Theorem ablfac2 20072

Description: Choose generators for each cyclic group in ablfac 20071. (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 14536	. . . . . . . 8 ⊢ (𝑠 ∈ Word 𝐶 → 𝑠:(0..^(♯‘𝑠))⟶𝐶)
2	1	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝑠:(0..^(♯‘𝑠))⟶𝐶)
3	2	fdmd 6716	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → dom 𝑠 = (0..^(♯‘𝑠)))
4		fzofi 13992	. . . . . 6 ⊢ (0..^(♯‘𝑠)) ∈ Fin
5	3, 4	eqeltrdi 2842	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → dom 𝑠 ∈ Fin)
6	2	ffdmd 6736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝑠:dom 𝑠⟶𝐶)
7	6	ffvelcdmda 7074	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠‘𝑘) ∈ 𝐶)
8		oveq2 7413	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑠‘𝑘) → (𝐺 ↾s 𝑟) = (𝐺 ↾s (𝑠‘𝑘)))
9	8	eleq1d 2819	. . . . . . . . . . 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 3674	. . . . . . . . . 10 ⊢ ((𝑠‘𝑘) ∈ 𝐶 ↔ ((𝑠‘𝑘) ∈ (SubGrp‘𝐺) ∧ (𝐺 ↾s (𝑠‘𝑘)) ∈ (CycGrp ∩ ran pGrp )))
12	11	simplbi 497	. . . . . . . . 9 ⊢ ((𝑠‘𝑘) ∈ 𝐶 → (𝑠‘𝑘) ∈ (SubGrp‘𝐺))
13	7, 12	syl 17	. . . . . . . 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 19110	. . . . . . . 8 ⊢ ((𝑠‘𝑘) ∈ (SubGrp‘𝐺) → (𝑠‘𝑘) ⊆ 𝐵)
16	13, 15	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠‘𝑘) ⊆ 𝐵)
17	11	simprbi 496	. . . . . . . . . . . 12 ⊢ ((𝑠‘𝑘) ∈ 𝐶 → (𝐺 ↾s (𝑠‘𝑘)) ∈ (CycGrp ∩ ran pGrp ))
18	7, 17	syl 17	. . . . . . . . . . 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 4179	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝐺 ↾s (𝑠‘𝑘)) ∈ CycGrp)
20		eqid 2735	. . . . . . . . . . . 12 ⊢ (Base‘(𝐺 ↾s (𝑠‘𝑘))) = (Base‘(𝐺 ↾s (𝑠‘𝑘)))
21		eqid 2735	. . . . . . . . . . . 12 ⊢ (.g‘(𝐺 ↾s (𝑠‘𝑘))) = (.g‘(𝐺 ↾s (𝑠‘𝑘)))
22	20, 21	iscyg 19860	. . . . . . . . . . 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 496	. . . . . . . . . 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 17	. . . . . . . . 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 2735	. . . . . . . . . . 11 ⊢ (𝐺 ↾s (𝑠‘𝑘)) = (𝐺 ↾s (𝑠‘𝑘))
26	25	subgbas 19113	. . . . . . . . . 10 ⊢ ((𝑠‘𝑘) ∈ (SubGrp‘𝐺) → (𝑠‘𝑘) = (Base‘(𝐺 ↾s (𝑠‘𝑘))))
27	13, 26	syl 17	. . . . . . . . 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 3310	. . . . . . . 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 726	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠‘𝑘)) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ)
31		simplr 768	. . . . . . . . . . . . 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 19123	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 5214	. . . . . . . . . . 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 5918	. . . . . . . . . 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 480	. . . . . . . . . 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 2751	. . . . . . . . 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 3162	. . . . . . . 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 257	. . . . . . 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 4028	. . . . . . 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 65	. . . . . 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 3132	. . . . 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 7413	. . . . . . . . 9 ⊢ (𝑥 = (𝑤‘𝑘) → (𝑛 · 𝑥) = (𝑛 · (𝑤‘𝑘)))
45	44	mpteq2dv 5215	. . . . . . . 8 ⊢ (𝑥 = (𝑤‘𝑘) → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))))
46	45	rneqd 5918	. . . . . . 7 ⊢ (𝑥 = (𝑤‘𝑘) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))))
47	46	eqeq1d 2737	. . . . . 6 ⊢ (𝑥 = (𝑤‘𝑘) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠‘𝑘) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘)))
48	47	ac6sfi 9292	. . . . 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 584	. . . 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 770	. . . . . . . . 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 480	. . . . . . . . . 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 6692	. . . . . . . . 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 232	. . . . . . . 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 14535	. . . . . . . 8 ⊢ (𝑤:(0..^(♯‘𝑠))⟶𝐵 → 𝑤 ∈ Word 𝐵)
55	53, 54	syl 17	. . . . . . 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 6716	. . . . . . . . . . . 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 2820	. . . . . . . . . . 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 476	. . . . . . . . . 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 772	. . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 = 𝑗 ∧ 𝑛 ∈ ℤ) → 𝑘 = 𝑗)
61	60	fveq2d 6880	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 = 𝑗 ∧ 𝑛 ∈ ℤ) → (𝑤‘𝑘) = (𝑤‘𝑗))
62	61	oveq2d 7421	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 = 𝑗 ∧ 𝑛 ∈ ℤ) → (𝑛 · (𝑤‘𝑘)) = (𝑛 · (𝑤‘𝑗)))
63	62	mpteq2dva 5214	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
64	63	rneqd 5918	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
65		fveq2 6876	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝑠‘𝑘) = (𝑠‘𝑗))
66	64, 65	eqeq12d 2751	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))) = (𝑠‘𝑗)))
67	66	rspccva 3600	. . . . . . . . . . . 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 580	. . . . . . . . . . 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 480	. . . . . . . . . . . 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 7074	. . . . . . . . . . 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 2834	. . . . . . . . . 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 591	. . . . . . . . 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 6876	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝑤‘𝑘) = (𝑤‘𝑗))
75	74	oveq2d 7421	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑛 · (𝑤‘𝑘)) = (𝑛 · (𝑤‘𝑗)))
76	75	mpteq2dv 5215	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
77	76	rneqd 5918	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
78	77	cbvmptv 5225	. . . . . . . . . 10 ⊢ (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘)))) = (𝑗 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
79	73, 78	eqtri 2758	. . . . . . . . 9 ⊢ 𝑆 = (𝑗 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
80	72, 79	fmptd 7104	. . . . . . . 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 770	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝐺dom DProd 𝑠)
82	81	adantr 480	. . . . . . . . 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 3309	. . . . . . . . . . . 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 5208	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 2782	. . . . . . . . . 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 19989	. . . . . . . . . . . 12 ⊢ (𝐺dom DProd 𝑠 → 𝑠:dom 𝑠⟶(SubGrp‘𝐺))
88	82, 87	syl 17	. . . . . . . . . . 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 6947	. . . . . . . . . 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 2773	. . . . . . . . 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 5147	. . . . . . . 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 7421	. . . . . . . . 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 777	. . . . . . . . 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 2770	. . . . . . . 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 1128	. . . . . . 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 511	. . . . . 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 412	. . . . 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 1917	. . . 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 15	. . 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 3061	. . 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 234	. 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 20071	. 2 ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))
105	101, 104	r19.29a 3148	1 ⊢ (𝜑 → ∃𝑤 ∈ Word 𝐵(𝑆:dom 𝑤⟶𝐶 ∧ 𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  {crab 3415   ∩ cin 3925   ⊆ wss 3926   class class class wbr 5119   ↦ cmpt 5201  dom cdm 5654  ran crn 5655  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  Fincfn 8959  0cc0 11129  ℤcz 12588  ..^cfzo 13671  ♯chash 14348  Word cword 14531  Basecbs 17228   ↾_s cress 17251  Grpcgrp 18916  ._gcmg 19050  SubGrpcsubg 19103   pGrp cpgp 19507  Abelcabl 19762  CycGrpccyg 19858   DProd cdprd 19976

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-inf2 9655  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-pre-sup 11207

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-disj 5087  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7671  df-rpss 7717  df-om 7862  df-1st 7988  df-2nd 7989  df-supp 8160  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-omul 8485  df-er 8719  df-ec 8721  df-qs 8725  df-map 8842  df-ixp 8912  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fsupp 9374  df-sup 9454  df-inf 9455  df-oi 9524  df-dju 9915  df-card 9953  df-acn 9956  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-div 11895  df-nn 12241  df-2 12303  df-3 12304  df-n0 12502  df-xnn0 12575  df-z 12589  df-uz 12853  df-q 12965  df-rp 13009  df-fz 13525  df-fzo 13672  df-fl 13809  df-mod 13887  df-seq 14020  df-exp 14080  df-fac 14292  df-bc 14321  df-hash 14349  df-word 14532  df-concat 14589  df-s1 14614  df-cj 15118  df-re 15119  df-im 15120  df-sqrt 15254  df-abs 15255  df-clim 15504  df-sum 15703  df-dvds 16273  df-gcd 16514  df-prm 16691  df-pc 16857  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-0g 17455  df-gsum 17456  df-mre 17598  df-mrc 17599  df-acs 17601  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-mhm 18761  df-submnd 18762  df-grp 18919  df-minusg 18920  df-sbg 18921  df-mulg 19051  df-subg 19106  df-eqg 19108  df-ghm 19196  df-gim 19242  df-ga 19273  df-cntz 19300  df-oppg 19329  df-od 19509  df-gex 19510  df-pgp 19511  df-lsm 19617  df-pj1 19618  df-cmn 19763  df-abl 19764  df-cyg 19859  df-dprd 19978

This theorem is referenced by:  dchrpt  27230