Theorem ablfaclem3 20336

Description: Lemma for ablfac 20337. (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 14146	. . . 4 ⊢ (𝜑 → (1...(♯‘𝐵)) ∈ Fin)
2		ablfac.a	. . . . 5 ⊢ 𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}
3		prmnn 16881	. . . . . . . 8 ⊢ (𝑤 ∈ ℙ → 𝑤 ∈ ℕ)
4	3	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ∈ ℕ)
5		prmz 16882	. . . . . . . . 9 ⊢ (𝑤 ∈ ℙ → 𝑤 ∈ ℤ)
6		ablfac.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ Abel)
7		ablgrp 20032	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
8		ablfac.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺)
9	8	grpbn0 19209	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
10	6, 7, 9	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ≠ ∅)
11		ablfac.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ Fin)
12		hashnncl 14540	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
13	11, 12	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
14	10, 13	mpbird 260	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ)
15		dvdsle 16512	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℤ ∧ (♯‘𝐵) ∈ ℕ) → (𝑤 ∥ (♯‘𝐵) → 𝑤 ≤ (♯‘𝐵)))
16	5, 14, 15	syl2anr 609	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ) → (𝑤 ∥ (♯‘𝐵) → 𝑤 ≤ (♯‘𝐵)))
17	16	3impia 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ≤ (♯‘𝐵))
18	14	nnzd 12749	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ)
19	18	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → (♯‘𝐵) ∈ ℤ)
20		fznn 13756	. . . . . . . 8 ⊢ ((♯‘𝐵) ∈ ℤ → (𝑤 ∈ (1...(♯‘𝐵)) ↔ (𝑤 ∈ ℕ ∧ 𝑤 ≤ (♯‘𝐵))))
21	19, 20	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → (𝑤 ∈ (1...(♯‘𝐵)) ↔ (𝑤 ∈ ℕ ∧ 𝑤 ≤ (♯‘𝐵))))
22	4, 17, 21	mpbir2and 726	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ∈ (1...(♯‘𝐵)))
23	22	rabssdv 4002	. . . . 5 ⊢ (𝜑 → {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} ⊆ (1...(♯‘𝐵)))
24	2, 23	eqsstrid 3951	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ (1...(♯‘𝐵)))
25	1, 24	ssfid 9291	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
26		dfin5 4095	. . . . . . . 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 4010	. . . . . . . . . . . . . . 15 ⊢ 𝐴 ⊆ ℙ
30	29	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ ℙ)
31	8, 27, 28, 6, 11, 30	ablfac1b 20319	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
32	8	fvexi 6925	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 ∈ V
33	32	rabex 5305	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V
34	33, 28	dmmpti 6708	. . . . . . . . . . . . . 14 ⊢ dom 𝑆 = 𝐴
35	34	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝑆 = 𝐴)
36	31, 35	dprdf2 20256	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
37	36	ffvelcdmda 7111	. . . . . . . . . . 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 20334	. . . . . . . . . . 11 ⊢ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) → (𝑊‘(𝑆‘𝑞)) = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))})
41	37, 40	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑊‘(𝑆‘𝑞)) = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))})
42		ssrab2 4008	. . . . . . . . . 10 ⊢ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))} ⊆ Word 𝐶
43	41, 42	eqsstrdi 3957	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑊‘(𝑆‘𝑞)) ⊆ Word 𝐶)
44		sseqin2 4123	. . . . . . . . 9 ⊢ ((𝑊‘(𝑆‘𝑞)) ⊆ Word 𝐶 ↔ (Word 𝐶 ∩ (𝑊‘(𝑆‘𝑞))) = (𝑊‘(𝑆‘𝑞)))
45	43, 44	sylib 221	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (Word 𝐶 ∩ (𝑊‘(𝑆‘𝑞))) = (𝑊‘(𝑆‘𝑞)))
46	26, 45	eqtr3id 2812	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ (𝑊‘(𝑆‘𝑞))} = (𝑊‘(𝑆‘𝑞)))
47	46, 41	eqtrd 2798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ (𝑊‘(𝑆‘𝑞))} = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))})
48		eqid 2763	. . . . . . . . 9 ⊢ (Base‘(𝐺 ↾s (𝑆‘𝑞))) = (Base‘(𝐺 ↾s (𝑆‘𝑞)))
49		eqid 2763	. . . . . . . . 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 2763	. . . . . . . . . . 11 ⊢ (𝐺 ↾s (𝑆‘𝑞)) = (𝐺 ↾s (𝑆‘𝑞))
51	50	subgabl 20083	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Abel ∧ (𝑆‘𝑞) ∈ (SubGrp‘𝐺)) → (𝐺 ↾s (𝑆‘𝑞)) ∈ Abel)
52	6, 37, 51	syl2an2r 698	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 ↾s (𝑆‘𝑞)) ∈ Abel)
53	30	sselda 3913	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℙ)
54	50	subgbas 19372	. . . . . . . . . . . . . 14 ⊢ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) → (𝑆‘𝑞) = (Base‘(𝐺 ↾s (𝑆‘𝑞))))
55	37, 54	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) = (Base‘(𝐺 ↾s (𝑆‘𝑞))))
56	55	fveq2d 6915	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) = (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))))
57	8, 27, 28, 6, 11, 30	ablfac1a 20318	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
58	56, 57	eqtr3d 2800	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
59	58	oveq2d 7462	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞))))) = (𝑞 pCnt (𝑞↑(𝑞 pCnt (♯‘𝐵)))))
60	14	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘𝐵) ∈ ℕ)
61	53, 60	pccld 17060	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℕ0)
62	61	nn0zd 12748	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℤ)
63		pcid 17083	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℙ ∧ (𝑞 pCnt (♯‘𝐵)) ∈ ℤ) → (𝑞 pCnt (𝑞↑(𝑞 pCnt (♯‘𝐵)))) = (𝑞 pCnt (♯‘𝐵)))
64	53, 62, 63	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (𝑞↑(𝑞 pCnt (♯‘𝐵)))) = (𝑞 pCnt (♯‘𝐵)))
65	59, 64	eqtrd 2798	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞))))) = (𝑞 pCnt (♯‘𝐵)))
66	65	oveq2d 7462	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))))) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
67	58, 66	eqtr4d 2801	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))) = (𝑞↑(𝑞 pCnt (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))))))
68	50	subggrp 19371	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) → (𝐺 ↾s (𝑆‘𝑞)) ∈ Grp)
69	37, 68	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 ↾s (𝑆‘𝑞)) ∈ Grp)
70	11	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐵 ∈ Fin)
71	8	subgss 19369	. . . . . . . . . . . . . 14 ⊢ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) → (𝑆‘𝑞) ⊆ 𝐵)
72	37, 71	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ⊆ 𝐵)
73	70, 72	ssfid 9291	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ Fin)
74	55, 73	eqeltrrd 2865	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (Base‘(𝐺 ↾s (𝑆‘𝑞))) ∈ Fin)
75	48	pgpfi2 19853	. . . . . . . . . . 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 596	. . . . . . . . . 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 726	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 pGrp (𝐺 ↾s (𝑆‘𝑞)))
78	48, 49, 52, 77, 74	pgpfac 20333	. . . . . . . 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 4008	. . . . . . . . . . . . . 14 ⊢ {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))
80		sswrd 14697	. . . . . . . . . . . . . 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 3909	. . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) → (𝑆‘𝑞) ∈ (SubGrp‘𝐺))
84	83	adantr 486	. . . . . . . . . . . . . . . . 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 20283	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) → ((𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠 ↔ (𝐺dom DProd 𝑠 ∧ ran 𝑠 ⊆ 𝒫 (𝑆‘𝑞))))
86	83, 85	syl 18	. . . . . . . . . . . . . . . . . 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 504	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) ∧ (𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠) → 𝐺dom DProd 𝑠)
88	86	simplbda 505	. . . . . . . . . . . . . . . . 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 20284	. . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 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 535	. . . . . . . . . . . . 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 459	. . . . . . . . . . . 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 605	. . . . . . . . . . 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 7454	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑦 → ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) = ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦))
98	97	eleq1d 2849	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑦 → (((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp ) ↔ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )))
99	98	cbvrabv 3426	. . . . . . . . . . . . . 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 19392	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) → (𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ↔ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑦 ⊆ (𝑆‘𝑞))))
101	37, 100	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ↔ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑦 ⊆ (𝑆‘𝑞))))
102	101	simprbda 504	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) → 𝑦 ∈ (SubGrp‘𝐺))
103	102	3adant3 1150	. . . . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . . . . . . 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 505	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) → 𝑦 ⊆ (𝑆‘𝑞))
106	105	3adant3 1150	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∧ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )) → 𝑦 ⊆ (𝑆‘𝑞))
107		ressabs 17458	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆‘𝑞) ∈ (SubGrp‘𝐺) ∧ 𝑦 ⊆ (𝑆‘𝑞)) → ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) = (𝐺 ↾s 𝑦))
108	104, 106, 107	syl2anc 596	. . . . . . . . . . . . . . . . 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 1156	. . . . . . . . . . . . . . . . 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 2865	. . . . . . . . . . . . . . . 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 7454	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = 𝑦 → (𝐺 ↾s 𝑟) = (𝐺 ↾s 𝑦))
112	111	eleq1d 2849	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑦 → ((𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp ) ↔ (𝐺 ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )))
113	112, 38	elrab2 3652	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐶 ↔ (𝑦 ∈ (SubGrp‘𝐺) ∧ (𝐺 ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )))
114	103, 110, 113	sylanbrc 595	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∧ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )) → 𝑦 ∈ 𝐶)
115	114	rabssdv 4002	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )} ⊆ 𝐶)
116	99, 115	eqsstrid 3951	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ 𝐶)
117		sswrd 14697	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 3913	. . . . . . . . . . 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 535	. . . . . . . . . 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 459	. . . . . . . . 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 3176	. . . . . . . 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 16	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞)))
124		rabn0 4342	. . . . . . 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 237	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))} ≠ ∅)
126	47, 125	eqnetrd 3026	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ (𝑊‘(𝑆‘𝑞))} ≠ ∅)
127		rabn0 4342	. . . . 5 ⊢ ({𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ (𝑊‘(𝑆‘𝑞))} ≠ ∅ ↔ ∃𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆‘𝑞)))
128	126, 127	sylib 221	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ∃𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆‘𝑞)))
129	128	ralrimiva 3158	. . 3 ⊢ (𝜑 → ∀𝑞 ∈ 𝐴 ∃𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆‘𝑞)))
130		eleq1 2852	. . . 4 ⊢ (𝑦 = (𝑓‘𝑞) → (𝑦 ∈ (𝑊‘(𝑆‘𝑞)) ↔ (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))))
131	130	ac6sfi 9306	. . 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 596	. 2 ⊢ (𝜑 → ∃𝑓(𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))))
133		sneq 4594	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → {𝑞} = {𝑦})
134		fveq2 6911	. . . . . . . . . 10 ⊢ (𝑞 = 𝑦 → (𝑓‘𝑞) = (𝑓‘𝑦))
135	134	dmeqd 5895	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → dom (𝑓‘𝑞) = dom (𝑓‘𝑦))
136	133, 135	xpeq12d 5687	. . . . . . . 8 ⊢ (𝑞 = 𝑦 → ({𝑞} × dom (𝑓‘𝑞)) = ({𝑦} × dom (𝑓‘𝑦)))
137	136	cbviunv 5000	. . . . . . 7 ⊢ ∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) = ∪ 𝑦 ∈ 𝐴 ({𝑦} × dom (𝑓‘𝑦))
138		snfi 9102	. . . . . . . . . 10 ⊢ {𝑦} ∈ Fin
139		simprl 783	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → 𝑓:𝐴⟶Word 𝐶)
140	139	ffvelcdmda 7111	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) ∧ 𝑦 ∈ 𝐴) → (𝑓‘𝑦) ∈ Word 𝐶)
141		wrdf 14693	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑦) ∈ Word 𝐶 → (𝑓‘𝑦):(0..^(♯‘(𝑓‘𝑦)))⟶𝐶)
142		fdm 6744	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑦):(0..^(♯‘(𝑓‘𝑦)))⟶𝐶 → dom (𝑓‘𝑦) = (0..^(♯‘(𝑓‘𝑦))))
143	140, 141, 142	3syl 19	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) ∧ 𝑦 ∈ 𝐴) → dom (𝑓‘𝑦) = (0..^(♯‘(𝑓‘𝑦))))
144		fzofi 14147	. . . . . . . . . . 11 ⊢ (0..^(♯‘(𝑓‘𝑦))) ∈ Fin
145	143, 144	eqeltrdi 2872	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) ∧ 𝑦 ∈ 𝐴) → dom (𝑓‘𝑦) ∈ Fin)
146		xpfi 9342	. . . . . . . . . 10 ⊢ (({𝑦} ∈ Fin ∧ dom (𝑓‘𝑦) ∈ Fin) → ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin)
147	138, 145, 146	sylancr 599	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) ∧ 𝑦 ∈ 𝐴) → ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin)
148	147	ralrimiva 3158	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → ∀𝑦 ∈ 𝐴 ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin)
149		iunfi 9363	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑦 ∈ 𝐴 ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin) → ∪ 𝑦 ∈ 𝐴 ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin)
150	25, 148, 149	syl2an2r 698	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → ∪ 𝑦 ∈ 𝐴 ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin)
151	137, 150	eqeltrid 2868	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → ∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) ∈ Fin)
152		hashcl 14530	. . . . . 6 ⊢ (∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) ∈ Fin → (♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))) ∈ ℕ0)
153		hashfzo0 14605	. . . . . 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 19	. . . . 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 14147	. . . . . 6 ⊢ (0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) ∈ Fin
156		hashen 14521	. . . . . 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 599	. . . . 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 235	. . . 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 9014	. . . 4 ⊢ ((0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) ≈ ∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) ↔ ∃ℎ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))⟶1-1onto∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))
160	158, 159	sylib 221	. . 3 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → ∃ℎ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))⟶1-1onto∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))
161	6	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))) ∧ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))⟶1-1onto∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) → 𝐺 ∈ Abel)
162	11	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))) ∧ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))⟶1-1onto∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) → 𝐵 ∈ Fin)
163		breq1 5109	. . . . . . . 8 ⊢ (𝑤 = 𝑎 → (𝑤 ∥ (♯‘𝐵) ↔ 𝑎 ∥ (♯‘𝐵)))
164	163	cbvrabv 3426	. . . . . . 7 ⊢ {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} = {𝑎 ∈ ℙ ∣ 𝑎 ∥ (♯‘𝐵)}
165	2, 164	eqtri 2786	. . . . . 6 ⊢ 𝐴 = {𝑎 ∈ ℙ ∣ 𝑎 ∥ (♯‘𝐵)}
166		fveq2 6911	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑐 → (𝑂‘𝑥) = (𝑂‘𝑐))
167	166	breq1d 5116	. . . . . . . . . 10 ⊢ (𝑥 = 𝑐 → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂‘𝑐) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))))
168	167	cbvrabv 3426	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}
169		id 23	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑏 → 𝑝 = 𝑏)
170		oveq1 7453	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑏 → (𝑝 pCnt (♯‘𝐵)) = (𝑏 pCnt (♯‘𝐵)))
171	169, 170	oveq12d 7464	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑏 → (𝑝↑(𝑝 pCnt (♯‘𝐵))) = (𝑏↑(𝑏 pCnt (♯‘𝐵))))
172	171	breq2d 5118	. . . . . . . . . 10 ⊢ (𝑝 = 𝑏 → ((𝑂‘𝑐) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂‘𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))))
173	172	rabbidv 3423	. . . . . . . . 9 ⊢ (𝑝 = 𝑏 → {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))})
174	168, 173	eqtrid 2810	. . . . . . . 8 ⊢ (𝑝 = 𝑏 → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))})
175	174	cbvmptv 5213	. . . . . . 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 2786	. . . . . 6 ⊢ 𝑆 = (𝑏 ∈ 𝐴 ↦ {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))})
177		breq2 5110	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑡))
178		oveq2 7454	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑡))
179	178	eqeq1d 2765	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → ((𝐺 DProd 𝑠) = 𝑔 ↔ (𝐺 DProd 𝑡) = 𝑔))
180	177, 179	anbi12d 644	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔) ↔ (𝐺dom DProd 𝑡 ∧ (𝐺 DProd 𝑡) = 𝑔)))
181	180	cbvrabv 3426	. . . . . . . 8 ⊢ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)} = {𝑡 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑡 ∧ (𝐺 DProd 𝑡) = 𝑔)}
182	181	mpteq2i 5205	. . . . . . 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 2786	. . . . . 6 ⊢ 𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑡 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑡 ∧ (𝐺 DProd 𝑡) = 𝑔)})
184		simprll 791	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))) ∧ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))⟶1-1onto∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) → 𝑓:𝐴⟶Word 𝐶)
185		simprlr 792	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))) ∧ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))⟶1-1onto∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) → ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))
186		2fveq3 6916	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → (𝑊‘(𝑆‘𝑞)) = (𝑊‘(𝑆‘𝑦)))
187	134, 186	eleq12d 2858	. . . . . . . 8 ⊢ (𝑞 = 𝑦 → ((𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)) ↔ (𝑓‘𝑦) ∈ (𝑊‘(𝑆‘𝑦))))
188	187	cbvralvw 3244	. . . . . . 7 ⊢ (∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)) ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ (𝑊‘(𝑆‘𝑦)))
189	185, 188	sylib 221	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))) ∧ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))⟶1-1onto∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) → ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ (𝑊‘(𝑆‘𝑦)))
190		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))) ∧ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))⟶1-1onto∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) → ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))⟶1-1onto∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))
191	8, 38, 161, 162, 27, 165, 176, 183, 184, 189, 137, 190	ablfaclem2 20335	. . . . 5 ⊢ ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))) ∧ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))⟶1-1onto∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) → (𝑊‘𝐵) ≠ ∅)
192	191	expr 462	. . . 4 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → (ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))⟶1-1onto∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) → (𝑊‘𝐵) ≠ ∅))
193	192	exlimdv 1968	. . 3 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → (∃ℎ ℎ:(0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))))⟶1-1onto∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) → (𝑊‘𝐵) ≠ ∅))
194	160, 193	mpd 16	. 2 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → (𝑊‘𝐵) ≠ ∅)
195	132, 194	exlimddv 1970	1 ⊢ (𝜑 → (𝑊‘𝐵) ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  𝒫 cpw 4560  {csn 4588  ∪ ciun 4954   class class class wbr 5106   ↦ cmpt 5190   × cxp 5654  dom cdm 5656  ran crn 5657  ⟶wf 6557  ⟶1-1ontowf1o 6560  ‘cfv 6561  (class class class)co 7446   ≈ cen 9001  Fincfn 9004  0cc0 11230  1c1 11231   ≤ cle 11374  ℕcn 12365  ℕ₀cn0 12636  ℤcz 12723  ...cfz 13669  ..^cfzo 13818  ↑cexp 14234  ♯chash 14504  Word cword 14688   ∥ cdvds 16454  ℙcprime 16878   pCnt cpc 17046  Basecbs 17419   ↾_s cress 17440  Grpcgrp 19176  SubGrpcsubg 19362  odcod 19771   pGrp cpgp 19773  Abelcabl 20028  CycGrpccyg 20124   DProd cdprd 20242

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-inf2 9672  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307  ax-pre-sup 11308

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-disj 5074  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-of 7720  df-rpss 7766  df-om 7905  df-1st 8028  df-2nd 8029  df-supp 8208  df-tpos 8274  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-2o 8508  df-oadd 8511  df-omul 8512  df-er 8748  df-ec 8750  df-qs 8754  df-map 8880  df-ixp 8957  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-fsupp 9385  df-sup 9464  df-inf 9465  df-oi 9534  df-dju 10012  df-card 10050  df-acn 10053  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-div 12004  df-nn 12366  df-2 12435  df-3 12436  df-n0 12637  df-xnn0 12710  df-z 12724  df-uz 12996  df-q 13106  df-rp 13151  df-fz 13670  df-fzo 13819  df-fl 13962  df-mod 14040  df-seq 14175  df-exp 14235  df-fac 14448  df-bc 14477  df-hash 14505  df-word 14689  df-concat 14746  df-s1 14773  df-cj 15298  df-re 15299  df-im 15300  df-sqrt 15434  df-abs 15435  df-clim 15687  df-sum 15886  df-dvds 16455  df-gcd 16697  df-prm 16879  df-pc 17047  df-sets 17374  df-slot 17392  df-ndx 17404  df-base 17420  df-ress 17441  df-plusg 17473  df-0g 17644  df-gsum 17645  df-mre 17788  df-mrc 17789  df-acs 17791  df-mgm 18848  df-sgrp 18940  df-mnd 18956  df-mhm 19010  df-submnd 19011  df-grp 19179  df-minusg 19180  df-sbg 19181  df-mulg 19310  df-subg 19365  df-eqg 19367  df-ghm 19460  df-gim 19505  df-ga 19537  df-cntz 19564  df-oppg 19593  df-od 19775  df-gex 19776  df-pgp 19777  df-lsm 19883  df-pj1 19884  df-cmn 20029  df-abl 20030  df-cyg 20125  df-dprd 20244

This theorem is used by:  ablfac  20337