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Theorem ablfac2 19207
Description: Choose generators for each cyclic group in ablfac 19206. (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 13866 . . . . . . . 8 (𝑠 ∈ Word 𝐶𝑠:(0..^(♯‘𝑠))⟶𝐶)
21ad2antlr 726 . . . . . . 7 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝑠:(0..^(♯‘𝑠))⟶𝐶)
32fdmd 6501 . . . . . 6 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → dom 𝑠 = (0..^(♯‘𝑠)))
4 fzofi 13341 . . . . . 6 (0..^(♯‘𝑠)) ∈ Fin
53, 4eqeltrdi 2901 . . . . 5 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → dom 𝑠 ∈ Fin)
62ffdmd 6515 . . . . . . . . . 10 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝑠:dom 𝑠𝐶)
76ffvelrnda 6832 . . . . . . . . 9 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠𝑘) ∈ 𝐶)
8 oveq2 7147 . . . . . . . . . . . 12 (𝑟 = (𝑠𝑘) → (𝐺s 𝑟) = (𝐺s (𝑠𝑘)))
98eleq1d 2877 . . . . . . . . . . 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 3634 . . . . . . . . . 10 ((𝑠𝑘) ∈ 𝐶 ↔ ((𝑠𝑘) ∈ (SubGrp‘𝐺) ∧ (𝐺s (𝑠𝑘)) ∈ (CycGrp ∩ ran pGrp )))
1211simplbi 501 . . . . . . . . 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 18275 . . . . . . . 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 500 . . . . . . . . . . . 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 4128 . . . . . . . . . 10 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝐺s (𝑠𝑘)) ∈ CycGrp)
20 eqid 2801 . . . . . . . . . . . 12 (Base‘(𝐺s (𝑠𝑘))) = (Base‘(𝐺s (𝑠𝑘)))
21 eqid 2801 . . . . . . . . . . . 12 (.g‘(𝐺s (𝑠𝑘))) = (.g‘(𝐺s (𝑠𝑘)))
2220, 21iscyg 18994 . . . . . . . . . . 11 ((𝐺s (𝑠𝑘)) ∈ CycGrp ↔ ((𝐺s (𝑠𝑘)) ∈ Grp ∧ ∃𝑥 ∈ (Base‘(𝐺s (𝑠𝑘)))ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺s (𝑠𝑘)))𝑥)) = (Base‘(𝐺s (𝑠𝑘)))))
2322simprbi 500 . . . . . . . . . 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 2801 . . . . . . . . . . . 12 (𝐺s (𝑠𝑘)) = (𝐺s (𝑠𝑘))
2625subgbas 18278 . . . . . . . . . . 11 ((𝑠𝑘) ∈ (SubGrp‘𝐺) → (𝑠𝑘) = (Base‘(𝐺s (𝑠𝑘))))
2713, 26syl 17 . . . . . . . . . 10 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠𝑘) = (Base‘(𝐺s (𝑠𝑘))))
2827rexeqdv 3368 . . . . . . . . 9 ((((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))) ↔ ∃𝑥 ∈ (Base‘(𝐺s (𝑠𝑘)))ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺s (𝑠𝑘)))𝑥)) = (Base‘(𝐺s (𝑠𝑘)))))
2924, 28mpbird 260 . . . . . . . 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))))
3013ad2antrr 725 . . . . . . . . . . . . 13 ((((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠𝑘)) ∧ 𝑛 ∈ ℤ) → (𝑠𝑘) ∈ (SubGrp‘𝐺))
31 simpr 488 . . . . . . . . . . . . 13 ((((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠𝑘)) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ)
32 simplr 768 . . . . . . . . . . . . 13 ((((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠𝑘)) ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ (𝑠𝑘))
33 ablfac2.m . . . . . . . . . . . . . 14 · = (.g𝐺)
3433, 25, 21subgmulg 18288 . . . . . . . . . . . . 13 (((𝑠𝑘) ∈ (SubGrp‘𝐺) ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ (𝑠𝑘)) → (𝑛 · 𝑥) = (𝑛(.g‘(𝐺s (𝑠𝑘)))𝑥))
3530, 31, 32, 34syl3anc 1368 . . . . . . . . . . . 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))
3635mpteq2dva 5128 . . . . . . . . . . 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)))
3736rneqd 5776 . . . . . . . . . 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)))
3827adantr 484 . . . . . . . . . 10 (((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠𝑘)) → (𝑠𝑘) = (Base‘(𝐺s (𝑠𝑘))))
3937, 38eqeq12d 2817 . . . . . . . . 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)))))
4039rexbidva 3258 . . . . . . . 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)))))
4129, 40mpbird 260 . . . . . . 7 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → ∃𝑥 ∈ (𝑠𝑘)ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘))
42 ssrexv 3985 . . . . . . 7 ((𝑠𝑘) ⊆ 𝐵 → (∃𝑥 ∈ (𝑠𝑘)ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘) → ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘)))
4316, 41, 42sylc 65 . . . . . 6 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘))
4443ralrimiva 3152 . . . . 5 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → ∀𝑘 ∈ dom 𝑠𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘))
45 oveq2 7147 . . . . . . . . 9 (𝑥 = (𝑤𝑘) → (𝑛 · 𝑥) = (𝑛 · (𝑤𝑘)))
4645mpteq2dv 5129 . . . . . . . 8 (𝑥 = (𝑤𝑘) → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))))
4746rneqd 5776 . . . . . . 7 (𝑥 = (𝑤𝑘) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))))
4847eqeq1d 2803 . . . . . 6 (𝑥 = (𝑤𝑘) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘)))
4948ac6sfi 8750 . . . . 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)))
505, 44, 49syl2anc 587 . . . 4 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → ∃𝑤(𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘)))
51 simprl 770 . . . . . . . . 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)
523adantr 484 . . . . . . . . . 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)))
5352feq2d 6477 . . . . . . . . 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))
5451, 53mpbid 235 . . . . . . . 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)
55 iswrdi 13865 . . . . . . . 8 (𝑤:(0..^(♯‘𝑠))⟶𝐵𝑤 ∈ Word 𝐵)
5654, 55syl 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)
5751fdmd 6501 . . . . . . . . . . . 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)
5857eleq2d 2878 . . . . . . . . . . 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))
5958biimpa 480 . . . . . . . . . 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)
60 simprr 772 . . . . . . . . . . . 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))
61 simpl 486 . . . . . . . . . . . . . . . . . 18 ((𝑘 = 𝑗𝑛 ∈ ℤ) → 𝑘 = 𝑗)
6261fveq2d 6653 . . . . . . . . . . . . . . . . 17 ((𝑘 = 𝑗𝑛 ∈ ℤ) → (𝑤𝑘) = (𝑤𝑗))
6362oveq2d 7155 . . . . . . . . . . . . . . . 16 ((𝑘 = 𝑗𝑛 ∈ ℤ) → (𝑛 · (𝑤𝑘)) = (𝑛 · (𝑤𝑗)))
6463mpteq2dva 5128 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
6564rneqd 5776 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
66 fveq2 6649 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝑠𝑘) = (𝑠𝑗))
6765, 66eqeq12d 2817 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → (ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))) = (𝑠𝑗)))
6867rspccva 3573 . . . . . . . . . . . 12 ((∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘) ∧ 𝑗 ∈ dom 𝑠) → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))) = (𝑠𝑗))
6960, 68sylan 583 . . . . . . . . . . 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))
706adantr 484 . . . . . . . . . . . 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)
7170ffvelrnda 6832 . . . . . . . . . . 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)
7269, 71eqeltrd 2893 . . . . . . . . . 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)
7359, 72syldan 594 . . . . . . . . 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)
74 ablfac2.s . . . . . . . . . 10 𝑆 = (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))))
75 fveq2 6649 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝑤𝑘) = (𝑤𝑗))
7675oveq2d 7155 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → (𝑛 · (𝑤𝑘)) = (𝑛 · (𝑤𝑗)))
7776mpteq2dv 5129 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
7877rneqd 5776 . . . . . . . . . . 11 (𝑘 = 𝑗 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
7978cbvmptv 5136 . . . . . . . . . 10 (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘)))) = (𝑗 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
8074, 79eqtri 2824 . . . . . . . . 9 𝑆 = (𝑗 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
8173, 80fmptd 6859 . . . . . . . 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)
82 simprl 770 . . . . . . . . . 10 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝐺dom DProd 𝑠)
8382adantr 484 . . . . . . . . 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)
8457raleqdv 3367 . . . . . . . . . . . . 13 ((((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) ↔ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘)))
8560, 84mpbird 260 . . . . . . . . . . . 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))
86 mpteq12 5120 . . . . . . . . . . . 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)))
8757, 85, 86syl2anc 587 . . . . . . . . . . 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)))
8874, 87syl5eq 2848 . . . . . . . . . 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)))
89 dprdf 19124 . . . . . . . . . . . 12 (𝐺dom DProd 𝑠𝑠:dom 𝑠⟶(SubGrp‘𝐺))
9083, 89syl 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))
9190feqmptd 6712 . . . . . . . . . 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)))
9288, 91eqtr4d 2839 . . . . . . . . 9 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → 𝑆 = 𝑠)
9383, 92breqtrrd 5061 . . . . . . . 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)
9492oveq2d 7155 . . . . . . . . 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))
95 simplrr 777 . . . . . . . . 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)
9694, 95eqtrd 2836 . . . . . . . 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)
9781, 93, 963jca 1125 . . . . . . 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))
9856, 97jca 515 . . . . . 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)))
9998ex 416 . . . . 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))))
10099eximdv 1918 . . . 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))))
10150, 100mpd 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)))
102 df-rex 3115 . . 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)))
103101, 102sylibr 237 . 2 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → ∃𝑤 ∈ Word 𝐵(𝑆:dom 𝑤𝐶𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵))
104 ablfac.1 . . 3 (𝜑𝐺 ∈ Abel)
105 ablfac.2 . . 3 (𝜑𝐵 ∈ Fin)
10614, 10, 104, 105ablfac 19206 . 2 (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))
107103, 106r19.29a 3251 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 399  w3a 1084   = wceq 1538  wex 1781  wcel 2112  wral 3109  wrex 3110  {crab 3113  cin 3883  wss 3884   class class class wbr 5033  cmpt 5113  dom cdm 5523  ran crn 5524  wf 6324  cfv 6328  (class class class)co 7139  Fincfn 8496  0cc0 10530  cz 11973  ..^cfzo 13032  chash 13690  Word cword 13861  Basecbs 16478  s cress 16479  Grpcgrp 18098  .gcmg 18219  SubGrpcsubg 18268   pGrp cpgp 18649  Abelcabl 18902  CycGrpccyg 18992   DProd cdprd 19111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-disj 4999  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-rpss 7433  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-tpos 7879  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-omul 8094  df-er 8276  df-ec 8278  df-qs 8282  df-map 8395  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-acn 9359  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-fz 12890  df-fzo 13033  df-fl 13161  df-mod 13237  df-seq 13369  df-exp 13430  df-fac 13634  df-bc 13663  df-hash 13691  df-word 13862  df-concat 13918  df-s1 13945  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-sum 15038  df-dvds 15603  df-gcd 15837  df-prm 16009  df-pc 16167  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-0g 16710  df-gsum 16711  df-mre 16852  df-mrc 16853  df-acs 16855  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-mhm 17951  df-submnd 17952  df-grp 18101  df-minusg 18102  df-sbg 18103  df-mulg 18220  df-subg 18271  df-eqg 18273  df-ghm 18351  df-gim 18394  df-ga 18415  df-cntz 18442  df-oppg 18469  df-od 18651  df-gex 18652  df-pgp 18653  df-lsm 18756  df-pj1 18757  df-cmn 18903  df-abl 18904  df-cyg 18993  df-dprd 19113
This theorem is referenced by:  dchrpt  25854
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