Theorem ablfaclem3 20075

Description: Lemma for ablfac 20076. (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 13996	. . . 4 ⊢ (𝜑 → (1...(♯‘𝐵)) ∈ Fin)
2		ablfac.a	. . . . 5 ⊢ 𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}
3		prmnn 16698	. . . . . . . 8 ⊢ (𝑤 ∈ ℙ → 𝑤 ∈ ℕ)
4	3	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ∈ ℕ)
5		prmz 16699	. . . . . . . . 9 ⊢ (𝑤 ∈ ℙ → 𝑤 ∈ ℤ)
6		ablfac.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ Abel)
7		ablgrp 19771	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
8		ablfac.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺)
9	8	grpbn0 18954	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
10	6, 7, 9	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ≠ ∅)
11		ablfac.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ Fin)
12		hashnncl 14389	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
14	10, 13	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ)
15		dvdsle 16334	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℤ ∧ (♯‘𝐵) ∈ ℕ) → (𝑤 ∥ (♯‘𝐵) → 𝑤 ≤ (♯‘𝐵)))
16	5, 14, 15	syl2anr 597	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ) → (𝑤 ∥ (♯‘𝐵) → 𝑤 ≤ (♯‘𝐵)))
17	16	3impia 1117	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ≤ (♯‘𝐵))
18	14	nnzd 12620	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ)
19	18	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → (♯‘𝐵) ∈ ℤ)
20		fznn 13614	. . . . . . . 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 713	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ∈ (1...(♯‘𝐵)))
23	22	rabssdv 4055	. . . . 5 ⊢ (𝜑 → {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} ⊆ (1...(♯‘𝐵)))
24	2, 23	eqsstrid 4002	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ (1...(♯‘𝐵)))
25	1, 24	ssfid 9278	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
26		dfin5 3939	. . . . . . . 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 4062	. . . . . . . . . . . . . . 15 ⊢ 𝐴 ⊆ ℙ
30	29	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ ℙ)
31	8, 27, 28, 6, 11, 30	ablfac1b 20058	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
32	8	fvexi 6895	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 ∈ V
33	32	rabex 5314	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V
34	33, 28	dmmpti 6687	. . . . . . . . . . . . . 14 ⊢ dom 𝑆 = 𝐴
35	34	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝑆 = 𝐴)
36	31, 35	dprdf2 19995	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
37	36	ffvelcdmda 7079	. . . . . . . . . . 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 20073	. . . . . . . . . . 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 4060	. . . . . . . . . 10 ⊢ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))} ⊆ Word 𝐶
43	41, 42	eqsstrdi 4008	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑊‘(𝑆‘𝑞)) ⊆ Word 𝐶)
44		sseqin2 4203	. . . . . . . . 9 ⊢ ((𝑊‘(𝑆‘𝑞)) ⊆ Word 𝐶 ↔ (Word 𝐶 ∩ (𝑊‘(𝑆‘𝑞))) = (𝑊‘(𝑆‘𝑞)))
45	43, 44	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (Word 𝐶 ∩ (𝑊‘(𝑆‘𝑞))) = (𝑊‘(𝑆‘𝑞)))
46	26, 45	eqtr3id 2785	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ (𝑊‘(𝑆‘𝑞))} = (𝑊‘(𝑆‘𝑞)))
47	46, 41	eqtrd 2771	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ (𝑊‘(𝑆‘𝑞))} = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))})
48		eqid 2736	. . . . . . . . 9 ⊢ (Base‘(𝐺 ↾s (𝑆‘𝑞))) = (Base‘(𝐺 ↾s (𝑆‘𝑞)))
49		eqid 2736	. . . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ (𝐺 ↾s (𝑆‘𝑞)) = (𝐺 ↾s (𝑆‘𝑞))
51	50	subgabl 19822	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Abel ∧ (𝑆‘𝑞) ∈ (SubGrp‘𝐺)) → (𝐺 ↾s (𝑆‘𝑞)) ∈ Abel)
52	6, 37, 51	syl2an2r 685	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 ↾s (𝑆‘𝑞)) ∈ Abel)
53	30	sselda 3963	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℙ)
54	50	subgbas 19118	. . . . . . . . . . . . . 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 6885	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) = (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))))
57	8, 27, 28, 6, 11, 30	ablfac1a 20057	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
58	56, 57	eqtr3d 2773	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
59	58	oveq2d 7426	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞))))) = (𝑞 pCnt (𝑞↑(𝑞 pCnt (♯‘𝐵)))))
60	14	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘𝐵) ∈ ℕ)
61	53, 60	pccld 16875	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℕ0)
62	61	nn0zd 12619	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℤ)
63		pcid 16898	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℙ ∧ (𝑞 pCnt (♯‘𝐵)) ∈ ℤ) → (𝑞 pCnt (𝑞↑(𝑞 pCnt (♯‘𝐵)))) = (𝑞 pCnt (♯‘𝐵)))
64	53, 62, 63	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (𝑞↑(𝑞 pCnt (♯‘𝐵)))) = (𝑞 pCnt (♯‘𝐵)))
65	59, 64	eqtrd 2771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞))))) = (𝑞 pCnt (♯‘𝐵)))
66	65	oveq2d 7426	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))))) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
67	58, 66	eqtr4d 2774	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))) = (𝑞↑(𝑞 pCnt (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))))))
68	50	subggrp 19117	. . . . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐵 ∈ Fin)
71	8	subgss 19115	. . . . . . . . . . . . . 14 ⊢ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) → (𝑆‘𝑞) ⊆ 𝐵)
72	37, 71	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ⊆ 𝐵)
73	70, 72	ssfid 9278	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ Fin)
74	55, 73	eqeltrrd 2836	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (Base‘(𝐺 ↾s (𝑆‘𝑞))) ∈ Fin)
75	48	pgpfi2 19592	. . . . . . . . . . 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 584	. . . . . . . . . 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 713	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 pGrp (𝐺 ↾s (𝑆‘𝑞)))
78	48, 49, 52, 77, 74	pgpfac 20072	. . . . . . . 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 4060	. . . . . . . . . . . . . 14 ⊢ {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))
80		sswrd 14545	. . . . . . . . . . . . . 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 3959	. . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) → (𝑆‘𝑞) ∈ (SubGrp‘𝐺))
84	83	adantr 480	. . . . . . . . . . . . . . . . 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 20022	. . . . . . . . . . . . . . . . . . 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 498	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) ∧ (𝐺 ↾s (𝑆‘𝑞))dom DProd 𝑠) → 𝐺dom DProd 𝑠)
88	86	simplbda 499	. . . . . . . . . . . . . . . . 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 20023	. . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . 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 525	. . . . . . . . . . . . 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 453	. . . . . . . . . . . 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 593	. . . . . . . . . . 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 7418	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑦 → ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) = ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦))
98	97	eleq1d 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑦 → (((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp ) ↔ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )))
99	98	cbvrabv 3431	. . . . . . . . . . . . . 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 19137	. . . . . . . . . . . . . . . . . . 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 498	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) → 𝑦 ∈ (SubGrp‘𝐺))
103	102	3adant3 1132	. . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . . . . 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 499	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))) → 𝑦 ⊆ (𝑆‘𝑞))
106	105	3adant3 1132	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∧ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )) → 𝑦 ⊆ (𝑆‘𝑞))
107		ressabs 17274	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆‘𝑞) ∈ (SubGrp‘𝐺) ∧ 𝑦 ⊆ (𝑆‘𝑞)) → ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) = (𝐺 ↾s 𝑦))
108	104, 106, 107	syl2anc 584	. . . . . . . . . . . . . . . . 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 1138	. . . . . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . . . . 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 7418	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = 𝑦 → (𝐺 ↾s 𝑟) = (𝐺 ↾s 𝑦))
112	111	eleq1d 2820	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑦 → ((𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp ) ↔ (𝐺 ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )))
113	112, 38	elrab2 3679	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐶 ↔ (𝑦 ∈ (SubGrp‘𝐺) ∧ (𝐺 ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )))
114	103, 110, 113	sylanbrc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑞 ∈ 𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∧ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )) → 𝑦 ∈ 𝐶)
115	114	rabssdv 4055	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )} ⊆ 𝐶)
116	99, 115	eqsstrid 4002	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ 𝐶)
117		sswrd 14545	. . . . . . . . . . . . 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 3963	. . . . . . . . . . 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 525	. . . . . . . . . 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 453	. . . . . . . . 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 3151	. . . . . . . 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 4369	. . . . . . 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 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))} ≠ ∅)
126	47, 125	eqnetrd 3000	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ (𝑊‘(𝑆‘𝑞))} ≠ ∅)
127		rabn0 4369	. . . . 5 ⊢ ({𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ (𝑊‘(𝑆‘𝑞))} ≠ ∅ ↔ ∃𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆‘𝑞)))
128	126, 127	sylib 218	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ∃𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆‘𝑞)))
129	128	ralrimiva 3133	. . 3 ⊢ (𝜑 → ∀𝑞 ∈ 𝐴 ∃𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆‘𝑞)))
130		eleq1 2823	. . . 4 ⊢ (𝑦 = (𝑓‘𝑞) → (𝑦 ∈ (𝑊‘(𝑆‘𝑞)) ↔ (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))))
131	130	ac6sfi 9297	. . 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 584	. 2 ⊢ (𝜑 → ∃𝑓(𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))))
133		sneq 4616	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → {𝑞} = {𝑦})
134		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑞 = 𝑦 → (𝑓‘𝑞) = (𝑓‘𝑦))
135	134	dmeqd 5890	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → dom (𝑓‘𝑞) = dom (𝑓‘𝑦))
136	133, 135	xpeq12d 5690	. . . . . . . 8 ⊢ (𝑞 = 𝑦 → ({𝑞} × dom (𝑓‘𝑞)) = ({𝑦} × dom (𝑓‘𝑦)))
137	136	cbviunv 5021	. . . . . . 7 ⊢ ∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) = ∪ 𝑦 ∈ 𝐴 ({𝑦} × dom (𝑓‘𝑦))
138		snfi 9062	. . . . . . . . . 10 ⊢ {𝑦} ∈ Fin
139		simprl 770	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → 𝑓:𝐴⟶Word 𝐶)
140	139	ffvelcdmda 7079	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) ∧ 𝑦 ∈ 𝐴) → (𝑓‘𝑦) ∈ Word 𝐶)
141		wrdf 14541	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑦) ∈ Word 𝐶 → (𝑓‘𝑦):(0..^(♯‘(𝑓‘𝑦)))⟶𝐶)
142		fdm 6720	. . . . . . . . . . . 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 13997	. . . . . . . . . . 11 ⊢ (0..^(♯‘(𝑓‘𝑦))) ∈ Fin
145	143, 144	eqeltrdi 2843	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) ∧ 𝑦 ∈ 𝐴) → dom (𝑓‘𝑦) ∈ Fin)
146		xpfi 9335	. . . . . . . . . 10 ⊢ (({𝑦} ∈ Fin ∧ dom (𝑓‘𝑦) ∈ Fin) → ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin)
147	138, 145, 146	sylancr 587	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) ∧ 𝑦 ∈ 𝐴) → ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin)
148	147	ralrimiva 3133	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → ∀𝑦 ∈ 𝐴 ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin)
149		iunfi 9360	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑦 ∈ 𝐴 ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin) → ∪ 𝑦 ∈ 𝐴 ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin)
150	25, 148, 149	syl2an2r 685	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → ∪ 𝑦 ∈ 𝐴 ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin)
151	137, 150	eqeltrid 2839	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → ∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) ∈ Fin)
152		hashcl 14379	. . . . . 6 ⊢ (∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) ∈ Fin → (♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))) ∈ ℕ0)
153		hashfzo0 14453	. . . . . 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 13997	. . . . . 6 ⊢ (0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) ∈ Fin
156		hashen 14370	. . . . . 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 587	. . . . 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 232	. . . 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 8974	. . . 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 218	. . 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 480	. . . . . 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 480	. . . . . 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 5127	. . . . . . . 8 ⊢ (𝑤 = 𝑎 → (𝑤 ∥ (♯‘𝐵) ↔ 𝑎 ∥ (♯‘𝐵)))
164	163	cbvrabv 3431	. . . . . . 7 ⊢ {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} = {𝑎 ∈ ℙ ∣ 𝑎 ∥ (♯‘𝐵)}
165	2, 164	eqtri 2759	. . . . . 6 ⊢ 𝐴 = {𝑎 ∈ ℙ ∣ 𝑎 ∥ (♯‘𝐵)}
166		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑐 → (𝑂‘𝑥) = (𝑂‘𝑐))
167	166	breq1d 5134	. . . . . . . . . 10 ⊢ (𝑥 = 𝑐 → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂‘𝑐) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))))
168	167	cbvrabv 3431	. . . . . . . . 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 7417	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑏 → (𝑝 pCnt (♯‘𝐵)) = (𝑏 pCnt (♯‘𝐵)))
171	169, 170	oveq12d 7428	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑏 → (𝑝↑(𝑝 pCnt (♯‘𝐵))) = (𝑏↑(𝑏 pCnt (♯‘𝐵))))
172	171	breq2d 5136	. . . . . . . . . 10 ⊢ (𝑝 = 𝑏 → ((𝑂‘𝑐) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂‘𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))))
173	172	rabbidv 3428	. . . . . . . . 9 ⊢ (𝑝 = 𝑏 → {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))})
174	168, 173	eqtrid 2783	. . . . . . . 8 ⊢ (𝑝 = 𝑏 → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))})
175	174	cbvmptv 5230	. . . . . . 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 2759	. . . . . 6 ⊢ 𝑆 = (𝑏 ∈ 𝐴 ↦ {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))})
177		breq2 5128	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑡))
178		oveq2 7418	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑡))
179	178	eqeq1d 2738	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → ((𝐺 DProd 𝑠) = 𝑔 ↔ (𝐺 DProd 𝑡) = 𝑔))
180	177, 179	anbi12d 632	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔) ↔ (𝐺dom DProd 𝑡 ∧ (𝐺 DProd 𝑡) = 𝑔)))
181	180	cbvrabv 3431	. . . . . . . 8 ⊢ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)} = {𝑡 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑡 ∧ (𝐺 DProd 𝑡) = 𝑔)}
182	181	mpteq2i 5222	. . . . . . 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 2759	. . . . . 6 ⊢ 𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑡 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑡 ∧ (𝐺 DProd 𝑡) = 𝑔)})
184		simprll 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)))) → 𝑓:𝐴⟶Word 𝐶)
185		simprlr 779	. . . . . . 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 6886	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → (𝑊‘(𝑆‘𝑞)) = (𝑊‘(𝑆‘𝑦)))
187	134, 186	eleq12d 2829	. . . . . . . 8 ⊢ (𝑞 = 𝑦 → ((𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)) ↔ (𝑓‘𝑦) ∈ (𝑊‘(𝑆‘𝑦))))
188	187	cbvralvw 3224	. . . . . . 7 ⊢ (∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)) ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ (𝑊‘(𝑆‘𝑦)))
189	185, 188	sylib 218	. . . . . 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 772	. . . . . 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 20074	. . . . 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 456	. . . 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 1933	. . 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 1935	1 ⊢ (𝜑 → (𝑊‘𝐵) ≠ ∅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  {crab 3420   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  𝒫 cpw 4580  {csn 4606  ∪ ciun 4972   class class class wbr 5124   ↦ cmpt 5206   × cxp 5657  dom cdm 5659  ran crn 5660  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410   ≈ cen 8961  Fincfn 8964  0cc0 11134  1c1 11135   ≤ cle 11275  ℕcn 12245  ℕ₀cn0 12506  ℤcz 12593  ...cfz 13529  ..^cfzo 13676  ↑cexp 14084  ♯chash 14353  Word cword 14536   ∥ cdvds 16277  ℙcprime 16695   pCnt cpc 16861  Basecbs 17233   ↾_s cress 17256  Grpcgrp 18921  SubGrpcsubg 19108  odcod 19510   pGrp cpgp 19512  Abelcabl 19767  CycGrpccyg 19863   DProd cdprd 19981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-disj 5092  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7676  df-rpss 7722  df-om 7867  df-1st 7993  df-2nd 7994  df-supp 8165  df-tpos 8230  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-omul 8490  df-er 8724  df-ec 8726  df-qs 8730  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9379  df-sup 9459  df-inf 9460  df-oi 9529  df-dju 9920  df-card 9958  df-acn 9961  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-div 11900  df-nn 12246  df-2 12308  df-3 12309  df-n0 12507  df-xnn0 12580  df-z 12594  df-uz 12858  df-q 12970  df-rp 13014  df-fz 13530  df-fzo 13677  df-fl 13814  df-mod 13892  df-seq 14025  df-exp 14085  df-fac 14297  df-bc 14326  df-hash 14354  df-word 14537  df-concat 14594  df-s1 14619  df-cj 15123  df-re 15124  df-im 15125  df-sqrt 15259  df-abs 15260  df-clim 15509  df-sum 15708  df-dvds 16278  df-gcd 16519  df-prm 16696  df-pc 16862  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-ress 17257  df-plusg 17289  df-0g 17460  df-gsum 17461  df-mre 17603  df-mrc 17604  df-acs 17606  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-mhm 18766  df-submnd 18767  df-grp 18924  df-minusg 18925  df-sbg 18926  df-mulg 19056  df-subg 19111  df-eqg 19113  df-ghm 19201  df-gim 19247  df-ga 19278  df-cntz 19305  df-oppg 19334  df-od 19514  df-gex 19515  df-pgp 19516  df-lsm 19622  df-pj1 19623  df-cmn 19768  df-abl 19769  df-cyg 19864  df-dprd 19983

This theorem is referenced by:  ablfac  20076