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Theorem ablfac2 20109
Description: Choose generators for each cyclic group in ablfac 20108. (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 14557 . . . . . . . 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 6746 . . . . . 6 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → dom 𝑠 = (0..^(♯‘𝑠)))
4 fzofi 14015 . . . . . 6 (0..^(♯‘𝑠)) ∈ Fin
53, 4eqeltrdi 2849 . . . . 5 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → dom 𝑠 ∈ Fin)
62ffdmd 6766 . . . . . . . . . 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 2826 . . . . . . . . . . 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 3695 . . . . . . . . . 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 19145 . . . . . . . 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 4204 . . . . . . . . . 10 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝐺s (𝑠𝑘)) ∈ CycGrp)
20 eqid 2737 . . . . . . . . . . . 12 (Base‘(𝐺s (𝑠𝑘))) = (Base‘(𝐺s (𝑠𝑘)))
21 eqid 2737 . . . . . . . . . . . 12 (.g‘(𝐺s (𝑠𝑘))) = (.g‘(𝐺s (𝑠𝑘)))
2220, 21iscyg 19897 . . . . . . . . . . 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 2737 . . . . . . . . . . 11 (𝐺s (𝑠𝑘)) = (𝐺s (𝑠𝑘))
2625subgbas 19148 . . . . . . . . . 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 3331 . . . . . . . 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 19158 . . . . . . . . . . . . 13 (((𝑠𝑘) ∈ (SubGrp‘𝐺) ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ (𝑠𝑘)) → (𝑛 · 𝑥) = (𝑛(.g‘(𝐺s (𝑠𝑘)))𝑥))
3429, 30, 31, 33syl3anc 1373 . . . . . . . . . . . 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 5242 . . . . . . . . . . 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 5949 . . . . . . . . . 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 2753 . . . . . . . . 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 3177 . . . . . . . 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 4053 . . . . . . 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 3146 . . . . 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 5244 . . . . . . . 8 (𝑥 = (𝑤𝑘) → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))))
4645rneqd 5949 . . . . . . 7 (𝑥 = (𝑤𝑘) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))))
4746eqeq1d 2739 . . . . . 6 (𝑥 = (𝑤𝑘) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘)))
4847ac6sfi 9320 . . . . 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 6722 . . . . . . . . 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 14556 . . . . . . . 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 6746 . . . . . . . . . . . 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 2827 . . . . . . . . . . 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 6910 . . . . . . . . . . . . . . . . 17 ((𝑘 = 𝑗𝑛 ∈ ℤ) → (𝑤𝑘) = (𝑤𝑗))
6261oveq2d 7447 . . . . . . . . . . . . . . . 16 ((𝑘 = 𝑗𝑛 ∈ ℤ) → (𝑛 · (𝑤𝑘)) = (𝑛 · (𝑤𝑗)))
6362mpteq2dva 5242 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
6463rneqd 5949 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
65 fveq2 6906 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝑠𝑘) = (𝑠𝑗))
6664, 65eqeq12d 2753 . . . . . . . . . . . . 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 2841 . . . . . . . . . 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 6906 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝑤𝑘) = (𝑤𝑗))
7574oveq2d 7447 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → (𝑛 · (𝑤𝑘)) = (𝑛 · (𝑤𝑗)))
7675mpteq2dv 5244 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
7776rneqd 5949 . . . . . . . . . . 11 (𝑘 = 𝑗 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
7877cbvmptv 5255 . . . . . . . . . 10 (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘)))) = (𝑗 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
7973, 78eqtri 2765 . . . . . . . . 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 3330 . . . . . . . . . . . 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 5234 . . . . . . . . . . . 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 2789 . . . . . . . . . 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 20026 . . . . . . . . . . . 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 2780 . . . . . . . . 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 5171 . . . . . . . 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 2777 . . . . . . . 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 1129 . . . . . . 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 1917 . . . 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 3071 . . 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 20108 . 2 (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))
105101, 104r19.29a 3162 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 1087   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  {crab 3436  cin 3950  wss 3951   class class class wbr 5143  cmpt 5225  dom cdm 5685  ran crn 5686  wf 6557  cfv 6561  (class class class)co 7431  Fincfn 8985  0cc0 11155  cz 12613  ..^cfzo 13694  chash 14369  Word cword 14552  Basecbs 17247  s cress 17274  Grpcgrp 18951  .gcmg 19085  SubGrpcsubg 19138   pGrp cpgp 19544  Abelcabl 19799  CycGrpccyg 19895   DProd cdprd 20013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-rpss 7743  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-acn 9982  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-fac 14313  df-bc 14342  df-hash 14370  df-word 14553  df-concat 14609  df-s1 14634  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-dvds 16291  df-gcd 16532  df-prm 16709  df-pc 16875  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-gsum 17487  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-eqg 19143  df-ghm 19231  df-gim 19277  df-ga 19308  df-cntz 19335  df-oppg 19364  df-od 19546  df-gex 19547  df-pgp 19548  df-lsm 19654  df-pj1 19655  df-cmn 19800  df-abl 19801  df-cyg 19896  df-dprd 20015
This theorem is referenced by:  dchrpt  27311
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