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Theorem ablfac2 20007

Description: Choose generators for each cyclic group in ablfac 20006. (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 14476	. . . . . . . 8 ⊢ (𝑠 ∈ Word 𝐶 → 𝑠:(0..^(♯‘𝑠))⟶𝐶)
2	1	ad2antlr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝑠:(0..^(♯‘𝑠))⟶𝐶)
3	2	fdmd 6728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → dom 𝑠 = (0..^(♯‘𝑠)))
4		fzofi 13946	. . . . . 6 ⊢ (0..^(♯‘𝑠)) ∈ Fin
5	3, 4	eqeltrdi 2840	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → dom 𝑠 ∈ Fin)
6	2	ffdmd 6748	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝑠:dom 𝑠⟶𝐶)
7	6	ffvelcdmda 7086	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝑠‘𝑘) ∈ 𝐶)
8		oveq2 7420	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑠‘𝑘) → (𝐺 ↾_s 𝑟) = (𝐺 ↾_s (𝑠‘𝑘)))
9	8	eleq1d 2817	. . . . . . . . . . 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 3686	. . . . . . . . . 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 19050	. . . . . . . 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 4198	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) → (𝐺 ↾_s (𝑠‘𝑘)) ∈ CycGrp)
20		eqid 2731	. . . . . . . . . . . 12 ⊢ (Base‘(𝐺 ↾_s (𝑠‘𝑘))) = (Base‘(𝐺 ↾_s (𝑠‘𝑘)))
21		eqid 2731	. . . . . . . . . . . 12 ⊢ (._g‘(𝐺 ↾_s (𝑠‘𝑘))) = (._g‘(𝐺 ↾_s (𝑠‘𝑘)))
22	20, 21	iscyg 19795	. . . . . . . . . . 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 2731	. . . . . . . . . . . 12 ⊢ (𝐺 ↾_s (𝑠‘𝑘)) = (𝐺 ↾_s (𝑠‘𝑘))
26	25	subgbas 19053	. . . . . . . . . . 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 3325	. . . . . . . . 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 257	. . . . . . . 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 723	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) ∧ 𝑘 ∈ dom 𝑠) ∧ 𝑥 ∈ (𝑠‘𝑘)) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ)
32		simplr 766	. . . . . . . . . . . . 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 19063	. . . . . . . . . . . . 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 1370	. . . . . . . . . . . 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 5248	. . . . . . . . . . 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 5937	. . . . . . . . . 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 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))))
39	37, 38	eqeq12d 2747	. . . . . . . . 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 3175	. . . . . . . 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 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))
42		ssrexv 4051	. . . . . . 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 3145	. . . . 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 7420	. . . . . . . . 9 ⊢ (𝑥 = (𝑤‘𝑘) → (𝑛 · 𝑥) = (𝑛 · (𝑤‘𝑘)))
46	45	mpteq2dv 5250	. . . . . . . 8 ⊢ (𝑥 = (𝑤‘𝑘) → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))))
47	46	rneqd 5937	. . . . . . 7 ⊢ (𝑥 = (𝑤‘𝑘) → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))))
48	47	eqeq1d 2733	. . . . . 6 ⊢ (𝑥 = (𝑤‘𝑘) → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑠‘𝑘) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑠‘𝑘)))
49	48	ac6sfi 9293	. . . . 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 583	. . . 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 768	. . . . . . . . 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 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)))
53	52	feq2d 6703	. . . . . . . . 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 14475	. . . . . . . 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 6728	. . . . . . . . . . . 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 2818	. . . . . . . . . . 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 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)
60		simprr 770	. . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 = 𝑗 ∧ 𝑛 ∈ ℤ) → 𝑘 = 𝑗)
62	61	fveq2d 6895	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 = 𝑗 ∧ 𝑛 ∈ ℤ) → (𝑤‘𝑘) = (𝑤‘𝑗))
63	62	oveq2d 7428	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 = 𝑗 ∧ 𝑛 ∈ ℤ) → (𝑛 · (𝑤‘𝑘)) = (𝑛 · (𝑤‘𝑗)))
64	63	mpteq2dva 5248	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
65	64	rneqd 5937	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
66		fveq2 6891	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝑠‘𝑘) = (𝑠‘𝑗))
67	65, 66	eqeq12d 2747	. . . . . . . . . . . . 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 579	. . . . . . . . . . 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 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)
71	70	ffvelcdmda 7086	. . . . . . . . . . 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 2832	. . . . . . . . . 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 590	. . . . . . . . 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 6891	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝑤‘𝑘) = (𝑤‘𝑗))
76	75	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑛 · (𝑤‘𝑘)) = (𝑛 · (𝑤‘𝑗)))
77	76	mpteq2dv 5250	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
78	77	rneqd 5937	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘))) = ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
79	78	cbvmptv 5261	. . . . . . . . . 10 ⊢ (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘)))) = (𝑗 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
80	74, 79	eqtri 2759	. . . . . . . . 9 ⊢ 𝑆 = (𝑗 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑗))))
81	73, 80	fmptd 7115	. . . . . . . 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 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ Word 𝐶) ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) → 𝐺dom DProd 𝑠)
83	82	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)
84	57	raleqdv 3324	. . . . . . . . . . . . 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 257	. . . . . . . . . . . 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 5240	. . . . . . . . . . . 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 583	. . . . . . . . . . 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 2783	. . . . . . . . . 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 19924	. . . . . . . . . . . 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 6960	. . . . . . . . . 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 2774	. . . . . . . . 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 5176	. . . . . . . 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 7428	. . . . . . . . 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 775	. . . . . . . . 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 2771	. . . . . . . 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 1127	. . . . . . 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 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)))
99	98	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))))
100	99	eximdv 1919	. . . 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 3070	. . 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 20006	. 2 ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))
107	103, 106	r19.29a 3161	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 395 ∧ w3a 1086 = wceq 1540 ∃wex 1780 ∈ wcel 2105 ∀wral 3060 ∃wrex 3069 {crab 3431 ∩ cin 3947 ⊆ wss 3948 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ran crn 5677 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 Fincfn 8945 0cc0 11116 ℤcz 12565 ..^cfzo 13634 ♯chash 14297 Word cword 14471 Basecbs 17151 ↾_s cress 17180 Grpcgrp 18861 ._gcmg 18993 SubGrpcsubg 19043 pGrp cpgp 19442 Abelcabl 19697 CycGrpccyg 19793 DProd cdprd 19911

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-rpss 7717 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-oadd 8476 df-omul 8477 df-er 8709 df-ec 8711 df-qs 8715 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-inf 9444 df-oi 9511 df-dju 9902 df-card 9940 df-acn 9943 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-xnn0 12552 df-z 12566 df-uz 12830 df-q 12940 df-rp 12982 df-fz 13492 df-fzo 13635 df-fl 13764 df-mod 13842 df-seq 13974 df-exp 14035 df-fac 14241 df-bc 14270 df-hash 14298 df-word 14472 df-concat 14528 df-s1 14553 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-sum 15640 df-dvds 16205 df-gcd 16443 df-prm 16616 df-pc 16777 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-0g 17394 df-gsum 17395 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-subg 19046 df-eqg 19048 df-ghm 19135 df-gim 19180 df-ga 19202 df-cntz 19229 df-oppg 19258 df-od 19444 df-gex 19445 df-pgp 19446 df-lsm 19552 df-pj1 19553 df-cmn 19698 df-abl 19699 df-cyg 19794 df-dprd 19913

This theorem is referenced by: dchrpt 27115

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