Theorem ablfaclem3 20056

Description: Lemma for ablfac 20057. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-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)
ablfac.o	⊢ 𝑂 = (od‘𝐺)
ablfac.a	⊢ 𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}
ablfac.s	⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})
ablfac.w	⊢ 𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})

Assertion

Ref	Expression
ablfaclem3	⊢ (𝜑 → (𝑊‘𝐵) ≠ ∅)

Distinct variable groups:   𝑠,𝑝,𝑥,𝐴   𝑔,𝑟,𝑠,𝑆   𝑔,𝑝,𝑤,𝑥,𝐵,𝑟,𝑠   𝑂,𝑝,𝑥   𝐶,𝑔,𝑝,𝑠,𝑤,𝑥   𝑊,𝑝,𝑤,𝑥   𝜑,𝑝,𝑠,𝑤,𝑥   𝑔,𝐺,𝑝,𝑟,𝑠,𝑤,𝑥

Allowed substitution hints:   𝜑(𝑔,𝑟)   𝐴(𝑤,𝑔,𝑟)   𝐶(𝑟)   𝑆(𝑥,𝑤,𝑝)   𝑂(𝑤,𝑔,𝑠,𝑟)   𝑊(𝑔,𝑠,𝑟)

Proof of Theorem ablfaclem3

Dummy variables 𝑎 𝑏 𝑐 𝑓 ℎ 𝑞 𝑡 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 13927	. . . 4 ⊢ (𝜑 → (1...(♯‘𝐵)) ∈ Fin)
2		ablfac.a	. . . . 5 ⊢ 𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}
3		prmnn 16635	. . . . . . . 8 ⊢ (𝑤 ∈ ℙ → 𝑤 ∈ ℕ)
4	3	3ad2ant2 1140	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ∈ ℕ)
5		prmz 16636	. . . . . . . . 9 ⊢ (𝑤 ∈ ℙ → 𝑤 ∈ ℤ)
6		ablfac.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ Abel)
7		ablgrp 19752	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
8		ablfac.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺)
9	8	grpbn0 18934	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
10	6, 7, 9	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ≠ ∅)
11		ablfac.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ Fin)
12		hashnncl 14320	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
14	10, 13	mpbird 258	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ)
15		dvdsle 16271	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℤ ∧ (♯‘𝐵) ∈ ℕ) → (𝑤 ∥ (♯‘𝐵) → 𝑤 ≤ (♯‘𝐵)))
16	5, 14, 15	syl2anr 603	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ) → (𝑤 ∥ (♯‘𝐵) → 𝑤 ≤ (♯‘𝐵)))
17	16	3impia 1123	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ≤ (♯‘𝐵))
18	14	nnzd 12542	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ)
19	18	3ad2ant1 1139	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → (♯‘𝐵) ∈ ℤ)
20		fznn 13538	. . . . . . . 8 ⊢ ((♯‘𝐵) ∈ ℤ → (𝑤 ∈ (1...(♯‘𝐵)) ↔ (𝑤 ∈ ℕ ∧ 𝑤 ≤ (♯‘𝐵))))
21	19, 20	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → (𝑤 ∈ (1...(♯‘𝐵)) ↔ (𝑤 ∈ ℕ ∧ 𝑤 ≤ (♯‘𝐵))))
22	4, 17, 21	mpbir2and 719	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ∈ (1...(♯‘𝐵)))
23	22	rabssdv 4006	. . . . 5 ⊢ (𝜑 → {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} ⊆ (1...(♯‘𝐵)))
24	2, 23	eqsstrid 3953	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ (1...(♯‘𝐵)))
25	1, 24	ssfid 9170	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
26		dfin5 3891	. . . . . . . 8 ⊢ (Word 𝐶 ∩ (𝑊‘(𝑆‘𝑞))) = {𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ (𝑊‘(𝑆‘𝑞))}
27		ablfac.o	. . . . . . . . . . . . . 14 ⊢ 𝑂 = (od‘𝐺)
28		ablfac.s	. . . . . . . . . . . . . 14 ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})
29	2	ssrab3 4014	. . . . . . . . . . . . . . 15 ⊢ 𝐴 ⊆ ℙ
30	29	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ ℙ)
31	8, 27, 28, 6, 11, 30	ablfac1b 20039	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
32	8	fvexi 6842	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 ∈ V
33	32	rabex 5268	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V
34	33, 28	dmmpti 6630	. . . . . . . . . . . . . 14 ⊢ dom 𝑆 = 𝐴
35	34	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝑆 = 𝐴)
36	31, 35	dprdf2 19976	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
37	36	ffvelcdmda 7026	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ (SubGrp‘𝐺))
38		ablfac.c	. . . . . . . . . . . 12 ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
39		ablfac.w	. . . . . . . . . . . 12 ⊢ 𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})
40	8, 38, 6, 11, 27, 2, 28, 39	ablfaclem1 20054	. . . . . . . . . . 11 ⊢ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) → (𝑊‘(𝑆‘𝑞)) = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))})
41	37, 40	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑊‘(𝑆‘𝑞)) = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))})
42		ssrab2 4012	. . . . . . . . . 10 ⊢ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))} ⊆ Word 𝐶
43	41, 42	eqsstrdi 3959	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑊‘(𝑆‘𝑞)) ⊆ Word 𝐶)
44		sseqin2 4153	. . . . . . . . 9 ⊢ ((𝑊‘(𝑆‘𝑞)) ⊆ Word 𝐶 ↔ (Word 𝐶 ∩ (𝑊‘(𝑆‘𝑞))) = (𝑊‘(𝑆‘𝑞)))
45	43, 44	sylib 219	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (Word 𝐶 ∩ (𝑊‘(𝑆‘𝑞))) = (𝑊‘(𝑆‘𝑞)))
46	26, 45	eqtr3id 2788	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ (𝑊‘(𝑆‘𝑞))} = (𝑊‘(𝑆‘𝑞)))
47	46, 41	eqtrd 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ (𝑊‘(𝑆‘𝑞))} = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))})
48		eqid 2739	. . . . . . . . 9 ⊢ (Base‘(𝐺 ↾s (𝑆‘𝑞))) = (Base‘(𝐺 ↾s (𝑆‘𝑞)))
49		eqid 2739	. . . . . . . . 9 ⊢ {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} = {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
50		eqid 2739	. . . . . . . . . . 11 ⊢ (𝐺 ↾s (𝑆‘𝑞)) = (𝐺 ↾s (𝑆‘𝑞))
51	50	subgabl 19803	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Abel ∧ (𝑆‘𝑞) ∈ (SubGrp‘𝐺)) → (𝐺 ↾s (𝑆‘𝑞)) ∈ Abel)
52	6, 37, 51	syl2an2r 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 ↾s (𝑆‘𝑞)) ∈ Abel)
53	30	sselda 3915	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℙ)
54	50	subgbas 19098	. . . . . . . . . . . . . 14 ⊢ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) → (𝑆‘𝑞) = (Base‘(𝐺 ↾s (𝑆‘𝑞))))
55	37, 54	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) = (Base‘(𝐺 ↾s (𝑆‘𝑞))))
56	55	fveq2d 6832	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) = (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))))
57	8, 27, 28, 6, 11, 30	ablfac1a 20038	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
58	56, 57	eqtr3d 2776	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
59	58	oveq2d 7373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞))))) = (𝑞 pCnt (𝑞↑(𝑞 pCnt (♯‘𝐵)))))
60	14	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘𝐵) ∈ ℕ)
61	53, 60	pccld 16813	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℕ0)
62	61	nn0zd 12541	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℤ)
63		pcid 16836	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℙ ∧ (𝑞 pCnt (♯‘𝐵)) ∈ ℤ) → (𝑞 pCnt (𝑞↑(𝑞 pCnt (♯‘𝐵)))) = (𝑞 pCnt (♯‘𝐵)))
64	53, 62, 63	syl2anc 590	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (𝑞↑(𝑞 pCnt (♯‘𝐵)))) = (𝑞 pCnt (♯‘𝐵)))
65	59, 64	eqtrd 2774	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞))))) = (𝑞 pCnt (♯‘𝐵)))
66	65	oveq2d 7373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))))) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
67	58, 66	eqtr4d 2777	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))) = (𝑞↑(𝑞 pCnt (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))))))
68	50	subggrp 19097	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) → (𝐺 ↾s (𝑆‘𝑞)) ∈ Grp)
69	37, 68	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 ↾s (𝑆‘𝑞)) ∈ Grp)
70	11	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐵 ∈ Fin)
71	8	subgss 19095	. . . . . . . . . . . . . 14 ⊢ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) → (𝑆‘𝑞) ⊆ 𝐵)
72	37, 71	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ⊆ 𝐵)
73	70, 72	ssfid 9170	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ Fin)
74	55, 73	eqeltrrd 2840	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (Base‘(𝐺 ↾s (𝑆‘𝑞))) ∈ Fin)
75	48	pgpfi2 19573	. . . . . . . . . . 11 ⊢ (((𝐺 ↾s (𝑆‘𝑞)) ∈ Grp ∧ (Base‘(𝐺 ↾s (𝑆‘𝑞))) ∈ Fin) → (𝑞 pGrp (𝐺 ↾s (𝑆‘𝑞)) ↔ (𝑞 ∈ ℙ ∧ (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))) = (𝑞↑(𝑞 pCnt (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))))))))
76	69, 74, 75	syl2anc 590	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pGrp (𝐺 ↾s (𝑆‘𝑞)) ↔ (𝑞 ∈ ℙ ∧ (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))) = (𝑞↑(𝑞 pCnt (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))))))))
77	53, 67, 76	mpbir2and 719	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 pGrp (𝐺 ↾s (𝑆‘𝑞)))
78	48, 49, 52, 77, 74	pgpfac 20053	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ∃𝑠 ∈ Word {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ((𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠 ∧ ((𝐺 ↾s (𝑆‘𝑞)) DProd 𝑠) = (Base‘(𝐺 ↾s (𝑆‘𝑞)))))
79		ssrab2 4012	. . . . . . . . . . . . . 14 ⊢ {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))
80		sswrd 14476	. . . . . . . . . . . . . 14 ⊢ ({𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) → Word {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))))
81	79, 80	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Word {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))
82	81	sseli 3911	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ Word {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} → 𝑠 ∈ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))))
83	37	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) → (𝑆‘𝑞) ∈ (SubGrp‘𝐺))
84	83	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) ∧ (𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠) → (𝑆‘𝑞) ∈ (SubGrp‘𝐺))
85	50	subgdmdprd 20003	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) → ((𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠 ↔ (𝐺dom DProd 𝑠 ∧ ran 𝑠 ⊆ 𝒫 (𝑆‘𝑞))))
86	83, 85	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) → ((𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠 ↔ (𝐺dom DProd 𝑠 ∧ ran 𝑠 ⊆ 𝒫 (𝑆‘𝑞))))
87	86	simprbda 499	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) ∧ (𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠) → 𝐺dom DProd 𝑠)
88	86	simplbda 500	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) ∧ (𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠) → ran 𝑠 ⊆ 𝒫 (𝑆‘𝑞))
89	50, 84, 87, 88	subgdprd 20004	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) ∧ (𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠) → ((𝐺 ↾s (𝑆‘𝑞)) DProd 𝑠) = (𝐺 DProd 𝑠))
90	55	ad2antrr 732	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) ∧ (𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠) → (𝑆‘𝑞) = (Base‘(𝐺 ↾s (𝑆‘𝑞))))
91	90	eqcomd 2745	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) ∧ (𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠) → (Base‘(𝐺 ↾s (𝑆‘𝑞))) = (𝑆‘𝑞))
92	89, 91	eqeq12d 2755	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) ∧ (𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠) → (((𝐺 ↾s (𝑆‘𝑞)) DProd 𝑠) = (Base‘(𝐺 ↾s (𝑆‘𝑞))) ↔ (𝐺 DProd 𝑠) = (𝑆‘𝑞)))
93	92	biimpd 230	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) ∧ (𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠) → (((𝐺 ↾s (𝑆‘𝑞)) DProd 𝑠) = (Base‘(𝐺 ↾s (𝑆‘𝑞))) → (𝐺 DProd 𝑠) = (𝑆‘𝑞)))
94	93, 87	jctild 530	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) ∧ (𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠) → (((𝐺 ↾s (𝑆‘𝑞)) DProd 𝑠) = (Base‘(𝐺 ↾s (𝑆‘𝑞))) → (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))))
95	94	expimpd 454	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) → (((𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠 ∧ ((𝐺 ↾s (𝑆‘𝑞)) DProd 𝑠) = (Base‘(𝐺 ↾s (𝑆‘𝑞)))) → (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))))
96	82, 95	sylan2 599	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}) → (((𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠 ∧ ((𝐺 ↾s (𝑆‘𝑞)) DProd 𝑠) = (Base‘(𝐺 ↾s (𝑆‘𝑞)))) → (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))))
97		oveq2 7365	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑦 → ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) = ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦))
98	97	eleq1d 2824	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑦 → (((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp ) ↔ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )))
99	98	cbvrabv 3401	. . . . . . . . . . . . . 14 ⊢ {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} = {𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )}
100	50	subsubg 19117	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) → (𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ↔ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑦 ⊆ (𝑆‘𝑞))))
101	37, 100	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ↔ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑦 ⊆ (𝑆‘𝑞))))
102	101	simprbda 499	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) → 𝑦 ∈ (SubGrp‘𝐺))
103	102	3adant3 1138	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∧ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )) → 𝑦 ∈ (SubGrp‘𝐺))
104	37	3ad2ant1 1139	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∧ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )) → (𝑆‘𝑞) ∈ (SubGrp‘𝐺))
105	101	simplbda 500	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) → 𝑦 ⊆ (𝑆‘𝑞))
106	105	3adant3 1138	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∧ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )) → 𝑦 ⊆ (𝑆‘𝑞))
107		ressabs 17210	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆‘𝑞) ∈ (SubGrp‘𝐺) ∧ 𝑦 ⊆ (𝑆‘𝑞)) → ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) = (𝐺 ↾s 𝑦))
108	104, 106, 107	syl2anc 590	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∧ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )) → ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) = (𝐺 ↾s 𝑦))
109		simp3 1144	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∧ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )) → ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp ))
110	108, 109	eqeltrrd 2840	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∧ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )) → (𝐺 ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp ))
111		oveq2 7365	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = 𝑦 → (𝐺 ↾s 𝑟) = (𝐺 ↾s 𝑦))
112	111	eleq1d 2824	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑦 → ((𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp ) ↔ (𝐺 ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )))
113	112, 38	elrab2 3632	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐶 ↔ (𝑦 ∈ (SubGrp‘𝐺) ∧ (𝐺 ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )))
114	103, 110, 113	sylanbrc 589	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∧ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )) → 𝑦 ∈ 𝐶)
115	114	rabssdv 4006	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )} ⊆ 𝐶)
116	99, 115	eqsstrid 3953	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ 𝐶)
117		sswrd 14476	. . . . . . . . . . . . 13 ⊢ ({𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ 𝐶 → Word {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ Word 𝐶)
118	116, 117	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → Word {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ Word 𝐶)
119	118	sselda 3915	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}) → 𝑠 ∈ Word 𝐶)
120	96, 119	jctild 530	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}) → (((𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠 ∧ ((𝐺 ↾s (𝑆‘𝑞)) DProd 𝑠) = (Base‘(𝐺 ↾s (𝑆‘𝑞)))) → (𝑠 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞)))))
121	120	expimpd 454	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑠 ∈ Word {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ∧ ((𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠 ∧ ((𝐺 ↾s (𝑆‘𝑞)) DProd 𝑠) = (Base‘(𝐺 ↾s (𝑆‘𝑞))))) → (𝑠 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞)))))
122	121	reximdv2 3149	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (∃𝑠 ∈ Word {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ((𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠 ∧ ((𝐺 ↾s (𝑆‘𝑞)) DProd 𝑠) = (Base‘(𝐺 ↾s (𝑆‘𝑞)))) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))))
123	78, 122	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞)))
124		rabn0 4318	. . . . . . 7 ⊢ ({𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))} ≠ ∅ ↔ ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞)))
125	123, 124	sylibr 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))} ≠ ∅)
126	47, 125	eqnetrd 3001	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ (𝑊‘(𝑆‘𝑞))} ≠ ∅)
127		rabn0 4318	. . . . 5 ⊢ ({𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ (𝑊‘(𝑆‘𝑞))} ≠ ∅ ↔ ∃𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆‘𝑞)))
128	126, 127	sylib 219	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ∃𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆‘𝑞)))
129	128	ralrimiva 3131	. . 3 ⊢ (𝜑 → ∀𝑞 ∈ 𝐴 ∃𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆‘𝑞)))
130		eleq1 2827	. . . 4 ⊢ (𝑦 = (𝑓‘𝑞) → (𝑦 ∈ (𝑊‘(𝑆‘𝑞)) ↔ (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))))
131	130	ac6sfi 9185	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑞 ∈ 𝐴 ∃𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆‘𝑞))) → ∃𝑓(𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))))
132	25, 129, 131	syl2anc 590	. 2 ⊢ (𝜑 → ∃𝑓(𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))))
133		sneq 4566	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → {𝑞} = {𝑦})
134		fveq2 6828	. . . . . . . . . 10 ⊢ (𝑞 = 𝑦 → (𝑓‘𝑞) = (𝑓‘𝑦))
135	134	dmeqd 5848	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → dom (𝑓‘𝑞) = dom (𝑓‘𝑦))
136	133, 135	xpeq12d 5650	. . . . . . . 8 ⊢ (𝑞 = 𝑦 → ({𝑞} × dom (𝑓‘𝑞)) = ({𝑦} × dom (𝑓‘𝑦)))
137	136	cbviunv 4969	. . . . . . 7 ⊢ ∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) = ∪ 𝑦 ∈ 𝐴 ({𝑦} × dom (𝑓‘𝑦))
138		snfi 8981	. . . . . . . . . 10 ⊢ {𝑦} ∈ Fin
139		simprl 776	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → 𝑓:𝐴⟶Word 𝐶)
140	139	ffvelcdmda 7026	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) ∧ 𝑦 ∈ 𝐴) → (𝑓‘𝑦) ∈ Word 𝐶)
141		wrdf 14472	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑦) ∈ Word 𝐶 → (𝑓‘𝑦):(0..^(♯‘(𝑓‘𝑦)))⟶𝐶)
142		fdm 6665	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑦):(0..^(♯‘(𝑓‘𝑦)))⟶𝐶 → dom (𝑓‘𝑦) = (0..^(♯‘(𝑓‘𝑦))))
143	140, 141, 142	3syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) ∧ 𝑦 ∈ 𝐴) → dom (𝑓‘𝑦) = (0..^(♯‘(𝑓‘𝑦))))
144		fzofi 13928	. . . . . . . . . . 11 ⊢ (0..^(♯‘(𝑓‘𝑦))) ∈ Fin
145	143, 144	eqeltrdi 2847	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) ∧ 𝑦 ∈ 𝐴) → dom (𝑓‘𝑦) ∈ Fin)
146		xpfi 9221	. . . . . . . . . 10 ⊢ (({𝑦} ∈ Fin ∧ dom (𝑓‘𝑦) ∈ Fin) → ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin)
147	138, 145, 146	sylancr 593	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) ∧ 𝑦 ∈ 𝐴) → ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin)
148	147	ralrimiva 3131	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → ∀𝑦 ∈ 𝐴 ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin)
149		iunfi 9244	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑦 ∈ 𝐴 ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin) → ∪ 𝑦 ∈ 𝐴 ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin)
150	25, 148, 149	syl2an2r 691	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → ∪ 𝑦 ∈ 𝐴 ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin)
151	137, 150	eqeltrid 2843	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → ∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) ∈ Fin)
152		hashcl 14310	. . . . . 6 ⊢ (∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) ∈ Fin → (♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))) ∈ ℕ0)
153		hashfzo0 14384	. . . . . 6 ⊢ ((♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))) ∈ ℕ0 → (♯‘(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))) = (♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))
154	151, 152, 153	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → (♯‘(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))) = (♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))
155		fzofi 13928	. . . . . 6 ⊢ (0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) ∈ Fin
156		hashen 14301	. . . . . 6 ⊢ (((0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) ∈ Fin ∧ ∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) ∈ Fin) → ((♯‘(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))) = (♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))) ↔ (0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) ≈ ∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))
157	155, 151, 156	sylancr 593	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → ((♯‘(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))) = (♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))) ↔ (0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) ≈ ∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))
158	154, 157	mpbid 233	. . . 4 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → (0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) ≈ ∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))
159		bren 8894	. . . 4 ⊢ ((0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) ≈ ∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) ↔ ∃ℎ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))–1-1-onto→∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))
160	158, 159	sylib 219	. . 3 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → ∃ℎ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))–1-1-onto→∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))
161	6	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))) ∧ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))–1-1-onto→∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) → 𝐺 ∈ Abel)
162	11	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))) ∧ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))–1-1-onto→∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) → 𝐵 ∈ Fin)
163		breq1 5076	. . . . . . . 8 ⊢ (𝑤 = 𝑎 → (𝑤 ∥ (♯‘𝐵) ↔ 𝑎 ∥ (♯‘𝐵)))
164	163	cbvrabv 3401	. . . . . . 7 ⊢ {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} = {𝑎 ∈ ℙ ∣ 𝑎 ∥ (♯‘𝐵)}
165	2, 164	eqtri 2762	. . . . . 6 ⊢ 𝐴 = {𝑎 ∈ ℙ ∣ 𝑎 ∥ (♯‘𝐵)}
166		fveq2 6828	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑐 → (𝑂‘𝑥) = (𝑂‘𝑐))
167	166	breq1d 5083	. . . . . . . . . 10 ⊢ (𝑥 = 𝑐 → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂‘𝑐) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))))
168	167	cbvrabv 3401	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}
169		id 22	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑏 → 𝑝 = 𝑏)
170		oveq1 7364	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑏 → (𝑝 pCnt (♯‘𝐵)) = (𝑏 pCnt (♯‘𝐵)))
171	169, 170	oveq12d 7375	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑏 → (𝑝↑(𝑝 pCnt (♯‘𝐵))) = (𝑏↑(𝑏 pCnt (♯‘𝐵))))
172	171	breq2d 5085	. . . . . . . . . 10 ⊢ (𝑝 = 𝑏 → ((𝑂‘𝑐) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂‘𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))))
173	172	rabbidv 3398	. . . . . . . . 9 ⊢ (𝑝 = 𝑏 → {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))})
174	168, 173	eqtrid 2786	. . . . . . . 8 ⊢ (𝑝 = 𝑏 → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))})
175	174	cbvmptv 5177	. . . . . . 7 ⊢ (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) = (𝑏 ∈ 𝐴 ↦ {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))})
176	28, 175	eqtri 2762	. . . . . 6 ⊢ 𝑆 = (𝑏 ∈ 𝐴 ↦ {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))})
177		breq2 5077	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑡))
178		oveq2 7365	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑡))
179	178	eqeq1d 2741	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → ((𝐺 DProd 𝑠) = 𝑔 ↔ (𝐺 DProd 𝑡) = 𝑔))
180	177, 179	anbi12d 638	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔) ↔ (𝐺dom DProd 𝑡 ∧ (𝐺 DProd 𝑡) = 𝑔)))
181	180	cbvrabv 3401	. . . . . . . 8 ⊢ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)} = {𝑡 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑡 ∧ (𝐺 DProd 𝑡) = 𝑔)}
182	181	mpteq2i 5169	. . . . . . 7 ⊢ (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)}) = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑡 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑡 ∧ (𝐺 DProd 𝑡) = 𝑔)})
183	39, 182	eqtri 2762	. . . . . 6 ⊢ 𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑡 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑡 ∧ (𝐺 DProd 𝑡) = 𝑔)})
184		simprll 784	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))) ∧ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))–1-1-onto→∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) → 𝑓:𝐴⟶Word 𝐶)
185		simprlr 785	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))) ∧ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))–1-1-onto→∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) → ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))
186		2fveq3 6833	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → (𝑊‘(𝑆‘𝑞)) = (𝑊‘(𝑆‘𝑦)))
187	134, 186	eleq12d 2833	. . . . . . . 8 ⊢ (𝑞 = 𝑦 → ((𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)) ↔ (𝑓‘𝑦) ∈ (𝑊‘(𝑆‘𝑦))))
188	187	cbvralvw 3217	. . . . . . 7 ⊢ (∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)) ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ (𝑊‘(𝑆‘𝑦)))
189	185, 188	sylib 219	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))) ∧ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))–1-1-onto→∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) → ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ (𝑊‘(𝑆‘𝑦)))
190		simprr 778	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))) ∧ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))–1-1-onto→∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) → ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))–1-1-onto→∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))
191	8, 38, 161, 162, 27, 165, 176, 183, 184, 189, 137, 190	ablfaclem2 20055	. . . . 5 ⊢ ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))) ∧ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))–1-1-onto→∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) → (𝑊‘𝐵) ≠ ∅)
192	191	expr 457	. . . 4 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → (ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))–1-1-onto→∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) → (𝑊‘𝐵) ≠ ∅))
193	192	exlimdv 1940	. . 3 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → (∃ℎ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))–1-1-onto→∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) → (𝑊‘𝐵) ≠ ∅))
194	160, 193	mpd 15	. 2 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → (𝑊‘𝐵) ≠ ∅)
195	132, 194	exlimddv 1942	1 ⊢ (𝜑 → (𝑊‘𝐵) ≠ ∅)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ∃wrex 3063  {crab 3391   ∩ cin 3882   ⊆ wss 3883  ∅c0 4262  𝒫 cpw 4530  {csn 4556  ∪ ciun 4922   class class class wbr 5073   ↦ cmpt 5154   × cxp 5617  dom cdm 5619  ran crn 5620  ⟶wf 6482  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7357   ≈ cen 8881  Fincfn 8884  0cc0 11030  1c1 11031   ≤ cle 11172  ℕcn 12166  ℕ₀cn0 12429  ℤcz 12516  ...cfz 13453  ..^cfzo 13600  ↑cexp 14015  ♯chash 14284  Word cword 14467   ∥ cdvds 16213  ℙcprime 16632   pCnt cpc 16799  Basecbs 17171   ↾_s cress 17192  Grpcgrp 18901  SubGrpcsubg 19088  odcod 19491   pGrp cpgp 19493  Abelcabl 19748  CycGrpccyg 19844   DProd cdprd 19962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-inf2 9554  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-int 4879  df-iun 4924  df-iin 4925  df-disj 5041  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7621  df-rpss 7667  df-om 7808  df-1st 7932  df-2nd 7933  df-supp 8102  df-tpos 8167  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-2o 8397  df-oadd 8400  df-omul 8401  df-er 8634  df-ec 8636  df-qs 8640  df-map 8766  df-ixp 8837  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-fsupp 9266  df-sup 9346  df-inf 9347  df-oi 9416  df-dju 9817  df-card 9855  df-acn 9858  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-div 11800  df-nn 12167  df-2 12236  df-3 12237  df-n0 12430  df-xnn0 12503  df-z 12517  df-uz 12781  df-q 12891  df-rp 12935  df-fz 13454  df-fzo 13601  df-fl 13743  df-mod 13821  df-seq 13956  df-exp 14016  df-fac 14228  df-bc 14257  df-hash 14285  df-word 14468  df-concat 14525  df-s1 14551  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-clim 15442  df-sum 15641  df-dvds 16214  df-gcd 16456  df-prm 16633  df-pc 16800  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17172  df-ress 17193  df-plusg 17225  df-0g 17396  df-gsum 17397  df-mre 17540  df-mrc 17541  df-acs 17543  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-mhm 18743  df-submnd 18744  df-grp 18904  df-minusg 18905  df-sbg 18906  df-mulg 19036  df-subg 19091  df-eqg 19093  df-ghm 19180  df-gim 19226  df-ga 19257  df-cntz 19284  df-oppg 19313  df-od 19495  df-gex 19496  df-pgp 19497  df-lsm 19603  df-pj1 19604  df-cmn 19749  df-abl 19750  df-cyg 19845  df-dprd 19964

This theorem is referenced by:  ablfac  20057