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Theorem ablfac2 20124
Description: Choose generators for each cyclic group in ablfac 20123. (Contributed by Mario Carneiro, 28-Apr-2016.)
Hypotheses
Ref Expression
ablfac.b 𝐵 = (Base‘𝐺)
ablfac.c 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
ablfac.1 (𝜑𝐺 ∈ Abel)
ablfac.2 (𝜑𝐵 ∈ Fin)
ablfac2.m · = (.g𝐺)
ablfac2.s 𝑆 = (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))))
Assertion
Ref Expression
ablfac2 (𝜑 → ∃𝑤 ∈ Word 𝐵(𝑆:dom 𝑤𝐶𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵))
Distinct variable groups:   𝑆,𝑟   𝑘,𝑛,𝑟,𝑤,𝐵   · ,𝑘,𝑤   𝐶,𝑘,𝑛,𝑤   𝜑,𝑘,𝑛,𝑤   𝑘,𝐺,𝑛,𝑟,𝑤
Allowed substitution hints:   𝜑(𝑟)   𝐶(𝑟)   𝑆(𝑤,𝑘,𝑛)   · (𝑛,𝑟)

Proof of Theorem ablfac2
Dummy variables 𝑠 𝑥 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wrdf 14554 . . . . . . . 8 (𝑠 ∈ Word 𝐶𝑠:(0..^(♯‘𝑠))⟶𝐶)
21ad2antlr 727 . . . . . . 7 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝑠:(0..^(♯‘𝑠))⟶𝐶)
32fdmd 6747 . . . . . 6 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → dom 𝑠 = (0..^(♯‘𝑠)))
4 fzofi 14012 . . . . . 6 (0..^(♯‘𝑠)) ∈ Fin
53, 4eqeltrdi 2847 . . . . 5 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → dom 𝑠 ∈ Fin)
62ffdmd 6767 . . . . . . . . . 10 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝑠:dom 𝑠𝐶)
76ffvelcdmda 7104 . . . . . . . . 9 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠𝑘) ∈ 𝐶)
8 oveq2 7439 . . . . . . . . . . . 12 (𝑟 = (𝑠𝑘) → (𝐺s 𝑟) = (𝐺s (𝑠𝑘)))
98eleq1d 2824 . . . . . . . . . . 11 (𝑟 = (𝑠𝑘) → ((𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp ) ↔ (𝐺s (𝑠𝑘)) ∈ (CycGrp ∩ ran pGrp )))
10 ablfac.c . . . . . . . . . . 11 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
119, 10elrab2 3698 . . . . . . . . . 10 ((𝑠𝑘) ∈ 𝐶 ↔ ((𝑠𝑘) ∈ (SubGrp‘𝐺) ∧ (𝐺s (𝑠𝑘)) ∈ (CycGrp ∩ ran pGrp )))
1211simplbi 497 . . . . . . . . 9 ((𝑠𝑘) ∈ 𝐶 → (𝑠𝑘) ∈ (SubGrp‘𝐺))
137, 12syl 17 . . . . . . . 8 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠𝑘) ∈ (SubGrp‘𝐺))
14 ablfac.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
1514subgss 19158 . . . . . . . 8 ((𝑠𝑘) ∈ (SubGrp‘𝐺) → (𝑠𝑘) ⊆ 𝐵)
1613, 15syl 17 . . . . . . 7 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠𝑘) ⊆ 𝐵)
1711simprbi 496 . . . . . . . . . . . 12 ((𝑠𝑘) ∈ 𝐶 → (𝐺s (𝑠𝑘)) ∈ (CycGrp ∩ ran pGrp ))
187, 17syl 17 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝐺s (𝑠𝑘)) ∈ (CycGrp ∩ ran pGrp ))
1918elin1d 4214 . . . . . . . . . 10 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝐺s (𝑠𝑘)) ∈ CycGrp)
20 eqid 2735 . . . . . . . . . . . 12 (Base‘(𝐺s (𝑠𝑘))) = (Base‘(𝐺s (𝑠𝑘)))
21 eqid 2735 . . . . . . . . . . . 12 (.g‘(𝐺s (𝑠𝑘))) = (.g‘(𝐺s (𝑠𝑘)))
2220, 21iscyg 19912 . . . . . . . . . . 11 ((𝐺s (𝑠𝑘)) ∈ CycGrp ↔ ((𝐺s (𝑠𝑘)) ∈ Grp ∧ ∃𝑥 ∈ (Base‘(𝐺s (𝑠𝑘)))ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺s (𝑠𝑘)))𝑥)) = (Base‘(𝐺s (𝑠𝑘)))))
2322simprbi 496 . . . . . . . . . 10 ((𝐺s (𝑠𝑘)) ∈ CycGrp → ∃𝑥 ∈ (Base‘(𝐺s (𝑠𝑘)))ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺s (𝑠𝑘)))𝑥)) = (Base‘(𝐺s (𝑠𝑘))))
2419, 23syl 17 . . . . . . . . 9 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → ∃𝑥 ∈ (Base‘(𝐺s (𝑠𝑘)))ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺s (𝑠𝑘)))𝑥)) = (Base‘(𝐺s (𝑠𝑘))))
25 eqid 2735 . . . . . . . . . . 11 (𝐺s (𝑠𝑘)) = (𝐺s (𝑠𝑘))
2625subgbas 19161 . . . . . . . . . 10 ((𝑠𝑘) ∈ (SubGrp‘𝐺) → (𝑠𝑘) = (Base‘(𝐺s (𝑠𝑘))))
2713, 26syl 17 . . . . . . . . 9 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠𝑘) = (Base‘(𝐺s (𝑠𝑘))))
2824, 27rexeqtrrdv 3329 . . . . . . . 8 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → ∃𝑥 ∈ (𝑠𝑘)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺s (𝑠𝑘)))𝑥)) = (Base‘(𝐺s (𝑠𝑘))))
2913ad2antrr 726 . . . . . . . . . . . . 13 ((((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠𝑘)) ∧ 𝑛 ∈ ℤ) → (𝑠𝑘) ∈ (SubGrp‘𝐺))
30 simpr 484 . . . . . . . . . . . . 13 ((((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠𝑘)) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ)
31 simplr 769 . . . . . . . . . . . . 13 ((((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠𝑘)) ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ (𝑠𝑘))
32 ablfac2.m . . . . . . . . . . . . . 14 · = (.g𝐺)
3332, 25, 21subgmulg 19171 . . . . . . . . . . . . 13 (((𝑠𝑘) ∈ (SubGrp‘𝐺) ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ (𝑠𝑘)) → (𝑛 · 𝑥) = (𝑛(.g‘(𝐺s (𝑠𝑘)))𝑥))
3429, 30, 31, 33syl3anc 1370 . . . . . . . . . . . 12 ((((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠𝑘)) ∧ 𝑛 ∈ ℤ) → (𝑛 · 𝑥) = (𝑛(.g‘(𝐺s (𝑠𝑘)))𝑥))
3534mpteq2dva 5248 . . . . . . . . . . 11 (((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠𝑘)) → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺s (𝑠𝑘)))𝑥)))
3635rneqd 5952 . . . . . . . . . 10 (((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠𝑘)) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺s (𝑠𝑘)))𝑥)))
3727adantr 480 . . . . . . . . . 10 (((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠𝑘)) → (𝑠𝑘) = (Base‘(𝐺s (𝑠𝑘))))
3836, 37eqeq12d 2751 . . . . . . . . 9 (((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠𝑘)) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺s (𝑠𝑘)))𝑥)) = (Base‘(𝐺s (𝑠𝑘)))))
3938rexbidva 3175 . . . . . . . 8 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (∃𝑥 ∈ (𝑠𝑘)ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘) ↔ ∃𝑥 ∈ (𝑠𝑘)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺s (𝑠𝑘)))𝑥)) = (Base‘(𝐺s (𝑠𝑘)))))
4028, 39mpbird 257 . . . . . . 7 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → ∃𝑥 ∈ (𝑠𝑘)ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘))
41 ssrexv 4065 . . . . . . 7 ((𝑠𝑘) ⊆ 𝐵 → (∃𝑥 ∈ (𝑠𝑘)ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘) → ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘)))
4216, 40, 41sylc 65 . . . . . 6 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘))
4342ralrimiva 3144 . . . . 5 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → ∀𝑘 ∈ dom 𝑠𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘))
44 oveq2 7439 . . . . . . . . 9 (𝑥 = (𝑤𝑘) → (𝑛 · 𝑥) = (𝑛 · (𝑤𝑘)))
4544mpteq2dv 5250 . . . . . . . 8 (𝑥 = (𝑤𝑘) → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))))
4645rneqd 5952 . . . . . . 7 (𝑥 = (𝑤𝑘) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))))
4746eqeq1d 2737 . . . . . 6 (𝑥 = (𝑤𝑘) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘)))
4847ac6sfi 9318 . . . . 5 ((dom 𝑠 ∈ Fin ∧ ∀𝑘 ∈ dom 𝑠𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘)) → ∃𝑤(𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘)))
495, 43, 48syl2anc 584 . . . 4 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → ∃𝑤(𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘)))
50 simprl 771 . . . . . . . . 9 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → 𝑤:dom 𝑠𝐵)
513adantr 480 . . . . . . . . . 10 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → dom 𝑠 = (0..^(♯‘𝑠)))
5251feq2d 6723 . . . . . . . . 9 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → (𝑤:dom 𝑠𝐵𝑤:(0..^(♯‘𝑠))⟶𝐵))
5350, 52mpbid 232 . . . . . . . 8 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → 𝑤:(0..^(♯‘𝑠))⟶𝐵)
54 iswrdi 14553 . . . . . . . 8 (𝑤:(0..^(♯‘𝑠))⟶𝐵𝑤 ∈ Word 𝐵)
5553, 54syl 17 . . . . . . 7 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → 𝑤 ∈ Word 𝐵)
5650fdmd 6747 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → dom 𝑤 = dom 𝑠)
5756eleq2d 2825 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → (𝑗 ∈ dom 𝑤𝑗 ∈ dom 𝑠))
5857biimpa 476 . . . . . . . . . 10 (((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) ∧ 𝑗 ∈ dom 𝑤) → 𝑗 ∈ dom 𝑠)
59 simprr 773 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))
60 simpl 482 . . . . . . . . . . . . . . . . . 18 ((𝑘 = 𝑗𝑛 ∈ ℤ) → 𝑘 = 𝑗)
6160fveq2d 6911 . . . . . . . . . . . . . . . . 17 ((𝑘 = 𝑗𝑛 ∈ ℤ) → (𝑤𝑘) = (𝑤𝑗))
6261oveq2d 7447 . . . . . . . . . . . . . . . 16 ((𝑘 = 𝑗𝑛 ∈ ℤ) → (𝑛 · (𝑤𝑘)) = (𝑛 · (𝑤𝑗)))
6362mpteq2dva 5248 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
6463rneqd 5952 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
65 fveq2 6907 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝑠𝑘) = (𝑠𝑗))
6664, 65eqeq12d 2751 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → (ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))) = (𝑠𝑗)))
6766rspccva 3621 . . . . . . . . . . . 12 ((∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘) ∧ 𝑗 ∈ dom 𝑠) → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))) = (𝑠𝑗))
6859, 67sylan 580 . . . . . . . . . . 11 (((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) ∧ 𝑗 ∈ dom 𝑠) → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))) = (𝑠𝑗))
696adantr 480 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → 𝑠:dom 𝑠𝐶)
7069ffvelcdmda 7104 . . . . . . . . . . 11 (((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) ∧ 𝑗 ∈ dom 𝑠) → (𝑠𝑗) ∈ 𝐶)
7168, 70eqeltrd 2839 . . . . . . . . . 10 (((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) ∧ 𝑗 ∈ dom 𝑠) → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))) ∈ 𝐶)
7258, 71syldan 591 . . . . . . . . 9 (((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) ∧ 𝑗 ∈ dom 𝑤) → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))) ∈ 𝐶)
73 ablfac2.s . . . . . . . . . 10 𝑆 = (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))))
74 fveq2 6907 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝑤𝑘) = (𝑤𝑗))
7574oveq2d 7447 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → (𝑛 · (𝑤𝑘)) = (𝑛 · (𝑤𝑗)))
7675mpteq2dv 5250 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
7776rneqd 5952 . . . . . . . . . . 11 (𝑘 = 𝑗 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
7877cbvmptv 5261 . . . . . . . . . 10 (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘)))) = (𝑗 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
7973, 78eqtri 2763 . . . . . . . . 9 𝑆 = (𝑗 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
8072, 79fmptd 7134 . . . . . . . 8 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → 𝑆:dom 𝑤𝐶)
81 simprl 771 . . . . . . . . . 10 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝐺dom DProd 𝑠)
8281adantr 480 . . . . . . . . 9 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → 𝐺dom DProd 𝑠)
8359, 56raleqtrrdv 3328 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → ∀𝑘 ∈ dom 𝑤ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))
84 mpteq12 5240 . . . . . . . . . . . 12 ((dom 𝑤 = dom 𝑠 ∧ ∀𝑘 ∈ dom 𝑤ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘)) → (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘)))) = (𝑘 ∈ dom 𝑠 ↦ (𝑠𝑘)))
8556, 83, 84syl2anc 584 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘)))) = (𝑘 ∈ dom 𝑠 ↦ (𝑠𝑘)))
8673, 85eqtrid 2787 . . . . . . . . . 10 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → 𝑆 = (𝑘 ∈ dom 𝑠 ↦ (𝑠𝑘)))
87 dprdf 20041 . . . . . . . . . . . 12 (𝐺dom DProd 𝑠𝑠:dom 𝑠⟶(SubGrp‘𝐺))
8882, 87syl 17 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → 𝑠:dom 𝑠⟶(SubGrp‘𝐺))
8988feqmptd 6977 . . . . . . . . . 10 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → 𝑠 = (𝑘 ∈ dom 𝑠 ↦ (𝑠𝑘)))
9086, 89eqtr4d 2778 . . . . . . . . 9 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → 𝑆 = 𝑠)
9182, 90breqtrrd 5176 . . . . . . . 8 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → 𝐺dom DProd 𝑆)
9290oveq2d 7447 . . . . . . . . 9 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → (𝐺 DProd 𝑆) = (𝐺 DProd 𝑠))
93 simplrr 778 . . . . . . . . 9 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → (𝐺 DProd 𝑠) = 𝐵)
9492, 93eqtrd 2775 . . . . . . . 8 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → (𝐺 DProd 𝑆) = 𝐵)
9580, 91, 943jca 1127 . . . . . . 7 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → (𝑆:dom 𝑤𝐶𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵))
9655, 95jca 511 . . . . . 6 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → (𝑤 ∈ Word 𝐵 ∧ (𝑆:dom 𝑤𝐶𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵)))
9796ex 412 . . . . 5 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → ((𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘)) → (𝑤 ∈ Word 𝐵 ∧ (𝑆:dom 𝑤𝐶𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵))))
9897eximdv 1915 . . . 4 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → (∃𝑤(𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘)) → ∃𝑤(𝑤 ∈ Word 𝐵 ∧ (𝑆:dom 𝑤𝐶𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵))))
9949, 98mpd 15 . . 3 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → ∃𝑤(𝑤 ∈ Word 𝐵 ∧ (𝑆:dom 𝑤𝐶𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵)))
100 df-rex 3069 . . 3 (∃𝑤 ∈ Word 𝐵(𝑆:dom 𝑤𝐶𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵) ↔ ∃𝑤(𝑤 ∈ Word 𝐵 ∧ (𝑆:dom 𝑤𝐶𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵)))
10199, 100sylibr 234 . 2 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → ∃𝑤 ∈ Word 𝐵(𝑆:dom 𝑤𝐶𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵))
102 ablfac.1 . . 3 (𝜑𝐺 ∈ Abel)
103 ablfac.2 . . 3 (𝜑𝐵 ∈ Fin)
10414, 10, 102, 103ablfac 20123 . 2 (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))
105101, 104r19.29a 3160 1 (𝜑 → ∃𝑤 ∈ Word 𝐵(𝑆:dom 𝑤𝐶𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  {crab 3433  cin 3962  wss 3963   class class class wbr 5148  cmpt 5231  dom cdm 5689  ran crn 5690  wf 6559  cfv 6563  (class class class)co 7431  Fincfn 8984  0cc0 11153  cz 12611  ..^cfzo 13691  chash 14366  Word cword 14549  Basecbs 17245  s cress 17274  Grpcgrp 18964  .gcmg 19098  SubGrpcsubg 19151   pGrp cpgp 19559  Abelcabl 19814  CycGrpccyg 19910   DProd cdprd 20028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-disj 5116  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-rpss 7742  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-er 8744  df-ec 8746  df-qs 8750  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-acn 9980  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-fz 13545  df-fzo 13692  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-fac 14310  df-bc 14339  df-hash 14367  df-word 14550  df-concat 14606  df-s1 14631  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720  df-dvds 16288  df-gcd 16529  df-prm 16706  df-pc 16871  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-0g 17488  df-gsum 17489  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-eqg 19156  df-ghm 19244  df-gim 19290  df-ga 19321  df-cntz 19348  df-oppg 19377  df-od 19561  df-gex 19562  df-pgp 19563  df-lsm 19669  df-pj1 19670  df-cmn 19815  df-abl 19816  df-cyg 19911  df-dprd 20030
This theorem is referenced by:  dchrpt  27326
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