Theorem ablfac2 20298

Description: Choose generators for each cyclic group in ablfac 20297. (Contributed by Mario Carneiro, 28-Apr-2016.)

Hypotheses

Ref	Expression
ablfac.b	⊢ 𝐵 = (Base‘𝐺)
ablfac.c	⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
ablfac.1	⊢ (𝜑 → 𝐺 ∈ Abel)
ablfac.2	⊢ (𝜑 → 𝐵 ∈ Fin)
ablfac2.m	⊢ · = (.g‘𝐺)
ablfac2.s	⊢ 𝑆 = (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))))

Assertion

Ref	Expression
ablfac2	⊢ (𝜑 → ∃𝑤 ∈ Word 𝐵(𝑆:dom 𝑤⟶𝐶 ∧ 𝐺dom 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.

Step	Hyp	Ref	Expression
1		wrdf 14656	. . . . . . . 8 ⊢ (𝑠 ∈ Word 𝐶 → 𝑠:(0..^(♯‘𝑠))⟶𝐶)
2	1	ad2antlr 740	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝑠:(0..^(♯‘𝑠))⟶𝐶)
3	2	fdmd 6718	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → dom 𝑠 = (0..^(♯‘𝑠)))
4		fzofi 14110	. . . . . 6 ⊢ (0..^(♯‘𝑠)) ∈ Fin
5	3, 4	eqeltrdi 2869	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → dom 𝑠 ∈ Fin)
6	2	ffdmd 6738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝑠:dom 𝑠⟶𝐶)
7	6	ffvelcdmda 7082	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠‘𝑘) ∈ 𝐶)
8		oveq2 7426	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑠‘𝑘) → (𝐺 ↾s 𝑟) = (𝐺 ↾s (𝑠‘𝑘)))
9	8	eleq1d 2846	. . . . . . . . . . 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 3649	. . . . . . . . . 10 ⊢ ((𝑠‘𝑘) ∈ 𝐶 ↔ ((𝑠‘𝑘) ∈ (SubGrp‘𝐺) ∧ (𝐺 ↾s (𝑠‘𝑘)) ∈ (CycGrp ∩ ran pGrp )))
12	11	simplbi 502	. . . . . . . . 9 ⊢ ((𝑠‘𝑘) ∈ 𝐶 → (𝑠‘𝑘) ∈ (SubGrp‘𝐺))
13	7, 12	syl 18	. . . . . . . 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 19330	. . . . . . . 8 ⊢ ((𝑠‘𝑘) ∈ (SubGrp‘𝐺) → (𝑠‘𝑘) ⊆ 𝐵)
16	13, 15	syl 18	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠‘𝑘) ⊆ 𝐵)
17	11	simprbi 503	. . . . . . . . . . . 12 ⊢ ((𝑠‘𝑘) ∈ 𝐶 → (𝐺 ↾s (𝑠‘𝑘)) ∈ (CycGrp ∩ ran pGrp ))
18	7, 17	syl 18	. . . . . . . . . . 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 4150	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝐺 ↾s (𝑠‘𝑘)) ∈ CycGrp)
20		eqid 2761	. . . . . . . . . . . 12 ⊢ (Base‘(𝐺 ↾s (𝑠‘𝑘))) = (Base‘(𝐺 ↾s (𝑠‘𝑘)))
21		eqid 2761	. . . . . . . . . . . 12 ⊢ (.g‘(𝐺 ↾s (𝑠‘𝑘))) = (.g‘(𝐺 ↾s (𝑠‘𝑘)))
22	20, 21	iscyg 20086	. . . . . . . . . . 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 503	. . . . . . . . . 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 18	. . . . . . . . 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 2761	. . . . . . . . . . 11 ⊢ (𝐺 ↾s (𝑠‘𝑘)) = (𝐺 ↾s (𝑠‘𝑘))
26	25	subgbas 19333	. . . . . . . . . 10 ⊢ ((𝑠‘𝑘) ∈ (SubGrp‘𝐺) → (𝑠‘𝑘) = (Base‘(𝐺 ↾s (𝑠‘𝑘))))
27	13, 26	syl 18	. . . . . . . . 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 3325	. . . . . . . 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 739	. . . . . . . . . . . . 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 490	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠‘𝑘)) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ)
31		simplr 781	. . . . . . . . . . . . 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 19344	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 5198	. . . . . . . . . . 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 5920	. . . . . . . . . 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 486	. . . . . . . . . 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 2777	. . . . . . . . 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 3185	. . . . . . . 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 260	. . . . . . 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 4001	. . . . . . 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 66	. . . . . 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 3155	. . . . 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 7426	. . . . . . . . 9 ⊢ (𝑥 = (𝑤‘𝑘) → (𝑛 · 𝑥) = (𝑛 · (𝑤‘𝑘)))
45	44	mpteq2dv 5199	. . . . . . . 8 ⊢ (𝑥 = (𝑤‘𝑘) → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))))
46	45	rneqd 5920	. . . . . . 7 ⊢ (𝑥 = (𝑤‘𝑘) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))))
47	46	eqeq1d 2763	. . . . . 6 ⊢ (𝑥 = (𝑤‘𝑘) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠‘𝑘) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘)))
48	47	ac6sfi 9268	. . . . 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 596	. . . 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 783	. . . . . . . . 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 486	. . . . . . . . . 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 6691	. . . . . . . . 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 235	. . . . . . . 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 14655	. . . . . . . 8 ⊢ (𝑤:(0..^(♯‘𝑠))⟶𝐵 → 𝑤 ∈ Word 𝐵)
55	53, 54	syl 18	. . . . . . 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 6718	. . . . . . . . . . . 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 2847	. . . . . . . . . . 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 482	. . . . . . . . . 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 785	. . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 = 𝑗 ∧ 𝑛 ∈ ℤ) → 𝑘 = 𝑗)
61	60	fveq2d 6887	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 = 𝑗 ∧ 𝑛 ∈ ℤ) → (𝑤‘𝑘) = (𝑤‘𝑗))
62	61	oveq2d 7434	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 = 𝑗 ∧ 𝑛 ∈ ℤ) → (𝑛 · (𝑤‘𝑘)) = (𝑛 · (𝑤‘𝑗)))
63	62	mpteq2dva 5198	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
64	63	rneqd 5920	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
65		fveq2 6883	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝑠‘𝑘) = (𝑠‘𝑗))
66	64, 65	eqeq12d 2777	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))) = (𝑠‘𝑗)))
67	66	rspccva 3576	. . . . . . . . . . . 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 592	. . . . . . . . . . 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 486	. . . . . . . . . . . 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 7082	. . . . . . . . . . 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 2861	. . . . . . . . . 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 603	. . . . . . . . 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 6883	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝑤‘𝑘) = (𝑤‘𝑗))
75	74	oveq2d 7434	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑛 · (𝑤‘𝑘)) = (𝑛 · (𝑤‘𝑗)))
76	75	mpteq2dv 5199	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
77	76	rneqd 5920	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
78	77	cbvmptv 5209	. . . . . . . . . 10 ⊢ (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘)))) = (𝑗 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
79	73, 78	eqtri 2784	. . . . . . . . 9 ⊢ 𝑆 = (𝑗 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
80	72, 79	fmptd 7112	. . . . . . . 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 783	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝐺dom DProd 𝑠)
82	81	adantr 486	. . . . . . . . 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 3324	. . . . . . . . . . . 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 5193	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 2808	. . . . . . . . . 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 20215	. . . . . . . . . . . 12 ⊢ (𝐺dom DProd 𝑠 → 𝑠:dom 𝑠⟶(SubGrp‘𝐺))
88	82, 87	syl 18	. . . . . . . . . . 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 6951	. . . . . . . . . 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 2799	. . . . . . . . 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 5133	. . . . . . . 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 7434	. . . . . . . . 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 790	. . . . . . . . 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 2796	. . . . . . . 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 1146	. . . . . . 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 521	. . . . . 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 418	. . . . 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 1950	. . . 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 16	. . 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 3088	. . 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 237	. 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 20297	. 2 ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))
105	101, 104	r19.29a 3171	1 ⊢ (𝜑 → ∃𝑤 ∈ Word 𝐵(𝑆:dom 𝑤⟶𝐶 ∧ 𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  {crab 3413   ∩ cin 3898   ⊆ wss 3899   class class class wbr 5103   ↦ cmpt 5186  dom cdm 5651  ran crn 5652  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  Fincfn 8966  0cc0 11193  ℤcz 12686  ..^cfzo 13781  ♯chash 14467  Word cword 14651  Basecbs 17380   ↾_s cress 17401  Grpcgrp 19137  ._gcmg 19270  SubGrpcsubg 19323   pGrp cpgp 19733  Abelcabl 19988  CycGrpccyg 20084   DProd cdprd 20202

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-inf2 9635  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270  ax-pre-sup 11271

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-disj 5071  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-of 7691  df-rpss 7737  df-om 7876  df-1st 7999  df-2nd 8000  df-supp 8171  df-tpos 8236  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-oadd 8473  df-omul 8474  df-er 8710  df-ec 8712  df-qs 8716  df-map 8842  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fsupp 9347  df-sup 9427  df-inf 9428  df-oi 9497  df-dju 9975  df-card 10013  df-acn 10016  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-div 11967  df-nn 12329  df-2 12398  df-3 12399  df-n0 12600  df-xnn0 12673  df-z 12687  df-uz 12959  df-q 13069  df-rp 13114  df-fz 13633  df-fzo 13782  df-fl 13925  df-mod 14003  df-seq 14138  df-exp 14198  df-fac 14411  df-bc 14440  df-hash 14468  df-word 14652  df-concat 14709  df-s1 14736  df-cj 15259  df-re 15260  df-im 15261  df-sqrt 15395  df-abs 15396  df-clim 15648  df-sum 15847  df-dvds 16416  df-gcd 16658  df-prm 16840  df-pc 17008  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-ress 17402  df-plusg 17434  df-0g 17605  df-gsum 17606  df-mre 17749  df-mrc 17750  df-acs 17752  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-mhm 18971  df-submnd 18972  df-grp 19140  df-minusg 19141  df-sbg 19142  df-mulg 19271  df-subg 19326  df-eqg 19328  df-ghm 19421  df-gim 19466  df-ga 19497  df-cntz 19524  df-oppg 19553  df-od 19735  df-gex 19736  df-pgp 19737  df-lsm 19843  df-pj1 19844  df-cmn 19989  df-abl 19990  df-cyg 20085  df-dprd 20204

This theorem is used by:  dchrpt  27587