Theorem ablfaclem3 20308

Description: Lemma for ablfac 20309. (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 14119	. . . 4 ⊢ (𝜑 → (1...(♯‘𝐵)) ∈ Fin)
2		ablfac.a	. . . . 5 ⊢ 𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}
3		prmnn 16854	. . . . . . . 8 ⊢ (𝑤 ∈ ℙ → 𝑤 ∈ ℕ)
4	3	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ∈ ℕ)
5		prmz 16855	. . . . . . . . 9 ⊢ (𝑤 ∈ ℙ → 𝑤 ∈ ℤ)
6		ablfac.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ Abel)
7		ablgrp 20004	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
8		ablfac.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺)
9	8	grpbn0 19182	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
10	6, 7, 9	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ≠ ∅)
11		ablfac.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ Fin)
12		hashnncl 14513	. . . . . . . . . . 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 16485	. . . . . . . . 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 12722	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ)
19	18	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → (♯‘𝐵) ∈ ℤ)
20		fznn 13729	. . . . . . . 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 3999	. . . . 5 ⊢ (𝜑 → {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} ⊆ (1...(♯‘𝐵)))
24	2, 23	eqsstrid 3946	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ (1...(♯‘𝐵)))
25	1, 24	ssfid 9263	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
26		dfin5 4092	. . . . . . . 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 4007	. . . . . . . . . . . . . . 15 ⊢ 𝐴 ⊆ ℙ
30	29	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ ℙ)
31	8, 27, 28, 6, 11, 30	ablfac1b 20291	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
32	8	fvexi 6899	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 ∈ V
33	32	rabex 5299	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V
34	33, 28	dmmpti 6683	. . . . . . . . . . . . . 14 ⊢ dom 𝑆 = 𝐴
35	34	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝑆 = 𝐴)
36	31, 35	dprdf2 20228	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
37	36	ffvelcdmda 7084	. . . . . . . . . . 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 20306	. . . . . . . . . . 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 4005	. . . . . . . . . 10 ⊢ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))} ⊆ Word 𝐶
43	41, 42	eqsstrdi 3952	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑊‘(𝑆‘𝑞)) ⊆ Word 𝐶)
44		sseqin2 4120	. . . . . . . . 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 2809	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ (𝑊‘(𝑆‘𝑞))} = (𝑊‘(𝑆‘𝑞)))
47	46, 41	eqtrd 2795	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ (𝑊‘(𝑆‘𝑞))} = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆‘𝑞))})
48		eqid 2760	. . . . . . . . 9 ⊢ (Base‘(𝐺 ↾s (𝑆‘𝑞))) = (Base‘(𝐺 ↾s (𝑆‘𝑞)))
49		eqid 2760	. . . . . . . . 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 2760	. . . . . . . . . . 11 ⊢ (𝐺 ↾s (𝑆‘𝑞)) = (𝐺 ↾s (𝑆‘𝑞))
51	50	subgabl 20055	. . . . . . . . . 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 3908	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℙ)
54	50	subgbas 19345	. . . . . . . . . . . . . 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 6889	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) = (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))))
57	8, 27, 28, 6, 11, 30	ablfac1a 20290	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑆‘𝑞)) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
58	56, 57	eqtr3d 2797	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
59	58	oveq2d 7436	. . . . . . . . . . . . 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 17033	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℕ0)
62	61	nn0zd 12721	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℤ)
63		pcid 17056	. . . . . . . . . . . . . 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 2795	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞))))) = (𝑞 pCnt (♯‘𝐵)))
66	65	oveq2d 7436	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))))) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
67	58, 66	eqtr4d 2798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))) = (𝑞↑(𝑞 pCnt (♯‘(Base‘(𝐺 ↾s (𝑆‘𝑞)))))))
68	50	subggrp 19344	. . . . . . . . . . . 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 19342	. . . . . . . . . . . . . 14 ⊢ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) → (𝑆‘𝑞) ⊆ 𝐵)
72	37, 71	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ⊆ 𝐵)
73	70, 72	ssfid 9263	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ Fin)
74	55, 73	eqeltrrd 2861	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (Base‘(𝐺 ↾s (𝑆‘𝑞))) ∈ Fin)
75	48	pgpfi2 19825	. . . . . . . . . . 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 20305	. . . . . . . 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 4005	. . . . . . . . . . . . . 14 ⊢ {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞)))
80		sswrd 14670	. . . . . . . . . . . . . 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 3904	. . . . . . . . . . . 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 20255	. . . . . . . . . . . . . . . . . . 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 20256	. . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑦 → ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) = ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦))
98	97	eleq1d 2845	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑦 → (((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp ) ↔ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )))
99	98	cbvrabv 3422	. . . . . . . . . . . . . 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 19365	. . . . . . . . . . . . . . . . . . 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 17431	. . . . . . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = 𝑦 → (𝐺 ↾s 𝑟) = (𝐺 ↾s 𝑦))
112	111	eleq1d 2845	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑦 → ((𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp ) ↔ (𝐺 ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )))
113	112, 38	elrab2 3648	. . . . . . . . . . . . . . . 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 3999	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑦 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )} ⊆ 𝐶)
116	99, 115	eqsstrid 3946	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑟 ∈ (SubGrp‘(𝐺 ↾s (𝑆‘𝑞))) ∣ ((𝐺 ↾s (𝑆‘𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ 𝐶)
117		sswrd 14670	. . . . . . . . . . . . 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 3908	. . . . . . . . . . 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 3172	. . . . . . . 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 4338	. . . . . . 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 3022	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → {𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ (𝑊‘(𝑆‘𝑞))} ≠ ∅)
127		rabn0 4338	. . . . 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 3154	. . 3 ⊢ (𝜑 → ∀𝑞 ∈ 𝐴 ∃𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆‘𝑞)))
130		eleq1 2848	. . . 4 ⊢ (𝑦 = (𝑓‘𝑞) → (𝑦 ∈ (𝑊‘(𝑆‘𝑞)) ↔ (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞))))
131	130	ac6sfi 9278	. . 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 4590	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → {𝑞} = {𝑦})
134		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑞 = 𝑦 → (𝑓‘𝑞) = (𝑓‘𝑦))
135	134	dmeqd 5886	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → dom (𝑓‘𝑞) = dom (𝑓‘𝑦))
136	133, 135	xpeq12d 5681	. . . . . . . 8 ⊢ (𝑞 = 𝑦 → ({𝑞} × dom (𝑓‘𝑞)) = ({𝑦} × dom (𝑓‘𝑦)))
137	136	cbviunv 4996	. . . . . . 7 ⊢ ∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) = ∪ 𝑦 ∈ 𝐴 ({𝑦} × dom (𝑓‘𝑦))
138		snfi 9074	. . . . . . . . . 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 7084	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) ∧ 𝑦 ∈ 𝐴) → (𝑓‘𝑦) ∈ Word 𝐶)
141		wrdf 14666	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑦) ∈ Word 𝐶 → (𝑓‘𝑦):(0..^(♯‘(𝑓‘𝑦)))⟶𝐶)
142		fdm 6719	. . . . . . . . . . . 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 14120	. . . . . . . . . . 11 ⊢ (0..^(♯‘(𝑓‘𝑦))) ∈ Fin
145	143, 144	eqeltrdi 2868	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) ∧ 𝑦 ∈ 𝐴) → dom (𝑓‘𝑦) ∈ Fin)
146		xpfi 9314	. . . . . . . . . 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 3154	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → ∀𝑦 ∈ 𝐴 ({𝑦} × dom (𝑓‘𝑦)) ∈ Fin)
149		iunfi 9335	. . . . . . . 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 2864	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → ∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) ∈ Fin)
152		hashcl 14503	. . . . . 6 ⊢ (∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)) ∈ Fin → (♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞))) ∈ ℕ0)
153		hashfzo0 14578	. . . . . 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 14120	. . . . . 6 ⊢ (0..^(♯‘∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) ∈ Fin
156		hashen 14494	. . . . . 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 8986	. . . 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 221	. . 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 486	. . . . . 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 486	. . . . . 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 5105	. . . . . . . 8 ⊢ (𝑤 = 𝑎 → (𝑤 ∥ (♯‘𝐵) ↔ 𝑎 ∥ (♯‘𝐵)))
164	163	cbvrabv 3422	. . . . . . 7 ⊢ {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} = {𝑎 ∈ ℙ ∣ 𝑎 ∥ (♯‘𝐵)}
165	2, 164	eqtri 2783	. . . . . 6 ⊢ 𝐴 = {𝑎 ∈ ℙ ∣ 𝑎 ∥ (♯‘𝐵)}
166		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑐 → (𝑂‘𝑥) = (𝑂‘𝑐))
167	166	breq1d 5112	. . . . . . . . . 10 ⊢ (𝑥 = 𝑐 → ((𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂‘𝑐) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))))
168	167	cbvrabv 3422	. . . . . . . . 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 7427	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑏 → (𝑝 pCnt (♯‘𝐵)) = (𝑏 pCnt (♯‘𝐵)))
171	169, 170	oveq12d 7438	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑏 → (𝑝↑(𝑝 pCnt (♯‘𝐵))) = (𝑏↑(𝑏 pCnt (♯‘𝐵))))
172	171	breq2d 5114	. . . . . . . . . 10 ⊢ (𝑝 = 𝑏 → ((𝑂‘𝑐) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂‘𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))))
173	172	rabbidv 3419	. . . . . . . . 9 ⊢ (𝑝 = 𝑏 → {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))})
174	168, 173	eqtrid 2807	. . . . . . . 8 ⊢ (𝑝 = 𝑏 → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))})
175	174	cbvmptv 5208	. . . . . . 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 2783	. . . . . 6 ⊢ 𝑆 = (𝑏 ∈ 𝐴 ↦ {𝑐 ∈ 𝐵 ∣ (𝑂‘𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))})
177		breq2 5106	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑡))
178		oveq2 7428	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑡))
179	178	eqeq1d 2762	. . . . . . . . . 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 3422	. . . . . . . 8 ⊢ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)} = {𝑡 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑡 ∧ (𝐺 DProd 𝑡) = 𝑔)}
182	181	mpteq2i 5200	. . . . . . 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 2783	. . . . . 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))))–1-1-onto→∪ 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))))–1-1-onto→∪ 𝑞 ∈ 𝐴 ({𝑞} × dom (𝑓‘𝑞)))) → ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))
186		2fveq3 6890	. . . . . . . . 9 ⊢ (𝑞 = 𝑦 → (𝑊‘(𝑆‘𝑞)) = (𝑊‘(𝑆‘𝑦)))
187	134, 186	eleq12d 2854	. . . . . . . 8 ⊢ (𝑞 = 𝑦 → ((𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)) ↔ (𝑓‘𝑦) ∈ (𝑊‘(𝑆‘𝑦))))
188	187	cbvralvw 3240	. . . . . . 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))))–1-1-onto→∪ 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))))–1-1-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 20307	. . . . 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 462	. . . 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 1965	. . 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 16	. 2 ⊢ ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞 ∈ 𝐴 (𝑓‘𝑞) ∈ (𝑊‘(𝑆‘𝑞)))) → (𝑊‘𝐵) ≠ ∅)
195	132, 194	exlimddv 1967	1 ⊢ (𝜑 → (𝑊‘𝐵) ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278  𝒫 cpw 4556  {csn 4584  ∪ ciun 4950   class class class wbr 5102   ↦ cmpt 5185   × cxp 5648  dom cdm 5650  ran crn 5651  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420   ≈ cen 8973  Fincfn 8976  0cc0 11203  1c1 11204   ≤ cle 11347  ℕcn 12338  ℕ₀cn0 12609  ℤcz 12696  ...cfz 13642  ..^cfzo 13791  ↑cexp 14207  ♯chash 14477  Word cword 14661   ∥ cdvds 16427  ℙcprime 16851   pCnt cpc 17019  Basecbs 17392   ↾_s cress 17413  Grpcgrp 19149  SubGrpcsubg 19335  odcod 19743   pGrp cpgp 19745  Abelcabl 20000  CycGrpccyg 20096   DProd cdprd 20214

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9645  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280  ax-pre-sup 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-disj 5070  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-rpss 7739  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8180  df-tpos 8246  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-2o 8480  df-oadd 8483  df-omul 8484  df-er 8720  df-ec 8722  df-qs 8726  df-map 8852  df-ixp 8929  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-fsupp 9357  df-sup 9437  df-inf 9438  df-oi 9507  df-dju 9985  df-card 10023  df-acn 10026  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-div 11977  df-nn 12339  df-2 12408  df-3 12409  df-n0 12610  df-xnn0 12683  df-z 12697  df-uz 12969  df-q 13079  df-rp 13124  df-fz 13643  df-fzo 13792  df-fl 13935  df-mod 14013  df-seq 14148  df-exp 14208  df-fac 14421  df-bc 14450  df-hash 14478  df-word 14662  df-concat 14719  df-s1 14746  df-cj 15271  df-re 15272  df-im 15273  df-sqrt 15407  df-abs 15408  df-clim 15660  df-sum 15859  df-dvds 16428  df-gcd 16670  df-prm 16852  df-pc 17020  df-sets 17347  df-slot 17365  df-ndx 17377  df-base 17393  df-ress 17414  df-plusg 17446  df-0g 17617  df-gsum 17618  df-mre 17761  df-mrc 17762  df-acs 17764  df-mgm 18821  df-sgrp 18913  df-mnd 18929  df-mhm 18983  df-submnd 18984  df-grp 19152  df-minusg 19153  df-sbg 19154  df-mulg 19283  df-subg 19338  df-eqg 19340  df-ghm 19433  df-gim 19478  df-ga 19509  df-cntz 19536  df-oppg 19565  df-od 19747  df-gex 19748  df-pgp 19749  df-lsm 19855  df-pj1 19856  df-cmn 20001  df-abl 20002  df-cyg 20097  df-dprd 20216

This theorem is used by:  ablfac  20309