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Theorem ablfac2 19953

Description: Choose generators for each cyclic group in ablfac 19952. (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 14465	. . . . . . . 8 ⊢ (𝑠 ∈ Word 𝐶 → 𝑠:(0..^(♯‘𝑠))⟶𝐶)
2	1	ad2antlr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝑠:(0..^(♯‘𝑠))⟶𝐶)
3	2	fdmd 6725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → dom 𝑠 = (0..^(♯‘𝑠)))
4		fzofi 13935	. . . . . 6 ⊢ (0..^(♯‘𝑠)) ∈ Fin
5	3, 4	eqeltrdi 2841	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → dom 𝑠 ∈ Fin)
6	2	ffdmd 6745	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝑠:dom 𝑠⟶𝐶)
7	6	ffvelcdmda 7083	. . . . . . . . 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 2818	. . . . . . . . . . 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 3685	. . . . . . . . . 10 ⊢ ((𝑠‘𝑘) ∈ 𝐶 ↔ ((𝑠‘𝑘) ∈ (SubGrp‘𝐺) ∧ (𝐺 ↾_s (𝑠‘𝑘)) ∈ (CycGrp ∩ ran pGrp )))
12	11	simplbi 498	. . . . . . . . 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 19001	. . . . . . . 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 497	. . . . . . . . . . . 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 4197	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝐺 ↾_s (𝑠‘𝑘)) ∈ CycGrp)
20		eqid 2732	. . . . . . . . . . . 12 ⊢ (Base‘(𝐺 ↾_s (𝑠‘𝑘))) = (Base‘(𝐺 ↾_s (𝑠‘𝑘)))
21		eqid 2732	. . . . . . . . . . . 12 ⊢ (._g‘(𝐺 ↾_s (𝑠‘𝑘))) = (._g‘(𝐺 ↾_s (𝑠‘𝑘)))
22	20, 21	iscyg 19741	. . . . . . . . . . 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 497	. . . . . . . . . 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 2732	. . . . . . . . . . . 12 ⊢ (𝐺 ↾_s (𝑠‘𝑘)) = (𝐺 ↾_s (𝑠‘𝑘))
26	25	subgbas 19004	. . . . . . . . . . 11 ⊢ ((𝑠‘𝑘) ∈ (SubGrp‘𝐺) → (𝑠‘𝑘) = (Base‘(𝐺 ↾_s (𝑠‘𝑘))))
27	13, 26	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠‘𝑘) = (Base‘(𝐺 ↾_s (𝑠‘𝑘))))
28	27	rexeqdv 3326	. . . . . . . . 9 ⊢ ((((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))) ↔ ∃𝑥 ∈ (Base‘(𝐺 ↾_s (𝑠‘𝑘)))ran (𝑛 ∈ ℤ ↦ (𝑛(._g‘(𝐺 ↾_s (𝑠‘𝑘)))𝑥)) = (Base‘(𝐺 ↾_s (𝑠‘𝑘)))))
29	24, 28	mpbird 256	. . . . . . . 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))))
30	13	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠‘𝑘)) ∧ 𝑛 ∈ ℤ) → (𝑠‘𝑘) ∈ (SubGrp‘𝐺))
31		simpr 485	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠‘𝑘)) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ)
32		simplr 767	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠‘𝑘)) ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ (𝑠‘𝑘))
33		ablfac2.m	. . . . . . . . . . . . . 14 ⊢ · = (._g‘𝐺)
34	33, 25, 21	subgmulg 19014	. . . . . . . . . . . . 13 ⊢ (((𝑠‘𝑘) ∈ (SubGrp‘𝐺) ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ (𝑠‘𝑘)) → (𝑛 · 𝑥) = (𝑛(._g‘(𝐺 ↾_s (𝑠‘𝑘)))𝑥))
35	30, 31, 32, 34	syl3anc 1371	. . . . . . . . . . . 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))
36	35	mpteq2dva 5247	. . . . . . . . . . 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)))
37	36	rneqd 5935	. . . . . . . . . 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)))
38	27	adantr 481	. . . . . . . . . 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))))
39	37, 38	eqeq12d 2748	. . . . . . . . 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)))))
40	39	rexbidva 3176	. . . . . . . 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)))))
41	29, 40	mpbird 256	. . . . . . 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))
42		ssrexv 4050	. . . . . . 7 ⊢ ((𝑠‘𝑘) ⊆ 𝐵 → (∃𝑥 ∈ (𝑠‘𝑘)ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠‘𝑘) → ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠‘𝑘)))
43	16, 41, 42	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))
44	43	ralrimiva 3146	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → ∀𝑘 ∈ dom 𝑠∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠‘𝑘))
45		oveq2 7413	. . . . . . . . 9 ⊢ (𝑥 = (𝑤‘𝑘) → (𝑛 · 𝑥) = (𝑛 · (𝑤‘𝑘)))
46	45	mpteq2dv 5249	. . . . . . . 8 ⊢ (𝑥 = (𝑤‘𝑘) → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))))
47	46	rneqd 5935	. . . . . . 7 ⊢ (𝑥 = (𝑤‘𝑘) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))))
48	47	eqeq1d 2734	. . . . . 6 ⊢ (𝑥 = (𝑤‘𝑘) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠‘𝑘) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘)))
49	48	ac6sfi 9283	. . . . 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)))
50	5, 44, 49	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)))
51		simprl 769	. . . . . . . . 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)
52	3	adantr 481	. . . . . . . . . 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)))
53	52	feq2d 6700	. . . . . . . . 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))
54	51, 53	mpbid 231	. . . . . . . 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)
55		iswrdi 14464	. . . . . . . 8 ⊢ (𝑤:(0..^(♯‘𝑠))⟶𝐵 → 𝑤 ∈ Word 𝐵)
56	54, 55	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)
57	51	fdmd 6725	. . . . . . . . . . . 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)
58	57	eleq2d 2819	. . . . . . . . . . 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))
59	58	biimpa 477	. . . . . . . . . 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)
60		simprr 771	. . . . . . . . . . . 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))
61		simpl 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 = 𝑗 ∧ 𝑛 ∈ ℤ) → 𝑘 = 𝑗)
62	61	fveq2d 6892	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 = 𝑗 ∧ 𝑛 ∈ ℤ) → (𝑤‘𝑘) = (𝑤‘𝑗))
63	62	oveq2d 7421	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 = 𝑗 ∧ 𝑛 ∈ ℤ) → (𝑛 · (𝑤‘𝑘)) = (𝑛 · (𝑤‘𝑗)))
64	63	mpteq2dva 5247	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
65	64	rneqd 5935	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
66		fveq2 6888	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝑠‘𝑘) = (𝑠‘𝑗))
67	65, 66	eqeq12d 2748	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))) = (𝑠‘𝑗)))
68	67	rspccva 3611	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘) ∧ 𝑗 ∈ dom 𝑠) → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))) = (𝑠‘𝑗))
69	60, 68	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))
70	6	adantr 481	. . . . . . . . . . . 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)
71	70	ffvelcdmda 7083	. . . . . . . . . . 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)
72	69, 71	eqeltrd 2833	. . . . . . . . . 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)
73	59, 72	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)
74		ablfac2.s	. . . . . . . . . 10 ⊢ 𝑆 = (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))))
75		fveq2 6888	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝑤‘𝑘) = (𝑤‘𝑗))
76	75	oveq2d 7421	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑛 · (𝑤‘𝑘)) = (𝑛 · (𝑤‘𝑗)))
77	76	mpteq2dv 5249	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
78	77	rneqd 5935	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
79	78	cbvmptv 5260	. . . . . . . . . 10 ⊢ (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘)))) = (𝑗 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
80	74, 79	eqtri 2760	. . . . . . . . 9 ⊢ 𝑆 = (𝑗 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
81	73, 80	fmptd 7110	. . . . . . . 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)
82		simprl 769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝐺dom DProd 𝑠)
83	82	adantr 481	. . . . . . . . 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)
84	57	raleqdv 3325	. . . . . . . . . . . . 13 ⊢ ((((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) ↔ ∀𝑘 ∈ dom 𝑠ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘)))
85	60, 84	mpbird 256	. . . . . . . . . . . 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))
86		mpteq12 5239	. . . . . . . . . . . 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)))
87	57, 85, 86	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)))
88	74, 87	eqtrid 2784	. . . . . . . . . 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)))
89		dprdf 19870	. . . . . . . . . . . 12 ⊢ (𝐺dom DProd 𝑠 → 𝑠:dom 𝑠⟶(SubGrp‘𝐺))
90	83, 89	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))
91	90	feqmptd 6957	. . . . . . . . . 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)))
92	88, 91	eqtr4d 2775	. . . . . . . . 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)
93	83, 92	breqtrrd 5175	. . . . . . . 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)
94	92	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))
95		simplrr 776	. . . . . . . . 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)
96	94, 95	eqtrd 2772	. . . . . . . 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)
97	81, 93, 96	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))
98	56, 97	jca 512	. . . . . 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)))
99	98	ex 413	. . . . 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))))
100	99	eximdv 1920	. . . 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))))
101	50, 100	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)))
102		df-rex 3071	. . 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)))
103	101, 102	sylibr 233	. 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))
104		ablfac.1	. . 3 ⊢ (𝜑 → 𝐺 ∈ Abel)
105		ablfac.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
106	14, 10, 104, 105	ablfac 19952	. 2 ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))
107	103, 106	r19.29a 3162	1 ⊢ (𝜑 → ∃𝑤 ∈ Word 𝐵(𝑆:dom 𝑤⟶𝐶 ∧ 𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 {crab 3432 ∩ cin 3946 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ran crn 5676 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 0cc0 11106 ℤcz 12554 ..^cfzo 13623 ♯chash 14286 Word cword 14460 Basecbs 17140 ↾_s cress 17169 Grpcgrp 18815 ._gcmg 18944 SubGrpcsubg 18994 pGrp cpgp 19388 Abelcabl 19643 CycGrpccyg 19739 DProd cdprd 19857

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-rpss 7709 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8699 df-ec 8701 df-qs 8705 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-acn 9933 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-word 14461 df-concat 14517 df-s1 14542 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-dvds 16194 df-gcd 16432 df-prm 16605 df-pc 16766 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-0g 17383 df-gsum 17384 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-eqg 18999 df-ghm 19084 df-gim 19127 df-ga 19148 df-cntz 19175 df-oppg 19204 df-od 19390 df-gex 19391 df-pgp 19392 df-lsm 19498 df-pj1 19499 df-cmn 19644 df-abl 19645 df-cyg 19740 df-dprd 19859

This theorem is referenced by: dchrpt 26759

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