| Step | Hyp | Ref
| Expression |
| 1 | | wrdf 14557 |
. . . . . . . 8
⊢ (𝑠 ∈ Word 𝐶 → 𝑠:(0..^(♯‘𝑠))⟶𝐶) |
| 2 | 1 | ad2antlr 727 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝑠:(0..^(♯‘𝑠))⟶𝐶) |
| 3 | 2 | fdmd 6746 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → dom 𝑠 = (0..^(♯‘𝑠))) |
| 4 | | fzofi 14015 |
. . . . . 6
⊢
(0..^(♯‘𝑠)) ∈ Fin |
| 5 | 3, 4 | eqeltrdi 2849 |
. . . . 5
⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → dom 𝑠 ∈ Fin) |
| 6 | 2 | ffdmd 6766 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝑠:dom 𝑠⟶𝐶) |
| 7 | 6 | ffvelcdmda 7104 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠‘𝑘) ∈ 𝐶) |
| 8 | | oveq2 7439 |
. . . . . . . . . . . 12
⊢ (𝑟 = (𝑠‘𝑘) → (𝐺 ↾s 𝑟) = (𝐺 ↾s (𝑠‘𝑘))) |
| 9 | 8 | eleq1d 2826 |
. . . . . . . . . . 11
⊢ (𝑟 = (𝑠‘𝑘) → ((𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp ) ↔
(𝐺 ↾s
(𝑠‘𝑘)) ∈ (CycGrp ∩ ran pGrp
))) |
| 10 | | ablfac.c |
. . . . . . . . . . 11
⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp
)} |
| 11 | 9, 10 | elrab2 3695 |
. . . . . . . . . 10
⊢ ((𝑠‘𝑘) ∈ 𝐶 ↔ ((𝑠‘𝑘) ∈ (SubGrp‘𝐺) ∧ (𝐺 ↾s (𝑠‘𝑘)) ∈ (CycGrp ∩ ran pGrp
))) |
| 12 | 11 | simplbi 497 |
. . . . . . . . 9
⊢ ((𝑠‘𝑘) ∈ 𝐶 → (𝑠‘𝑘) ∈ (SubGrp‘𝐺)) |
| 13 | 7, 12 | syl 17 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠‘𝑘) ∈ (SubGrp‘𝐺)) |
| 14 | | ablfac.b |
. . . . . . . . 9
⊢ 𝐵 = (Base‘𝐺) |
| 15 | 14 | subgss 19145 |
. . . . . . . 8
⊢ ((𝑠‘𝑘) ∈ (SubGrp‘𝐺) → (𝑠‘𝑘) ⊆ 𝐵) |
| 16 | 13, 15 | syl 17 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠‘𝑘) ⊆ 𝐵) |
| 17 | 11 | simprbi 496 |
. . . . . . . . . . . 12
⊢ ((𝑠‘𝑘) ∈ 𝐶 → (𝐺 ↾s (𝑠‘𝑘)) ∈ (CycGrp ∩ ran pGrp
)) |
| 18 | 7, 17 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝐺 ↾s (𝑠‘𝑘)) ∈ (CycGrp ∩ ran pGrp
)) |
| 19 | 18 | elin1d 4204 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝐺 ↾s (𝑠‘𝑘)) ∈ CycGrp) |
| 20 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(Base‘(𝐺
↾s (𝑠‘𝑘))) = (Base‘(𝐺 ↾s (𝑠‘𝑘))) |
| 21 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(.g‘(𝐺 ↾s (𝑠‘𝑘))) = (.g‘(𝐺 ↾s (𝑠‘𝑘))) |
| 22 | 20, 21 | iscyg 19897 |
. . . . . . . . . . 11
⊢ ((𝐺 ↾s (𝑠‘𝑘)) ∈ CycGrp ↔ ((𝐺 ↾s (𝑠‘𝑘)) ∈ Grp ∧ ∃𝑥 ∈ (Base‘(𝐺 ↾s (𝑠‘𝑘)))ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺 ↾s (𝑠‘𝑘)))𝑥)) = (Base‘(𝐺 ↾s (𝑠‘𝑘))))) |
| 23 | 22 | simprbi 496 |
. . . . . . . . . 10
⊢ ((𝐺 ↾s (𝑠‘𝑘)) ∈ CycGrp → ∃𝑥 ∈ (Base‘(𝐺 ↾s (𝑠‘𝑘)))ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺 ↾s (𝑠‘𝑘)))𝑥)) = (Base‘(𝐺 ↾s (𝑠‘𝑘)))) |
| 24 | 19, 23 | syl 17 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → ∃𝑥 ∈ (Base‘(𝐺 ↾s (𝑠‘𝑘)))ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺 ↾s (𝑠‘𝑘)))𝑥)) = (Base‘(𝐺 ↾s (𝑠‘𝑘)))) |
| 25 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (𝐺 ↾s (𝑠‘𝑘)) = (𝐺 ↾s (𝑠‘𝑘)) |
| 26 | 25 | subgbas 19148 |
. . . . . . . . . 10
⊢ ((𝑠‘𝑘) ∈ (SubGrp‘𝐺) → (𝑠‘𝑘) = (Base‘(𝐺 ↾s (𝑠‘𝑘)))) |
| 27 | 13, 26 | syl 17 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠‘𝑘) = (Base‘(𝐺 ↾s (𝑠‘𝑘)))) |
| 28 | 24, 27 | rexeqtrrdv 3331 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → ∃𝑥 ∈ (𝑠‘𝑘)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺 ↾s (𝑠‘𝑘)))𝑥)) = (Base‘(𝐺 ↾s (𝑠‘𝑘)))) |
| 29 | 13 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢
((((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠‘𝑘)) ∧ 𝑛 ∈ ℤ) → (𝑠‘𝑘) ∈ (SubGrp‘𝐺)) |
| 30 | | simpr 484 |
. . . . . . . . . . . . 13
⊢
((((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠‘𝑘)) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ) |
| 31 | | simplr 769 |
. . . . . . . . . . . . 13
⊢
((((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠‘𝑘)) ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ (𝑠‘𝑘)) |
| 32 | | ablfac2.m |
. . . . . . . . . . . . . 14
⊢ · =
(.g‘𝐺) |
| 33 | 32, 25, 21 | subgmulg 19158 |
. . . . . . . . . . . . 13
⊢ (((𝑠‘𝑘) ∈ (SubGrp‘𝐺) ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ (𝑠‘𝑘)) → (𝑛 · 𝑥) = (𝑛(.g‘(𝐺 ↾s (𝑠‘𝑘)))𝑥)) |
| 34 | 29, 30, 31, 33 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢
((((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠‘𝑘)) ∧ 𝑛 ∈ ℤ) → (𝑛 · 𝑥) = (𝑛(.g‘(𝐺 ↾s (𝑠‘𝑘)))𝑥)) |
| 35 | 34 | mpteq2dva 5242 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠‘𝑘)) → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺 ↾s (𝑠‘𝑘)))𝑥))) |
| 36 | 35 | rneqd 5949 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠‘𝑘)) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺 ↾s (𝑠‘𝑘)))𝑥))) |
| 37 | 27 | adantr 480 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠‘𝑘)) → (𝑠‘𝑘) = (Base‘(𝐺 ↾s (𝑠‘𝑘)))) |
| 38 | 36, 37 | eqeq12d 2753 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠‘𝑘)) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠‘𝑘) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺 ↾s (𝑠‘𝑘)))𝑥)) = (Base‘(𝐺 ↾s (𝑠‘𝑘))))) |
| 39 | 38 | rexbidva 3177 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (∃𝑥 ∈ (𝑠‘𝑘)ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠‘𝑘) ↔ ∃𝑥 ∈ (𝑠‘𝑘)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺 ↾s (𝑠‘𝑘)))𝑥)) = (Base‘(𝐺 ↾s (𝑠‘𝑘))))) |
| 40 | 28, 39 | mpbird 257 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → ∃𝑥 ∈ (𝑠‘𝑘)ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠‘𝑘)) |
| 41 | | ssrexv 4053 |
. . . . . . 7
⊢ ((𝑠‘𝑘) ⊆ 𝐵 → (∃𝑥 ∈ (𝑠‘𝑘)ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠‘𝑘) → ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠‘𝑘))) |
| 42 | 16, 40, 41 | sylc 65 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠‘𝑘)) |
| 43 | 42 | ralrimiva 3146 |
. . . . 5
⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → ∀𝑘 ∈ dom 𝑠∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠‘𝑘)) |
| 44 | | oveq2 7439 |
. . . . . . . . 9
⊢ (𝑥 = (𝑤‘𝑘) → (𝑛 · 𝑥) = (𝑛 · (𝑤‘𝑘))) |
| 45 | 44 | mpteq2dv 5244 |
. . . . . . . 8
⊢ (𝑥 = (𝑤‘𝑘) → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘)))) |
| 46 | 45 | rneqd 5949 |
. . . . . . 7
⊢ (𝑥 = (𝑤‘𝑘) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘)))) |
| 47 | 46 | eqeq1d 2739 |
. . . . . 6
⊢ (𝑥 = (𝑤‘𝑘) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠‘𝑘) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) |
| 48 | 47 | ac6sfi 9320 |
. . . . 5
⊢ ((dom
𝑠 ∈ Fin ∧
∀𝑘 ∈ dom 𝑠∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠‘𝑘)) → ∃𝑤(𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) |
| 49 | 5, 43, 48 | syl2anc 584 |
. . . 4
⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → ∃𝑤(𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) |
| 50 | | simprl 771 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → 𝑤:dom 𝑠⟶𝐵) |
| 51 | 3 | adantr 480 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → dom 𝑠 = (0..^(♯‘𝑠))) |
| 52 | 51 | feq2d 6722 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → (𝑤:dom 𝑠⟶𝐵 ↔ 𝑤:(0..^(♯‘𝑠))⟶𝐵)) |
| 53 | 50, 52 | mpbid 232 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → 𝑤:(0..^(♯‘𝑠))⟶𝐵) |
| 54 | | iswrdi 14556 |
. . . . . . . 8
⊢ (𝑤:(0..^(♯‘𝑠))⟶𝐵 → 𝑤 ∈ Word 𝐵) |
| 55 | 53, 54 | syl 17 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → 𝑤 ∈ Word 𝐵) |
| 56 | 50 | fdmd 6746 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → dom 𝑤 = dom 𝑠) |
| 57 | 56 | eleq2d 2827 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → (𝑗 ∈ dom 𝑤 ↔ 𝑗 ∈ dom 𝑠)) |
| 58 | 57 | biimpa 476 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) ∧ 𝑗 ∈ dom 𝑤) → 𝑗 ∈ dom 𝑠) |
| 59 | | simprr 773 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘)) |
| 60 | | simpl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 = 𝑗 ∧ 𝑛 ∈ ℤ) → 𝑘 = 𝑗) |
| 61 | 60 | fveq2d 6910 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘 = 𝑗 ∧ 𝑛 ∈ ℤ) → (𝑤‘𝑘) = (𝑤‘𝑗)) |
| 62 | 61 | oveq2d 7447 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 = 𝑗 ∧ 𝑛 ∈ ℤ) → (𝑛 · (𝑤‘𝑘)) = (𝑛 · (𝑤‘𝑗))) |
| 63 | 62 | mpteq2dva 5242 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑗 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗)))) |
| 64 | 63 | rneqd 5949 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑗 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗)))) |
| 65 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑗 → (𝑠‘𝑘) = (𝑠‘𝑗)) |
| 66 | 64, 65 | eqeq12d 2753 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑗 → (ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))) = (𝑠‘𝑗))) |
| 67 | 66 | rspccva 3621 |
. . . . . . . . . . . 12
⊢
((∀𝑘 ∈
dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘) ∧ 𝑗 ∈ dom 𝑠) → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))) = (𝑠‘𝑗)) |
| 68 | 59, 67 | sylan 580 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) ∧ 𝑗 ∈ dom 𝑠) → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))) = (𝑠‘𝑗)) |
| 69 | 6 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → 𝑠:dom 𝑠⟶𝐶) |
| 70 | 69 | ffvelcdmda 7104 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) ∧ 𝑗 ∈ dom 𝑠) → (𝑠‘𝑗) ∈ 𝐶) |
| 71 | 68, 70 | eqeltrd 2841 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) ∧ 𝑗 ∈ dom 𝑠) → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))) ∈ 𝐶) |
| 72 | 58, 71 | syldan 591 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) ∧ 𝑗 ∈ dom 𝑤) → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))) ∈ 𝐶) |
| 73 | | ablfac2.s |
. . . . . . . . . 10
⊢ 𝑆 = (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘)))) |
| 74 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑗 → (𝑤‘𝑘) = (𝑤‘𝑗)) |
| 75 | 74 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑗 → (𝑛 · (𝑤‘𝑘)) = (𝑛 · (𝑤‘𝑗))) |
| 76 | 75 | mpteq2dv 5244 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑗 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗)))) |
| 77 | 76 | rneqd 5949 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑗 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗)))) |
| 78 | 77 | cbvmptv 5255 |
. . . . . . . . . 10
⊢ (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘)))) = (𝑗 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗)))) |
| 79 | 73, 78 | eqtri 2765 |
. . . . . . . . 9
⊢ 𝑆 = (𝑗 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗)))) |
| 80 | 72, 79 | fmptd 7134 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → 𝑆:dom 𝑤⟶𝐶) |
| 81 | | simprl 771 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝐺dom DProd 𝑠) |
| 82 | 81 | adantr 480 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → 𝐺dom DProd 𝑠) |
| 83 | 59, 56 | raleqtrrdv 3330 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → ∀𝑘 ∈ dom 𝑤ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘)) |
| 84 | | mpteq12 5234 |
. . . . . . . . . . . 12
⊢ ((dom
𝑤 = dom 𝑠 ∧ ∀𝑘 ∈ dom 𝑤ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘)) → (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘)))) = (𝑘 ∈ dom 𝑠 ↦ (𝑠‘𝑘))) |
| 85 | 56, 83, 84 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘)))) = (𝑘 ∈ dom 𝑠 ↦ (𝑠‘𝑘))) |
| 86 | 73, 85 | eqtrid 2789 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → 𝑆 = (𝑘 ∈ dom 𝑠 ↦ (𝑠‘𝑘))) |
| 87 | | dprdf 20026 |
. . . . . . . . . . . 12
⊢ (𝐺dom DProd 𝑠 → 𝑠:dom 𝑠⟶(SubGrp‘𝐺)) |
| 88 | 82, 87 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → 𝑠:dom 𝑠⟶(SubGrp‘𝐺)) |
| 89 | 88 | feqmptd 6977 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → 𝑠 = (𝑘 ∈ dom 𝑠 ↦ (𝑠‘𝑘))) |
| 90 | 86, 89 | eqtr4d 2780 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → 𝑆 = 𝑠) |
| 91 | 82, 90 | breqtrrd 5171 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → 𝐺dom DProd 𝑆) |
| 92 | 90 | oveq2d 7447 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → (𝐺 DProd 𝑆) = (𝐺 DProd 𝑠)) |
| 93 | | simplrr 778 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → (𝐺 DProd 𝑠) = 𝐵) |
| 94 | 92, 93 | eqtrd 2777 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → (𝐺 DProd 𝑆) = 𝐵) |
| 95 | 80, 91, 94 | 3jca 1129 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → (𝑆:dom 𝑤⟶𝐶 ∧ 𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵)) |
| 96 | 55, 95 | jca 511 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘))) → (𝑤 ∈ Word 𝐵 ∧ (𝑆:dom 𝑤⟶𝐶 ∧ 𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵))) |
| 97 | 96 | ex 412 |
. . . . 5
⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → ((𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘)) → (𝑤 ∈ Word 𝐵 ∧ (𝑆:dom 𝑤⟶𝐶 ∧ 𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵)))) |
| 98 | 97 | eximdv 1917 |
. . . 4
⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → (∃𝑤(𝑤:dom 𝑠⟶𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘)) → ∃𝑤(𝑤 ∈ Word 𝐵 ∧ (𝑆:dom 𝑤⟶𝐶 ∧ 𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵)))) |
| 99 | 49, 98 | mpd 15 |
. . 3
⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → ∃𝑤(𝑤 ∈ Word 𝐵 ∧ (𝑆:dom 𝑤⟶𝐶 ∧ 𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵))) |
| 100 | | df-rex 3071 |
. . 3
⊢
(∃𝑤 ∈
Word 𝐵(𝑆:dom 𝑤⟶𝐶 ∧ 𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵) ↔ ∃𝑤(𝑤 ∈ Word 𝐵 ∧ (𝑆:dom 𝑤⟶𝐶 ∧ 𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵))) |
| 101 | 99, 100 | sylibr 234 |
. 2
⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → ∃𝑤 ∈ Word 𝐵(𝑆:dom 𝑤⟶𝐶 ∧ 𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵)) |
| 102 | | ablfac.1 |
. . 3
⊢ (𝜑 → 𝐺 ∈ Abel) |
| 103 | | ablfac.2 |
. . 3
⊢ (𝜑 → 𝐵 ∈ Fin) |
| 104 | 14, 10, 102, 103 | ablfac 20108 |
. 2
⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) |
| 105 | 101, 104 | r19.29a 3162 |
1
⊢ (𝜑 → ∃𝑤 ∈ Word 𝐵(𝑆:dom 𝑤⟶𝐶 ∧ 𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵)) |