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Theorem ablfac2 18311
Description: Choose generators for each cyclic group in ablfac 18310. (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 13165 . . . . . . . 8 (𝑠 ∈ Word 𝐶𝑠:(0..^(#‘𝑠))⟶𝐶)
21ad2antlr 759 . . . . . . 7 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝑠:(0..^(#‘𝑠))⟶𝐶)
3 fdm 5964 . . . . . . 7 (𝑠:(0..^(#‘𝑠))⟶𝐶 → dom 𝑠 = (0..^(#‘𝑠)))
42, 3syl 17 . . . . . 6 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → dom 𝑠 = (0..^(#‘𝑠)))
5 fzofi 12635 . . . . . 6 (0..^(#‘𝑠)) ∈ Fin
64, 5syl6eqel 2696 . . . . 5 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → dom 𝑠 ∈ Fin)
74feq2d 5944 . . . . . . . . . . 11 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → (𝑠:dom 𝑠𝐶𝑠:(0..^(#‘𝑠))⟶𝐶))
82, 7mpbird 246 . . . . . . . . . 10 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝑠:dom 𝑠𝐶)
98ffvelrnda 6267 . . . . . . . . 9 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠𝑘) ∈ 𝐶)
10 oveq2 6557 . . . . . . . . . . . 12 (𝑟 = (𝑠𝑘) → (𝐺s 𝑟) = (𝐺s (𝑠𝑘)))
1110eleq1d 2672 . . . . . . . . . . 11 (𝑟 = (𝑠𝑘) → ((𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp ) ↔ (𝐺s (𝑠𝑘)) ∈ (CycGrp ∩ ran pGrp )))
12 ablfac.c . . . . . . . . . . 11 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
1311, 12elrab2 3333 . . . . . . . . . 10 ((𝑠𝑘) ∈ 𝐶 ↔ ((𝑠𝑘) ∈ (SubGrp‘𝐺) ∧ (𝐺s (𝑠𝑘)) ∈ (CycGrp ∩ ran pGrp )))
1413simplbi 475 . . . . . . . . 9 ((𝑠𝑘) ∈ 𝐶 → (𝑠𝑘) ∈ (SubGrp‘𝐺))
159, 14syl 17 . . . . . . . 8 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠𝑘) ∈ (SubGrp‘𝐺))
16 ablfac.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
1716subgss 17418 . . . . . . . 8 ((𝑠𝑘) ∈ (SubGrp‘𝐺) → (𝑠𝑘) ⊆ 𝐵)
1815, 17syl 17 . . . . . . 7 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠𝑘) ⊆ 𝐵)
19 inss1 3795 . . . . . . . . . . 11 (CycGrp ∩ ran pGrp ) ⊆ CycGrp
2013simprbi 479 . . . . . . . . . . . 12 ((𝑠𝑘) ∈ 𝐶 → (𝐺s (𝑠𝑘)) ∈ (CycGrp ∩ ran pGrp ))
219, 20syl 17 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝐺s (𝑠𝑘)) ∈ (CycGrp ∩ ran pGrp ))
2219, 21sseldi 3566 . . . . . . . . . 10 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝐺s (𝑠𝑘)) ∈ CycGrp)
23 eqid 2610 . . . . . . . . . . . 12 (Base‘(𝐺s (𝑠𝑘))) = (Base‘(𝐺s (𝑠𝑘)))
24 eqid 2610 . . . . . . . . . . . 12 (.g‘(𝐺s (𝑠𝑘))) = (.g‘(𝐺s (𝑠𝑘)))
2523, 24iscyg 18104 . . . . . . . . . . 11 ((𝐺s (𝑠𝑘)) ∈ CycGrp ↔ ((𝐺s (𝑠𝑘)) ∈ Grp ∧ ∃𝑥 ∈ (Base‘(𝐺s (𝑠𝑘)))ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺s (𝑠𝑘)))𝑥)) = (Base‘(𝐺s (𝑠𝑘)))))
2625simprbi 479 . . . . . . . . . 10 ((𝐺s (𝑠𝑘)) ∈ CycGrp → ∃𝑥 ∈ (Base‘(𝐺s (𝑠𝑘)))ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘(𝐺s (𝑠𝑘)))𝑥)) = (Base‘(𝐺s (𝑠𝑘))))
2722, 26syl 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))))
28 eqid 2610 . . . . . . . . . . . 12 (𝐺s (𝑠𝑘)) = (𝐺s (𝑠𝑘))
2928subgbas 17421 . . . . . . . . . . 11 ((𝑠𝑘) ∈ (SubGrp‘𝐺) → (𝑠𝑘) = (Base‘(𝐺s (𝑠𝑘))))
3015, 29syl 17 . . . . . . . . . 10 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠𝑘) = (Base‘(𝐺s (𝑠𝑘))))
3130rexeqdv 3122 . . . . . . . . 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)))))
3227, 31mpbird 246 . . . . . . . 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))))
3315ad2antrr 758 . . . . . . . . . . . . 13 ((((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠𝑘)) ∧ 𝑛 ∈ ℤ) → (𝑠𝑘) ∈ (SubGrp‘𝐺))
34 simpr 476 . . . . . . . . . . . . 13 ((((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠𝑘)) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ)
35 simplr 788 . . . . . . . . . . . . 13 ((((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠𝑘)) ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ (𝑠𝑘))
36 ablfac2.m . . . . . . . . . . . . . 14 · = (.g𝐺)
3736, 28, 24subgmulg 17431 . . . . . . . . . . . . 13 (((𝑠𝑘) ∈ (SubGrp‘𝐺) ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ (𝑠𝑘)) → (𝑛 · 𝑥) = (𝑛(.g‘(𝐺s (𝑠𝑘)))𝑥))
3833, 34, 35, 37syl3anc 1318 . . . . . . . . . . . 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))
3938mpteq2dva 4672 . . . . . . . . . . 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)))
4039rneqd 5274 . . . . . . . . . 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)))
4130adantr 480 . . . . . . . . . 10 (((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠𝑘)) → (𝑠𝑘) = (Base‘(𝐺s (𝑠𝑘))))
4240, 41eqeq12d 2625 . . . . . . . . 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)))))
4342rexbidva 3031 . . . . . . . 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)))))
4432, 43mpbird 246 . . . . . . 7 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → ∃𝑥 ∈ (𝑠𝑘)ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘))
45 ssrexv 3630 . . . . . . 7 ((𝑠𝑘) ⊆ 𝐵 → (∃𝑥 ∈ (𝑠𝑘)ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘) → ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘)))
4618, 44, 45sylc 63 . . . . . 6 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘))
4746ralrimiva 2949 . . . . 5 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → ∀𝑘 ∈ dom 𝑠𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘))
48 oveq2 6557 . . . . . . . . 9 (𝑥 = (𝑤𝑘) → (𝑛 · 𝑥) = (𝑛 · (𝑤𝑘)))
4948mpteq2dv 4673 . . . . . . . 8 (𝑥 = (𝑤𝑘) → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))))
5049rneqd 5274 . . . . . . 7 (𝑥 = (𝑤𝑘) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))))
5150eqeq1d 2612 . . . . . 6 (𝑥 = (𝑤𝑘) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠𝑘) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘)))
5251ac6sfi 8089 . . . . 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)))
536, 47, 52syl2anc 691 . . . 4 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → ∃𝑤(𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘)))
54 simprl 790 . . . . . . . . 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)
554adantr 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)))
5655feq2d 5944 . . . . . . . . 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))
5754, 56mpbid 221 . . . . . . . 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)
58 iswrdi 13164 . . . . . . . 8 (𝑤:(0..^(#‘𝑠))⟶𝐵𝑤 ∈ Word 𝐵)
5957, 58syl 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)
60 fdm 5964 . . . . . . . . . . . . . 14 (𝑤:(0..^(#‘𝑠))⟶𝐵 → dom 𝑤 = (0..^(#‘𝑠)))
6157, 60syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → dom 𝑤 = (0..^(#‘𝑠)))
6261, 55eqtr4d 2647 . . . . . . . . . . . 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)
6362eleq2d 2673 . . . . . . . . . . 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))
6463biimpa 500 . . . . . . . . . 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)
65 simprr 792 . . . . . . . . . . . 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))
66 simpl 472 . . . . . . . . . . . . . . . . . 18 ((𝑘 = 𝑗𝑛 ∈ ℤ) → 𝑘 = 𝑗)
6766fveq2d 6107 . . . . . . . . . . . . . . . . 17 ((𝑘 = 𝑗𝑛 ∈ ℤ) → (𝑤𝑘) = (𝑤𝑗))
6867oveq2d 6565 . . . . . . . . . . . . . . . 16 ((𝑘 = 𝑗𝑛 ∈ ℤ) → (𝑛 · (𝑤𝑘)) = (𝑛 · (𝑤𝑗)))
6968mpteq2dva 4672 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
7069rneqd 5274 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
71 fveq2 6103 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝑠𝑘) = (𝑠𝑗))
7270, 71eqeq12d 2625 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → (ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))) = (𝑠𝑗)))
7372rspccva 3281 . . . . . . . . . . . 12 ((∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘) ∧ 𝑗 ∈ dom 𝑠) → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))) = (𝑠𝑗))
7465, 73sylan 487 . . . . . . . . . . 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))
758adantr 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)
7675ffvelrnda 6267 . . . . . . . . . . 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)
7774, 76eqeltrd 2688 . . . . . . . . . 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)
7864, 77syldan 486 . . . . . . . . 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)
79 ablfac2.s . . . . . . . . . 10 𝑆 = (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))))
80 fveq2 6103 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝑤𝑘) = (𝑤𝑗))
8180oveq2d 6565 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → (𝑛 · (𝑤𝑘)) = (𝑛 · (𝑤𝑗)))
8281mpteq2dv 4673 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
8382rneqd 5274 . . . . . . . . . . 11 (𝑘 = 𝑗 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
8483cbvmptv 4678 . . . . . . . . . 10 (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘)))) = (𝑗 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
8579, 84eqtri 2632 . . . . . . . . 9 𝑆 = (𝑗 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑗))))
8678, 85fmptd 6292 . . . . . . . 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)
87 simprl 790 . . . . . . . . . 10 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝐺dom DProd 𝑠)
8887adantr 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)
8962raleqdv 3121 . . . . . . . . . . . . 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)))
9065, 89mpbird 246 . . . . . . . . . . . 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))
91 mpteq12 4664 . . . . . . . . . . . 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)))
9262, 90, 91syl2anc 691 . . . . . . . . . . 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)))
9379, 92syl5eq 2656 . . . . . . . . . 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)))
94 dprdf 18228 . . . . . . . . . . . 12 (𝐺dom DProd 𝑠𝑠:dom 𝑠⟶(SubGrp‘𝐺))
9588, 94syl 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))
9695feqmptd 6159 . . . . . . . . . 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)))
9793, 96eqtr4d 2647 . . . . . . . . 9 ((((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ (𝑤:dom 𝑠𝐵 ∧ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))) = (𝑠𝑘))) → 𝑆 = 𝑠)
9888, 97breqtrrd 4611 . . . . . . . 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)
9997oveq2d 6565 . . . . . . . . 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))
100 simplrr 797 . . . . . . . . 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)
10199, 100eqtrd 2644 . . . . . . . 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)
10286, 98, 1013jca 1235 . . . . . . 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))
10359, 102jca 553 . . . . . 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)))
104103ex 449 . . . . 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))))
105104eximdv 1833 . . . 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))))
10653, 105mpd 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)))
107 df-rex 2902 . . 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)))
108106, 107sylibr 223 . 2 (((𝜑𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → ∃𝑤 ∈ Word 𝐵(𝑆:dom 𝑤𝐶𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵))
109 ablfac.1 . . 3 (𝜑𝐺 ∈ Abel)
110 ablfac.2 . . 3 (𝜑𝐵 ∈ Fin)
11116, 12, 109, 110ablfac 18310 . 2 (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))
112108, 111r19.29a 3060 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 383  w3a 1031   = wceq 1475  wex 1695  wcel 1977  wral 2896  wrex 2897  {crab 2900  cin 3539  wss 3540   class class class wbr 4583  cmpt 4643  dom cdm 5038  ran crn 5039  wf 5800  cfv 5804  (class class class)co 6549  Fincfn 7841  0cc0 9815  cz 11254  ..^cfzo 12334  #chash 12979  Word cword 13146  Basecbs 15695  s cress 15696  Grpcgrp 17245  .gcmg 17363  SubGrpcsubg 17411   pGrp cpgp 17769  Abelcabl 18017  CycGrpccyg 18102   DProd cdprd 18215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-rpss 6835  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-dvds 14822  df-gcd 15055  df-prm 15224  df-pc 15380  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-eqg 17416  df-ghm 17481  df-gim 17524  df-ga 17546  df-cntz 17573  df-oppg 17599  df-od 17771  df-gex 17772  df-pgp 17773  df-lsm 17874  df-pj1 17875  df-cmn 18018  df-abl 18019  df-cyg 18103  df-dprd 18217
This theorem is referenced by:  dchrpt  24792
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