| Step | Hyp | Ref
| Expression |
| 1 | | vex 3484 |
. . . . . 6
⊢ 𝑥 ∈ V |
| 2 | | eqid 2737 |
. . . . . . 7
⊢ (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) = (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) |
| 3 | 2 | elrnmpt 5969 |
. . . . . 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 3484 |
. . . . . . . 8
⊢ 𝑦 ∈ V |
| 6 | 2 | elrnmpt 5969 |
. . . . . . . 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 6906 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑙 → (𝑌‘𝑘) = (𝑌‘𝑙)) |
| 9 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑙 → (𝑉‘𝑘) = (𝑉‘𝑙)) |
| 10 | 8, 9 | oveq12d 7449 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑙 → ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) = ((𝑌‘𝑙) ↑ (𝑉‘𝑙))) |
| 11 | 10 | eqeq2d 2748 |
. . . . . . . . 9
⊢ (𝑘 = 𝑙 → (𝑦 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ↔ 𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙)))) |
| 12 | 11 | cbvrexvw 3238 |
. . . . . . . 8
⊢
(∃𝑘 ∈
𝑆 𝑦 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ↔ ∃𝑙 ∈ 𝑆 𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙))) |
| 13 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(Base‘𝑃) =
(Base‘𝑃) |
| 14 | | mplcoe2.g |
. . . . . . . . . . . . . . . 16
⊢ 𝐺 = (mulGrp‘𝑃) |
| 15 | | eqid 2737 |
. . . . . . . . . . . . . . . 16
⊢
(.r‘𝑃) = (.r‘𝑃) |
| 16 | 14, 15 | mgpplusg 20141 |
. . . . . . . . . . . . . . 15
⊢
(.r‘𝑃) = (+g‘𝐺) |
| 17 | 16 | eqcomi 2746 |
. . . . . . . . . . . . . 14
⊢
(+g‘𝐺) = (.r‘𝑃) |
| 18 | | mplcoe2.m |
. . . . . . . . . . . . . 14
⊢ ↑ =
(.g‘𝐺) |
| 19 | | mplcoe1.p |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑃 = (𝐼 mPoly 𝑅) |
| 20 | | mplcoe1.i |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐼 ∈ 𝑊) |
| 21 | | mplcoe5.r |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑅 ∈ Ring) |
| 22 | 19, 20, 21 | mplringd 22043 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑃 ∈ Ring) |
| 23 | | ringsrg 20294 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑃 ∈ Ring → 𝑃 ∈ SRing) |
| 24 | 22, 23 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑃 ∈ SRing) |
| 25 | 24 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → 𝑃 ∈ SRing) |
| 26 | 25 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → 𝑃 ∈ SRing) |
| 27 | 14, 13 | mgpbas 20142 |
. . . . . . . . . . . . . . . 16
⊢
(Base‘𝑃) =
(Base‘𝐺) |
| 28 | 14 | ringmgp 20236 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑃 ∈ Ring → 𝐺 ∈ Mnd) |
| 29 | 22, 28 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐺 ∈ Mnd) |
| 30 | 29 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → 𝐺 ∈ Mnd) |
| 31 | | mplcoe5.s |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑆 ⊆ 𝐼) |
| 32 | 31 | sseld 3982 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑙 ∈ 𝑆 → 𝑙 ∈ 𝐼)) |
| 33 | 32 | imdistani 568 |
. . . . . . . . . . . . . . . . 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 21937 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐼 ∈ 𝑊 → (𝑌 ∈ 𝐷 ↔ (𝑌:𝐼⟶ℕ0 ∧ (◡𝑌 “ ℕ) ∈
Fin))) |
| 37 | 20, 36 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑌 ∈ 𝐷 ↔ (𝑌:𝐼⟶ℕ0 ∧ (◡𝑌 “ ℕ) ∈
Fin))) |
| 38 | 34, 37 | mpbid 232 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑌:𝐼⟶ℕ0 ∧ (◡𝑌 “ ℕ) ∈
Fin)) |
| 39 | 38 | simpld 494 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑌:𝐼⟶ℕ0) |
| 40 | 39 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . . 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 | 20 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → 𝐼 ∈ 𝑊) |
| 44 | 21 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → 𝑅 ∈ Ring) |
| 45 | 31 | sselda 3983 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → 𝑙 ∈ 𝐼) |
| 46 | 19, 42, 13, 43, 44, 45 | mvrcl 22012 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → (𝑉‘𝑙) ∈ (Base‘𝑃)) |
| 47 | 27, 18, 30, 41, 46 | mulgnn0cld 19113 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑆) → ((𝑌‘𝑙) ↑ (𝑉‘𝑙)) ∈ (Base‘𝑃)) |
| 48 | 47 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → ((𝑌‘𝑙) ↑ (𝑉‘𝑙)) ∈ (Base‘𝑃)) |
| 49 | 20 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → 𝐼 ∈ 𝑊) |
| 50 | 21 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → 𝑅 ∈ Ring) |
| 51 | 31 | sselda 3983 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ 𝐼) |
| 52 | 19, 42, 13, 49, 50, 51 | mvrcl 22012 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑉‘𝑘) ∈ (Base‘𝑃)) |
| 53 | 52 | adantlr 715 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → (𝑉‘𝑘) ∈ (Base‘𝑃)) |
| 54 | 39 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑌‘𝑘) ∈
ℕ0) |
| 55 | 51, 54 | syldan 591 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑌‘𝑘) ∈
ℕ0) |
| 56 | 55 | adantlr 715 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → (𝑌‘𝑘) ∈
ℕ0) |
| 57 | 46 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → (𝑉‘𝑙) ∈ (Base‘𝑃)) |
| 58 | 41 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → (𝑌‘𝑙) ∈
ℕ0) |
| 59 | | mplcoe5.c |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦))) |
| 60 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑙 → (𝑉‘𝑥) = (𝑉‘𝑙)) |
| 61 | 60 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑙 → ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑙))) |
| 62 | 60 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑙 → ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦)) = ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑦))) |
| 63 | 61, 62 | eqeq12d 2753 |
. . . . . . . . . . . . . . . . . . . 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)))) |
| 64 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = 𝑘 → (𝑉‘𝑦) = (𝑉‘𝑘)) |
| 65 | 64 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 𝑘 → ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑙)) = ((𝑉‘𝑘)(+g‘𝐺)(𝑉‘𝑙))) |
| 66 | 64 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 𝑘 → ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑦)) = ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑘))) |
| 67 | 65, 66 | eqeq12d 2753 |
. . . . . . . . . . . . . . . . . . . 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)))) |
| 68 | 63, 67 | rspc2v 3633 |
. . . . . . . . . . . . . . . . . . 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)))) |
| 69 | 45, 51 | anim12dan 619 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑙 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) → (𝑙 ∈ 𝐼 ∧ 𝑘 ∈ 𝐼)) |
| 70 | 68, 69 | 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)))) |
| 71 | 70 | expd 415 |
. . . . . . . . . . . . . . . . 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))))) |
| 72 | 59, 71 | mpcom 38 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑙 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → ((𝑉‘𝑘)(+g‘𝐺)(𝑉‘𝑙)) = ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑘)))) |
| 73 | 72 | impl 455 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → ((𝑉‘𝑘)(+g‘𝐺)(𝑉‘𝑙)) = ((𝑉‘𝑙)(+g‘𝐺)(𝑉‘𝑘))) |
| 74 | 13, 17, 14, 18, 26, 53, 57, 58, 73 | srgpcomp 20215 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → (((𝑌‘𝑙) ↑ (𝑉‘𝑙))(+g‘𝐺)(𝑉‘𝑘)) = ((𝑉‘𝑘)(+g‘𝐺)((𝑌‘𝑙) ↑ (𝑉‘𝑙)))) |
| 75 | 13, 17, 14, 18, 26, 48, 53, 56, 74 | srgpcomp 20215 |
. . . . . . . . . . . . 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)))) |
| 76 | | oveq12 7440 |
. . . . . . . . . . . . . 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)))) |
| 77 | | oveq12 7440 |
. . . . . . . . . . . . . . 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)))) |
| 78 | 77 | ancoms 458 |
. . . . . . . . . . . . . 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)))) |
| 79 | 76, 78 | eqeq12d 2753 |
. . . . . . . . . . . . 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))))) |
| 80 | 75, 79 | syl5ibrcom 247 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑙 ∈ 𝑆) ∧ 𝑘 ∈ 𝑆) → ((𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ∧ 𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙))) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
| 81 | 80 | expd 415 |
. . . . . . . . . . 11
⊢ (((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 | rexlimdva 3155 |
. . . . . . . . . 10
⊢ ((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 | 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)))) |
| 84 | 83 | rexlimdva 3155 |
. . . . . . . 8
⊢ (𝜑 → (∃𝑙 ∈ 𝑆 𝑦 = ((𝑌‘𝑙) ↑ (𝑉‘𝑙)) → (∃𝑘 ∈ 𝑆 𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))) |
| 85 | 12, 84 | biimtrid 242 |
. . . . . . 7
⊢ (𝜑 → (∃𝑘 ∈ 𝑆 𝑦 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) → (∃𝑘 ∈ 𝑆 𝑥 = ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))) |
| 86 | 7, 85 | sylbid 240 |
. . . . . 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)))) |
| 87 | 86 | 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)))) |
| 88 | 4, 87 | sylbid 240 |
. . . 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)))) |
| 89 | 88 | imp32 418 |
. . 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)) |
| 90 | 89 | ralrimivva 3202 |
. 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)) |
| 91 | | eqid 2737 |
. . . . . 6
⊢
(Base‘𝐺) =
(Base‘𝐺) |
| 92 | 29 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → 𝐺 ∈ Mnd) |
| 93 | 31 | sseld 3982 |
. . . . . . . 8
⊢ (𝜑 → (𝑘 ∈ 𝑆 → 𝑘 ∈ 𝐼)) |
| 94 | 93 | imdistani 568 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝜑 ∧ 𝑘 ∈ 𝐼)) |
| 95 | 94, 54 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑌‘𝑘) ∈
ℕ0) |
| 96 | 52, 27 | eleqtrdi 2851 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑉‘𝑘) ∈ (Base‘𝐺)) |
| 97 | 91, 18, 92, 95, 96 | mulgnn0cld 19113 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ∈ (Base‘𝐺)) |
| 98 | 97 | fmpttd 7135 |
. . . 4
⊢ (𝜑 → (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))):𝑆⟶(Base‘𝐺)) |
| 99 | 98 | frnd 6744 |
. . 3
⊢ (𝜑 → ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ⊆ (Base‘𝐺)) |
| 100 | | eqid 2737 |
. . . 4
⊢
(+g‘𝐺) = (+g‘𝐺) |
| 101 | | eqid 2737 |
. . . 4
⊢
(Cntz‘𝐺) =
(Cntz‘𝐺) |
| 102 | 91, 100, 101 | sscntz 19344 |
. . 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))) |
| 103 | 99, 99, 102 | 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))) |
| 104 | 90, 103 | mpbird 257 |
1
⊢ (𝜑 → ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ⊆ ((Cntz‘𝐺)‘ran (𝑘 ∈ 𝑆 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))) |