “ ℕ) ∈
Fin))) |
38 | 34, 37 | mpbid 235 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑌:𝐼⟶ℕ0 ∧ (◡𝑌 “ ℕ) ∈
Fin)) |
39 | 38 | simpld 498 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑌:𝐼⟶ℕ0) |
40 | 39 | ffvelrnda 6864 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐼) → (𝑌‘𝑙) ∈
ℕ0) |
41 | 33, 40 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → (𝑌‘𝑙) ∈
ℕ0) |
42 | | mplcoe2.v |
. . . . . . . . . . . . . . . . 17
⊢ 𝑉 = (𝐼 mVar 𝑅) |
43 | 19 | adantr 484 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → 𝐼 ∈ 𝑊) |
44 | 20 | adantr 484 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → 𝑅 ∈ Ring) |
45 | 31 | sselda 3878 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → 𝑙 ∈ 𝐼) |
46 | 21, 42, 13, 43, 44, 45 | mvrcl 20834 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → (𝑉‘𝑙) ∈ (Base‘𝑃)) |
47 | 14, 13 | mgpbas 19367 |
. . . . . . . . . . . . . . . . 17
⊢
(Base‘𝑃) =
(Base‘𝐺) |
48 | 47, 18 | mulgnn0cl 18365 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺 ∈ Mnd ∧ (𝑌‘𝑙) ∈ ℕ0 ∧ (𝑉‘𝑙) ∈ (Base‘𝑃)) → ((𝑌‘𝑙) ↑ (𝑉‘𝑙)) ∈ (Base‘𝑃)) |
49 | 30, 41, 46, 48 | syl3anc 1372 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → ((𝑌‘𝑙) ↑ (𝑉‘𝑙)) ∈ (Base‘𝑃)) |
50 | 49 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → ((𝑌‘𝑙) ↑ (𝑉‘𝑙)) ∈ (Base‘𝑃)) |
51 | 19 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → 𝐼 ∈ 𝑊) |
52 | 20 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → 𝑅 ∈ Ring) |
53 | 31 | sselda 3878 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ 𝐼) |
54 | 21, 42, 13, 51, 52, 53 | mvrcl 20834 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑉‘𝑘) ∈ (Base‘𝑃)) |
55 | 54 | adantlr 715 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → (𝑉‘𝑘) ∈ (Base‘𝑃)) |
56 | 39 | ffvelrnda 6864 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑌‘𝑘) ∈
ℕ0) |
57 | 53, 56 | syldan 594 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑌‘𝑘) ∈
ℕ0) |
58 | 57 | adantlr 715 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → (𝑌‘𝑘) ∈
ℕ0) |
59 | 46 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → (𝑉‘𝑙) ∈ (Base‘𝑃)) |
60 | 41 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → (𝑌‘𝑙) ∈
ℕ0) |
61 | | mplcoe5.c |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦))) |
62 | | fveq2 6677 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑙 → (𝑉‘𝑥) = (𝑉‘𝑙)) |
63 | 62 | oveq2d 7189 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑙 → ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑙))) |
64 | 62 | oveq1d 7188 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑙 → ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦)) = ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑦))) |
65 | 63, 64 | eqeq12d 2755 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 𝑙 → (((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦)) ↔ ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑙)) = ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑦)))) |
66 | | fveq2 6677 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = 𝑘 → (𝑉‘𝑦) = (𝑉‘𝑘)) |
67 | 66 | oveq1d 7188 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 𝑘 → ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑙)) = ((𝑉‘𝑘)(+g‘𝐺)(𝑉‘𝑙))) |
68 | 66 | oveq2d 7189 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 𝑘 → ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑦)) = ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑘))) |
69 | 67, 68 | eqeq12d 2755 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑘 → (((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑙)) = ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑦)) ↔ ((𝑉‘𝑘)(+g‘𝐺)(𝑉‘𝑙)) = ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑘)))) |
70 | 65, 69 | rspc2v 3537 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑙 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼) → (∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦)) → ((𝑉‘𝑘)(+g‘𝐺)(𝑉‘𝑙)) = ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑘)))) |
71 | 45, 53 | anim12dan 622 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑙 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) → (𝑙 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼)) |
72 | 70, 71 | syl11 33 |
. . . . . . . . . . . . . . . . . 18
⊢
(∀𝑥 ∈
𝐼 ∀𝑦 ∈ 𝐼 ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦)) → ((𝜑 ∧ (𝑙 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) → ((𝑉‘𝑘)(+g‘𝐺)(𝑉‘𝑙)) = ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑘)))) |
73 | 72 | expd 419 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑥 ∈
𝐼 ∀𝑦 ∈ 𝐼 ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦)) → (𝜑 → ((𝑙 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → ((𝑉‘𝑘)(+g‘𝐺)(𝑉‘𝑙)) = ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑘))))) |
74 | 61, 73 | mpcom 38 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑙 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → ((𝑉‘𝑘)(+g‘𝐺)(𝑉‘𝑙)) = ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑘)))) |
75 | 74 | impl 459 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → ((𝑉‘𝑘)(+g‘𝐺)(𝑉‘𝑙)) = ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑘))) |
76 | 13, 17, 14, 18, 27, 55, 59, 60, 75 | srgpcomp 19404 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → (((𝑌‘𝑙) ↑ (𝑉‘𝑙))(+g‘𝐺)(𝑉‘𝑘)) = ((𝑉‘𝑘)(+g‘𝐺)((𝑌‘𝑙) ↑ (𝑉‘𝑙)))) |
77 | 13, 17, 14, 18, 27, 50, 55, 58, 76 | srgpcomp 19404 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → (((𝑌‘𝑘) ↑ (𝑉‘𝑘))(+g‘𝐺)((𝑌‘𝑙) ↑ (𝑉‘𝑙))) = (((𝑌‘𝑙) ↑ (𝑉‘𝑙))(+g‘𝐺)((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) |
78 | | oveq12 7182 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ∧ 𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙))) → (𝑥(+g‘𝐺)𝑦) = (((𝑌‘𝑘) ↑ (𝑉‘𝑘))(+g‘𝐺)((𝑌‘𝑙) ↑ (𝑉‘𝑙)))) |
79 | | oveq12 7182 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙)) ∧ 𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) → (𝑦(+g‘𝐺)𝑥) = (((𝑌‘𝑙) ↑ (𝑉‘𝑙))(+g‘𝐺)((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) |
80 | 79 | ancoms 462 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ∧ 𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙))) → (𝑦(+g‘𝐺)𝑥) = (((𝑌‘𝑙) ↑ (𝑉‘𝑙))(+g‘𝐺)((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) |
81 | 78, 80 | eqeq12d 2755 |
. . . . . . . . . . . . 13
⊢ ((𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ∧ 𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙))) → ((𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥) ↔ (((𝑌‘𝑘) ↑ (𝑉‘𝑘))(+g‘𝐺)((𝑌‘𝑙) ↑ (𝑉‘𝑙))) = (((𝑌‘𝑙) ↑ (𝑉‘𝑙))(+g‘𝐺)((𝑌‘𝑘) ↑ (𝑉‘𝑘))))) |
82 | 77, 81 | syl5ibrcom 250 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → ((𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ∧ 𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙))) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
83 | 82 | expd 419 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → (𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) → (𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))) |
84 | 83 | rexlimdva 3195 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → (∃𝑘 ∈ 𝑆 𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) → (𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))) |
85 | 84 | com23 86 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → (𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙)) → (∃𝑘 ∈ 𝑆 𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))) |
86 | 85 | rexlimdva 3195 |
. . . . . . . 8
⊢ (𝜑 → (∃𝑙 ∈ 𝑆 𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙)) → (∃𝑘 ∈ 𝑆 𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))) |
87 | 12, 86 | syl5bi 245 |
. . . . . . 7
⊢ (𝜑 → (∃𝑘 ∈ 𝑆 𝑦 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) → (∃𝑘 ∈ 𝑆 𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))) |
88 | 7, 87 | sylbid 243 |
. . . . . 6
⊢ (𝜑 → (𝑦 ∈ ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) → (∃𝑘 ∈ 𝑆 𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))) |
89 | 88 | com23 86 |
. . . . 5
⊢ (𝜑 → (∃𝑘 ∈ 𝑆 𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) → (𝑦 ∈ ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))) |
90 | 4, 89 | sylbid 243 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) → (𝑦 ∈ ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))) |
91 | 90 | imp32 422 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ∧ 𝑦 ∈ ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)) |
92 | 91 | ralrimivva 3104 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))∀𝑦 ∈ ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)) |
93 | 29 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → 𝐺 ∈ Mnd) |
94 | 31 | sseld 3877 |
. . . . . . . 8
⊢ (𝜑 → (𝑘 ∈ 𝑆 → 𝑘 ∈ 𝐼)) |
95 | 94 | imdistani 572 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝜑 ∧ 𝑘 ∈ 𝐼)) |
96 | 95, 56 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑌‘𝑘) ∈
ℕ0) |
97 | 54, 47 | eleqtrdi 2844 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑉‘𝑘) ∈ (Base‘𝐺)) |
98 | | eqid 2739 |
. . . . . . 7
⊢
(Base‘𝐺) =
(Base‘𝐺) |
99 | 98, 18 | mulgnn0cl 18365 |
. . . . . 6
⊢ ((𝐺 ∈ Mnd ∧ (𝑌‘𝑘) ∈ ℕ0 ∧ (𝑉‘𝑘) ∈ (Base‘𝐺)) → ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ∈ (Base‘𝐺)) |
100 | 93, 96, 97, 99 | syl3anc 1372 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ∈ (Base‘𝐺)) |
101 | 100 | fmpttd 6892 |
. . . 4
⊢ (𝜑 → (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))):𝑆⟶(Base‘𝐺)) |
102 | 101 | frnd 6513 |
. . 3
⊢ (𝜑 → ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ⊆ (Base‘𝐺)) |
103 | | eqid 2739 |
. . . 4
⊢
(+g‘𝐺) = (+g‘𝐺) |
104 | | eqid 2739 |
. . . 4
⊢
(Cntz‘𝐺) =
(Cntz‘𝐺) |
105 | 98, 103, 104 | sscntz 18577 |
. . 3
⊢ ((ran
(𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ⊆ (Base‘𝐺) ∧ ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ⊆ (Base‘𝐺)) → (ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ⊆ ((Cntz‘𝐺)‘ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) ↔ ∀𝑥 ∈ ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))∀𝑦 ∈ ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
106 | 102, 102,
105 | syl2anc 587 |
. 2
⊢ (𝜑 → (ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ⊆ ((Cntz‘𝐺)‘ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) ↔ ∀𝑥 ∈ ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))∀𝑦 ∈ ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
107 | 92, 106 | mpbird 260 |
1
⊢ (𝜑 → ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ⊆ ((Cntz‘𝐺)‘ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))) |