Step | Hyp | Ref
| Expression |
1 | | vex 3449 |
. . . . . 6
⊢ 𝑥 ∈ V |
2 | | eqid 2736 |
. . . . . . 7
⊢ (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) = (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) |
3 | 2 | elrnmpt 5911 |
. . . . . 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 3449 |
. . . . . . . 8
⊢ 𝑦 ∈ V |
6 | 2 | elrnmpt 5911 |
. . . . . . . 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 6842 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑙 → (𝑌‘𝑘) = (𝑌‘𝑙)) |
9 | | fveq2 6842 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑙 → (𝑉‘𝑘) = (𝑉‘𝑙)) |
10 | 8, 9 | oveq12d 7375 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑙 → ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) = ((𝑌‘𝑙) ↑ (𝑉‘𝑙))) |
11 | 10 | eqeq2d 2747 |
. . . . . . . . 9
⊢ (𝑘 = 𝑙 → (𝑦 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ↔ 𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙)))) |
12 | 11 | cbvrexvw 3226 |
. . . . . . . 8
⊢
(∃𝑘 ∈
𝑆 𝑦 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ↔ ∃𝑙 ∈ 𝑆 𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙))) |
13 | | eqid 2736 |
. . . . . . . . . . . . . 14
⊢
(Base‘𝑃) =
(Base‘𝑃) |
14 | | mplcoe2.g |
. . . . . . . . . . . . . . . 16
⊢ 𝐺 = (mulGrp‘𝑃) |
15 | | eqid 2736 |
. . . . . . . . . . . . . . . 16
⊢
(.r‘𝑃) = (.r‘𝑃) |
16 | 14, 15 | mgpplusg 19900 |
. . . . . . . . . . . . . . 15
⊢
(.r‘𝑃) = (+g‘𝐺) |
17 | 16 | eqcomi 2745 |
. . . . . . . . . . . . . 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 21424 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring) |
23 | 19, 20, 22 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑃 ∈ Ring) |
24 | | ringsrg 20013 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑃 ∈ Ring → 𝑃 ∈ SRing) |
25 | 23, 24 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑃 ∈ SRing) |
26 | 25 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → 𝑃 ∈ SRing) |
27 | 26 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → 𝑃 ∈ SRing) |
28 | 14, 13 | mgpbas 19902 |
. . . . . . . . . . . . . . . 16
⊢
(Base‘𝑃) =
(Base‘𝐺) |
29 | 14 | ringmgp 19970 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑃 ∈ Ring → 𝐺 ∈ Mnd) |
30 | 23, 29 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐺 ∈ Mnd) |
31 | 30 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → 𝐺 ∈ Mnd) |
32 | | mplcoe5.s |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑆 ⊆ 𝐼) |
33 | 32 | sseld 3943 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑙 ∈ 𝑆 → 𝑙 ∈ 𝐼)) |
34 | 33 | imdistani 569 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → (𝜑 ∧ 𝑙 ∈ 𝐼)) |
35 | | mplcoe5.y |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑌 ∈ 𝐷) |
36 | | mplcoe1.d |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝐷 = {𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈
Fin} |
37 | 36 | psrbag 21319 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐼 ∈ 𝑊 → (𝑌 ∈ 𝐷 ↔ (𝑌:𝐼⟶ℕ0 ∧ (◡𝑌 “ ℕ) ∈
Fin))) |
38 | 19, 37 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑌 ∈ 𝐷 ↔ (𝑌:𝐼⟶ℕ0 ∧ (◡𝑌 “ ℕ) ∈
Fin))) |
39 | 35, 38 | mpbid 231 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑌:𝐼⟶ℕ0 ∧ (◡𝑌 “ ℕ) ∈
Fin)) |
40 | 39 | simpld 495 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑌:𝐼⟶ℕ0) |
41 | 40 | ffvelcdmda 7035 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐼) → (𝑌‘𝑙) ∈
ℕ0) |
42 | 34, 41 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → (𝑌‘𝑙) ∈
ℕ0) |
43 | | mplcoe2.v |
. . . . . . . . . . . . . . . . 17
⊢ 𝑉 = (𝐼 mVar 𝑅) |
44 | 19 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → 𝐼 ∈ 𝑊) |
45 | 20 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → 𝑅 ∈ Ring) |
46 | 32 | sselda 3944 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → 𝑙 ∈ 𝐼) |
47 | 21, 43, 13, 44, 45, 46 | mvrcl 21421 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → (𝑉‘𝑙) ∈ (Base‘𝑃)) |
48 | 28, 18, 31, 42, 47 | mulgnn0cld 18897 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → ((𝑌‘𝑙) ↑ (𝑉‘𝑙)) ∈ (Base‘𝑃)) |
49 | 48 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → ((𝑌‘𝑙) ↑ (𝑉‘𝑙)) ∈ (Base‘𝑃)) |
50 | 19 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → 𝐼 ∈ 𝑊) |
51 | 20 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → 𝑅 ∈ Ring) |
52 | 32 | sselda 3944 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ 𝐼) |
53 | 21, 43, 13, 50, 51, 52 | mvrcl 21421 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑉‘𝑘) ∈ (Base‘𝑃)) |
54 | 53 | adantlr 713 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → (𝑉‘𝑘) ∈ (Base‘𝑃)) |
55 | 40 | ffvelcdmda 7035 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑌‘𝑘) ∈
ℕ0) |
56 | 52, 55 | syldan 591 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑌‘𝑘) ∈
ℕ0) |
57 | 56 | adantlr 713 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → (𝑌‘𝑘) ∈
ℕ0) |
58 | 47 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → (𝑉‘𝑙) ∈ (Base‘𝑃)) |
59 | 42 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → (𝑌‘𝑙) ∈
ℕ0) |
60 | | mplcoe5.c |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦))) |
61 | | fveq2 6842 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑙 → (𝑉‘𝑥) = (𝑉‘𝑙)) |
62 | 61 | oveq2d 7373 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑙 → ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑙))) |
63 | 61 | oveq1d 7372 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑙 → ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦)) = ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑦))) |
64 | 62, 63 | eqeq12d 2752 |
. . . . . . . . . . . . . . . . . . . 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)))) |
65 | | fveq2 6842 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = 𝑘 → (𝑉‘𝑦) = (𝑉‘𝑘)) |
66 | 65 | oveq1d 7372 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 𝑘 → ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑙)) = ((𝑉‘𝑘)(+g‘𝐺)(𝑉‘𝑙))) |
67 | 65 | oveq2d 7373 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 𝑘 → ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑦)) = ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑘))) |
68 | 66, 67 | eqeq12d 2752 |
. . . . . . . . . . . . . . . . . . . 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)))) |
69 | 64, 68 | rspc2v 3590 |
. . . . . . . . . . . . . . . . . . 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)))) |
70 | 46, 52 | anim12dan 619 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑙 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) → (𝑙 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼)) |
71 | 69, 70 | 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)))) |
72 | 71 | expd 416 |
. . . . . . . . . . . . . . . . 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))))) |
73 | 60, 72 | mpcom 38 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑙 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → ((𝑉‘𝑘)(+g‘𝐺)(𝑉‘𝑙)) = ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑘)))) |
74 | 73 | impl 456 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → ((𝑉‘𝑘)(+g‘𝐺)(𝑉‘𝑙)) = ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑘))) |
75 | 13, 17, 14, 18, 27, 54, 58, 59, 74 | srgpcomp 19949 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → (((𝑌‘𝑙) ↑ (𝑉‘𝑙))(+g‘𝐺)(𝑉‘𝑘)) = ((𝑉‘𝑘)(+g‘𝐺)((𝑌‘𝑙) ↑ (𝑉‘𝑙)))) |
76 | 13, 17, 14, 18, 27, 49, 54, 57, 75 | srgpcomp 19949 |
. . . . . . . . . . . . 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)))) |
77 | | oveq12 7366 |
. . . . . . . . . . . . . 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)))) |
78 | | oveq12 7366 |
. . . . . . . . . . . . . . 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)))) |
79 | 78 | ancoms 459 |
. . . . . . . . . . . . . 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)))) |
80 | 77, 79 | eqeq12d 2752 |
. . . . . . . . . . . . 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))))) |
81 | 76, 80 | syl5ibrcom 246 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → ((𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ∧ 𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙))) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
82 | 81 | expd 416 |
. . . . . . . . . . 11
⊢ (((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 | rexlimdva 3152 |
. . . . . . . . . 10
⊢ ((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 | 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)))) |
85 | 84 | rexlimdva 3152 |
. . . . . . . 8
⊢ (𝜑 → (∃𝑙 ∈ 𝑆 𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙)) → (∃𝑘 ∈ 𝑆 𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))) |
86 | 12, 85 | biimtrid 241 |
. . . . . . 7
⊢ (𝜑 → (∃𝑘 ∈ 𝑆 𝑦 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) → (∃𝑘 ∈ 𝑆 𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))) |
87 | 7, 86 | sylbid 239 |
. . . . . 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)))) |
88 | 87 | 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)))) |
89 | 4, 88 | sylbid 239 |
. . . 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)))) |
90 | 89 | imp32 419 |
. . 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)) |
91 | 90 | ralrimivva 3197 |
. 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)) |
92 | | eqid 2736 |
. . . . . 6
⊢
(Base‘𝐺) =
(Base‘𝐺) |
93 | 30 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → 𝐺 ∈ Mnd) |
94 | 32 | sseld 3943 |
. . . . . . . 8
⊢ (𝜑 → (𝑘 ∈ 𝑆 → 𝑘 ∈ 𝐼)) |
95 | 94 | imdistani 569 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝜑 ∧ 𝑘 ∈ 𝐼)) |
96 | 95, 55 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑌‘𝑘) ∈
ℕ0) |
97 | 53, 28 | eleqtrdi 2848 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑉‘𝑘) ∈ (Base‘𝐺)) |
98 | 92, 18, 93, 96, 97 | mulgnn0cld 18897 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ∈ (Base‘𝐺)) |
99 | 98 | fmpttd 7063 |
. . . 4
⊢ (𝜑 → (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))):𝑆⟶(Base‘𝐺)) |
100 | 99 | frnd 6676 |
. . 3
⊢ (𝜑 → ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ⊆ (Base‘𝐺)) |
101 | | eqid 2736 |
. . . 4
⊢
(+g‘𝐺) = (+g‘𝐺) |
102 | | eqid 2736 |
. . . 4
⊢
(Cntz‘𝐺) =
(Cntz‘𝐺) |
103 | 92, 101, 102 | sscntz 19106 |
. . 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))) |
104 | 100, 100,
103 | syl2anc 584 |
. 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))) |
105 | 91, 104 | mpbird 256 |
1
⊢ (𝜑 → ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ⊆ ((Cntz‘𝐺)‘ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))) |