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Theorem pgpfaclem1 18303

Description: Lemma for pgpfac 18306. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)

**Hypotheses**
Ref	Expression
pgpfac.b	⊢ 𝐵 = (Base‘𝐺)
pgpfac.c	⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾_s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
pgpfac.g	⊢ (𝜑 → 𝐺 ∈ Abel)
pgpfac.p	⊢ (𝜑 → 𝑃 pGrp 𝐺)
pgpfac.f	⊢ (𝜑 → 𝐵 ∈ Fin)
pgpfac.u	⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
pgpfac.a	⊢ (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))
pgpfac.h	⊢ 𝐻 = (𝐺 ↾_s 𝑈)
pgpfac.k	⊢ 𝐾 = (mrCls‘(SubGrp‘𝐻))
pgpfac.o	⊢ 𝑂 = (od‘𝐻)
pgpfac.e	⊢ 𝐸 = (gEx‘𝐻)
pgpfac.0	⊢ 0 = (0_g‘𝐻)
pgpfac.l	⊢ ⊕ = (LSSum‘𝐻)
pgpfac.1	⊢ (𝜑 → 𝐸 ≠ 1)
pgpfac.x	⊢ (𝜑 → 𝑋 ∈ 𝑈)
pgpfac.oe	⊢ (𝜑 → (𝑂‘𝑋) = 𝐸)
pgpfac.w	⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐻))
pgpfac.i	⊢ (𝜑 → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })
pgpfac.s	⊢ (𝜑 → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈)
pgpfac.2	⊢ (𝜑 → 𝑆 ∈ Word 𝐶)
pgpfac.4	⊢ (𝜑 → 𝐺dom DProd 𝑆)
pgpfac.5	⊢ (𝜑 → (𝐺 DProd 𝑆) = 𝑊)
pgpfac.t	⊢ 𝑇 = (𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)

**Assertion**
Ref	Expression
pgpfaclem1	⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))

Distinct variable groups: 𝑡,𝑠,𝐶 𝑠,𝑟,𝑡,𝐺 𝐾,𝑟,𝑠 𝜑,𝑡 𝐵,𝑠,𝑡 𝑈,𝑟,𝑠,𝑡 𝑊,𝑠,𝑡 𝑋,𝑟,𝑠 𝑇,𝑠 Allowed substitution hints: 𝜑(𝑠,𝑟) 𝐵(𝑟) 𝐶(𝑟) 𝑃(𝑡,𝑠,𝑟) ⊕ (𝑡,𝑠,𝑟) 𝑆(𝑡,𝑠,𝑟) 𝑇(𝑡,𝑟) 𝐸(𝑡,𝑠,𝑟) 𝐻(𝑡,𝑠,𝑟) 𝐾(𝑡) 𝑂(𝑡,𝑠,𝑟) 𝑊(𝑟) 𝑋(𝑡) 0 (𝑡,𝑠,𝑟)

**Proof of Theorem pgpfaclem1**
Step	Hyp	Ref	Expression
1		pgpfac.t	. . 3 ⊢ 𝑇 = (𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)
2		pgpfac.2	. . 3 ⊢ (𝜑 → 𝑆 ∈ Word 𝐶)
3		pgpfac.u	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
4		pgpfac.h	. . . . . . . . . 10 ⊢ 𝐻 = (𝐺 ↾_s 𝑈)
5	4	subggrp 17420	. . . . . . . . 9 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
6	3, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ Grp)
7		eqid 2610	. . . . . . . . 9 ⊢ (Base‘𝐻) = (Base‘𝐻)
8	7	subgacs 17452	. . . . . . . 8 ⊢ (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
9		acsmre 16136	. . . . . . . 8 ⊢ ((SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)) → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
10	6, 8, 9	3syl 18	. . . . . . 7 ⊢ (𝜑 → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
11		pgpfac.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
12	4	subgbas 17421	. . . . . . . . 9 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 = (Base‘𝐻))
13	3, 12	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 = (Base‘𝐻))
14	11, 13	eleqtrd 2690	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻))
15		pgpfac.k	. . . . . . . 8 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐻))
16	15	mrcsncl 16095	. . . . . . 7 ⊢ (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
17	10, 14, 16	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
18	4	subsubg 17440	. . . . . . 7 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → ((𝐾‘{𝑋}) ∈ (SubGrp‘𝐻) ↔ ((𝐾‘{𝑋}) ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝑋}) ⊆ 𝑈)))
19	3, 18	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝐾‘{𝑋}) ∈ (SubGrp‘𝐻) ↔ ((𝐾‘{𝑋}) ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝑋}) ⊆ 𝑈)))
20	17, 19	mpbid 221	. . . . 5 ⊢ (𝜑 → ((𝐾‘{𝑋}) ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝑋}) ⊆ 𝑈))
21	20	simpld 474	. . . 4 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐺))
22	4	oveq1i 6559	. . . . . . 7 ⊢ (𝐻 ↾_s (𝐾‘{𝑋})) = ((𝐺 ↾_s 𝑈) ↾_s (𝐾‘{𝑋}))
23	20	simprd 478	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ 𝑈)
24		ressabs 15766	. . . . . . . 8 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝑋}) ⊆ 𝑈) → ((𝐺 ↾_s 𝑈) ↾_s (𝐾‘{𝑋})) = (𝐺 ↾_s (𝐾‘{𝑋})))
25	3, 23, 24	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → ((𝐺 ↾_s 𝑈) ↾_s (𝐾‘{𝑋})) = (𝐺 ↾_s (𝐾‘{𝑋})))
26	22, 25	syl5eq 2656	. . . . . 6 ⊢ (𝜑 → (𝐻 ↾_s (𝐾‘{𝑋})) = (𝐺 ↾_s (𝐾‘{𝑋})))
27	7, 15	cycsubgcyg2 18126	. . . . . . 7 ⊢ ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐻 ↾_s (𝐾‘{𝑋})) ∈ CycGrp)
28	6, 14, 27	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (𝐻 ↾_s (𝐾‘{𝑋})) ∈ CycGrp)
29	26, 28	eqeltrrd 2689	. . . . 5 ⊢ (𝜑 → (𝐺 ↾_s (𝐾‘{𝑋})) ∈ CycGrp)
30		pgpfac.p	. . . . . . 7 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
31		pgpprm 17831	. . . . . . 7 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)
32	30, 31	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℙ)
33		subgpgp 17835	. . . . . . 7 ⊢ ((𝑃 pGrp 𝐺 ∧ (𝐾‘{𝑋}) ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺 ↾_s (𝐾‘{𝑋})))
34	30, 21, 33	syl2anc 691	. . . . . 6 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾_s (𝐾‘{𝑋})))
35		brelrng 5276	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐺 ↾_s (𝐾‘{𝑋})) ∈ CycGrp ∧ 𝑃 pGrp (𝐺 ↾_s (𝐾‘{𝑋}))) → (𝐺 ↾_s (𝐾‘{𝑋})) ∈ ran pGrp )
36	32, 29, 34, 35	syl3anc 1318	. . . . 5 ⊢ (𝜑 → (𝐺 ↾_s (𝐾‘{𝑋})) ∈ ran pGrp )
37	29, 36	elind 3760	. . . 4 ⊢ (𝜑 → (𝐺 ↾_s (𝐾‘{𝑋})) ∈ (CycGrp ∩ ran pGrp ))
38		oveq2 6557	. . . . . 6 ⊢ (𝑟 = (𝐾‘{𝑋}) → (𝐺 ↾_s 𝑟) = (𝐺 ↾_s (𝐾‘{𝑋})))
39	38	eleq1d 2672	. . . . 5 ⊢ (𝑟 = (𝐾‘{𝑋}) → ((𝐺 ↾_s 𝑟) ∈ (CycGrp ∩ ran pGrp ) ↔ (𝐺 ↾_s (𝐾‘{𝑋})) ∈ (CycGrp ∩ ran pGrp )))
40		pgpfac.c	. . . . 5 ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾_s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
41	39, 40	elrab2 3333	. . . 4 ⊢ ((𝐾‘{𝑋}) ∈ 𝐶 ↔ ((𝐾‘{𝑋}) ∈ (SubGrp‘𝐺) ∧ (𝐺 ↾_s (𝐾‘{𝑋})) ∈ (CycGrp ∩ ran pGrp )))
42	21, 37, 41	sylanbrc 695	. . 3 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ 𝐶)
43	1, 2, 42	cats1cld 13451	. 2 ⊢ (𝜑 → 𝑇 ∈ Word 𝐶)
44		wrdf 13165	. . . . 5 ⊢ (𝑇 ∈ Word 𝐶 → 𝑇:(0..^(#‘𝑇))⟶𝐶)
45	43, 44	syl 17	. . . 4 ⊢ (𝜑 → 𝑇:(0..^(#‘𝑇))⟶𝐶)
46		ssrab2 3650	. . . . 5 ⊢ {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾_s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ (SubGrp‘𝐺)
47	40, 46	eqsstri 3598	. . . 4 ⊢ 𝐶 ⊆ (SubGrp‘𝐺)
48		fss 5969	. . . 4 ⊢ ((𝑇:(0..^(#‘𝑇))⟶𝐶 ∧ 𝐶 ⊆ (SubGrp‘𝐺)) → 𝑇:(0..^(#‘𝑇))⟶(SubGrp‘𝐺))
49	45, 47, 48	sylancl 693	. . 3 ⊢ (𝜑 → 𝑇:(0..^(#‘𝑇))⟶(SubGrp‘𝐺))
50		fzodisj 12371	. . . 4 ⊢ ((0..^(#‘𝑆)) ∩ ((#‘𝑆)..^((#‘𝑆) + 1))) = ∅
51		lencl 13179	. . . . . . . 8 ⊢ (𝑆 ∈ Word 𝐶 → (#‘𝑆) ∈ ℕ₀)
52	2, 51	syl 17	. . . . . . 7 ⊢ (𝜑 → (#‘𝑆) ∈ ℕ₀)
53	52	nn0zd 11356	. . . . . 6 ⊢ (𝜑 → (#‘𝑆) ∈ ℤ)
54		fzosn 12405	. . . . . 6 ⊢ ((#‘𝑆) ∈ ℤ → ((#‘𝑆)..^((#‘𝑆) + 1)) = {(#‘𝑆)})
55	53, 54	syl 17	. . . . 5 ⊢ (𝜑 → ((#‘𝑆)..^((#‘𝑆) + 1)) = {(#‘𝑆)})
56	55	ineq2d 3776	. . . 4 ⊢ (𝜑 → ((0..^(#‘𝑆)) ∩ ((#‘𝑆)..^((#‘𝑆) + 1))) = ((0..^(#‘𝑆)) ∩ {(#‘𝑆)}))
57	50, 56	syl5reqr 2659	. . 3 ⊢ (𝜑 → ((0..^(#‘𝑆)) ∩ {(#‘𝑆)}) = ∅)
58	1	fveq2i 6106	. . . . . . 7 ⊢ (#‘𝑇) = (#‘(𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩))
59	42	s1cld 13236	. . . . . . . 8 ⊢ (𝜑 → ⟨“(𝐾‘{𝑋})”⟩ ∈ Word 𝐶)
60		ccatlen 13213	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐶 ∧ ⟨“(𝐾‘{𝑋})”⟩ ∈ Word 𝐶) → (#‘(𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)) = ((#‘𝑆) + (#‘⟨“(𝐾‘{𝑋})”⟩)))
61	2, 59, 60	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → (#‘(𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)) = ((#‘𝑆) + (#‘⟨“(𝐾‘{𝑋})”⟩)))
62	58, 61	syl5eq 2656	. . . . . 6 ⊢ (𝜑 → (#‘𝑇) = ((#‘𝑆) + (#‘⟨“(𝐾‘{𝑋})”⟩)))
63		s1len 13238	. . . . . . 7 ⊢ (#‘⟨“(𝐾‘{𝑋})”⟩) = 1
64	63	oveq2i 6560	. . . . . 6 ⊢ ((#‘𝑆) + (#‘⟨“(𝐾‘{𝑋})”⟩)) = ((#‘𝑆) + 1)
65	62, 64	syl6eq 2660	. . . . 5 ⊢ (𝜑 → (#‘𝑇) = ((#‘𝑆) + 1))
66	65	oveq2d 6565	. . . 4 ⊢ (𝜑 → (0..^(#‘𝑇)) = (0..^((#‘𝑆) + 1)))
67		nn0uz 11598	. . . . . 6 ⊢ ℕ₀ = (ℤ_≥‘0)
68	52, 67	syl6eleq 2698	. . . . 5 ⊢ (𝜑 → (#‘𝑆) ∈ (ℤ_≥‘0))
69		fzosplitsn 12442	. . . . 5 ⊢ ((#‘𝑆) ∈ (ℤ_≥‘0) → (0..^((#‘𝑆) + 1)) = ((0..^(#‘𝑆)) ∪ {(#‘𝑆)}))
70	68, 69	syl 17	. . . 4 ⊢ (𝜑 → (0..^((#‘𝑆) + 1)) = ((0..^(#‘𝑆)) ∪ {(#‘𝑆)}))
71	66, 70	eqtrd 2644	. . 3 ⊢ (𝜑 → (0..^(#‘𝑇)) = ((0..^(#‘𝑆)) ∪ {(#‘𝑆)}))
72		eqid 2610	. . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
73		eqid 2610	. . 3 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
74		pgpfac.4	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
75		cats1un 13327	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐶 ∧ (𝐾‘{𝑋}) ∈ 𝐶) → (𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩) = (𝑆 ∪ {⟨(#‘𝑆), (𝐾‘{𝑋})⟩}))
76	2, 42, 75	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → (𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩) = (𝑆 ∪ {⟨(#‘𝑆), (𝐾‘{𝑋})⟩}))
77	1, 76	syl5eq 2656	. . . . . 6 ⊢ (𝜑 → 𝑇 = (𝑆 ∪ {⟨(#‘𝑆), (𝐾‘{𝑋})⟩}))
78	77	reseq1d 5316	. . . . 5 ⊢ (𝜑 → (𝑇 ↾ (0..^(#‘𝑆))) = ((𝑆 ∪ {⟨(#‘𝑆), (𝐾‘{𝑋})⟩}) ↾ (0..^(#‘𝑆))))
79		wrdf 13165	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐶 → 𝑆:(0..^(#‘𝑆))⟶𝐶)
80		ffn 5958	. . . . . . 7 ⊢ (𝑆:(0..^(#‘𝑆))⟶𝐶 → 𝑆 Fn (0..^(#‘𝑆)))
81	2, 79, 80	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝑆 Fn (0..^(#‘𝑆)))
82		fzonel 12352	. . . . . 6 ⊢ ¬ (#‘𝑆) ∈ (0..^(#‘𝑆))
83		fsnunres 6359	. . . . . 6 ⊢ ((𝑆 Fn (0..^(#‘𝑆)) ∧ ¬ (#‘𝑆) ∈ (0..^(#‘𝑆))) → ((𝑆 ∪ {⟨(#‘𝑆), (𝐾‘{𝑋})⟩}) ↾ (0..^(#‘𝑆))) = 𝑆)
84	81, 82, 83	sylancl 693	. . . . 5 ⊢ (𝜑 → ((𝑆 ∪ {⟨(#‘𝑆), (𝐾‘{𝑋})⟩}) ↾ (0..^(#‘𝑆))) = 𝑆)
85	78, 84	eqtrd 2644	. . . 4 ⊢ (𝜑 → (𝑇 ↾ (0..^(#‘𝑆))) = 𝑆)
86	74, 85	breqtrrd 4611	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑇 ↾ (0..^(#‘𝑆))))
87		fvex 6113	. . . . . 6 ⊢ (#‘𝑆) ∈ V
88		dprdsn 18258	. . . . . 6 ⊢ (((#‘𝑆) ∈ V ∧ (𝐾‘{𝑋}) ∈ (SubGrp‘𝐺)) → (𝐺dom DProd {⟨(#‘𝑆), (𝐾‘{𝑋})⟩} ∧ (𝐺 DProd {⟨(#‘𝑆), (𝐾‘{𝑋})⟩}) = (𝐾‘{𝑋})))
89	87, 21, 88	sylancr 694	. . . . 5 ⊢ (𝜑 → (𝐺dom DProd {⟨(#‘𝑆), (𝐾‘{𝑋})⟩} ∧ (𝐺 DProd {⟨(#‘𝑆), (𝐾‘{𝑋})⟩}) = (𝐾‘{𝑋})))
90	89	simpld 474	. . . 4 ⊢ (𝜑 → 𝐺dom DProd {⟨(#‘𝑆), (𝐾‘{𝑋})⟩})
91		ffn 5958	. . . . . . 7 ⊢ (𝑇:(0..^(#‘𝑇))⟶𝐶 → 𝑇 Fn (0..^(#‘𝑇)))
92	43, 44, 91	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝑇 Fn (0..^(#‘𝑇)))
93		ssun2 3739	. . . . . . . 8 ⊢ {(#‘𝑆)} ⊆ ((0..^(#‘𝑆)) ∪ {(#‘𝑆)})
94	87	snss 4259	. . . . . . . 8 ⊢ ((#‘𝑆) ∈ ((0..^(#‘𝑆)) ∪ {(#‘𝑆)}) ↔ {(#‘𝑆)} ⊆ ((0..^(#‘𝑆)) ∪ {(#‘𝑆)}))
95	93, 94	mpbir 220	. . . . . . 7 ⊢ (#‘𝑆) ∈ ((0..^(#‘𝑆)) ∪ {(#‘𝑆)})
96	95, 71	syl5eleqr 2695	. . . . . 6 ⊢ (𝜑 → (#‘𝑆) ∈ (0..^(#‘𝑇)))
97		fnressn 6330	. . . . . 6 ⊢ ((𝑇 Fn (0..^(#‘𝑇)) ∧ (#‘𝑆) ∈ (0..^(#‘𝑇))) → (𝑇 ↾ {(#‘𝑆)}) = {⟨(#‘𝑆), (𝑇‘(#‘𝑆))⟩})
98	92, 96, 97	syl2anc 691	. . . . 5 ⊢ (𝜑 → (𝑇 ↾ {(#‘𝑆)}) = {⟨(#‘𝑆), (𝑇‘(#‘𝑆))⟩})
99	1	fveq1i 6104	. . . . . . . . 9 ⊢ (𝑇‘(#‘𝑆)) = ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(#‘𝑆))
100	52	nn0cnd 11230	. . . . . . . . . . . 12 ⊢ (𝜑 → (#‘𝑆) ∈ ℂ)
101	100	addid2d 10116	. . . . . . . . . . 11 ⊢ (𝜑 → (0 + (#‘𝑆)) = (#‘𝑆))
102	101	eqcomd 2616	. . . . . . . . . 10 ⊢ (𝜑 → (#‘𝑆) = (0 + (#‘𝑆)))
103	102	fveq2d 6107	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(#‘𝑆)) = ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(0 + (#‘𝑆))))
104	99, 103	syl5eq 2656	. . . . . . . 8 ⊢ (𝜑 → (𝑇‘(#‘𝑆)) = ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(0 + (#‘𝑆))))
105		1nn 10908	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
106	105	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℕ)
107	63, 106	syl5eqel 2692	. . . . . . . . . 10 ⊢ (𝜑 → (#‘⟨“(𝐾‘{𝑋})”⟩) ∈ ℕ)
108		lbfzo0 12375	. . . . . . . . . 10 ⊢ (0 ∈ (0..^(#‘⟨“(𝐾‘{𝑋})”⟩)) ↔ (#‘⟨“(𝐾‘{𝑋})”⟩) ∈ ℕ)
109	107, 108	sylibr 223	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (0..^(#‘⟨“(𝐾‘{𝑋})”⟩)))
110		ccatval3 13216	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐶 ∧ ⟨“(𝐾‘{𝑋})”⟩ ∈ Word 𝐶 ∧ 0 ∈ (0..^(#‘⟨“(𝐾‘{𝑋})”⟩))) → ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(0 + (#‘𝑆))) = (⟨“(𝐾‘{𝑋})”⟩‘0))
111	2, 59, 109, 110	syl3anc 1318	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(0 + (#‘𝑆))) = (⟨“(𝐾‘{𝑋})”⟩‘0))
112		fvex 6113	. . . . . . . . 9 ⊢ (𝐾‘{𝑋}) ∈ V
113		s1fv 13243	. . . . . . . . 9 ⊢ ((𝐾‘{𝑋}) ∈ V → (⟨“(𝐾‘{𝑋})”⟩‘0) = (𝐾‘{𝑋}))
114	112, 113	mp1i 13	. . . . . . . 8 ⊢ (𝜑 → (⟨“(𝐾‘{𝑋})”⟩‘0) = (𝐾‘{𝑋}))
115	104, 111, 114	3eqtrd 2648	. . . . . . 7 ⊢ (𝜑 → (𝑇‘(#‘𝑆)) = (𝐾‘{𝑋}))
116	115	opeq2d 4347	. . . . . 6 ⊢ (𝜑 → ⟨(#‘𝑆), (𝑇‘(#‘𝑆))⟩ = ⟨(#‘𝑆), (𝐾‘{𝑋})⟩)
117	116	sneqd 4137	. . . . 5 ⊢ (𝜑 → {⟨(#‘𝑆), (𝑇‘(#‘𝑆))⟩} = {⟨(#‘𝑆), (𝐾‘{𝑋})⟩})
118	98, 117	eqtrd 2644	. . . 4 ⊢ (𝜑 → (𝑇 ↾ {(#‘𝑆)}) = {⟨(#‘𝑆), (𝐾‘{𝑋})⟩})
119	90, 118	breqtrrd 4611	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑇 ↾ {(#‘𝑆)}))
120		pgpfac.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Abel)
121		dprdsubg 18246	. . . . 5 ⊢ (𝐺dom DProd (𝑇 ↾ (0..^(#‘𝑆))) → (𝐺 DProd (𝑇 ↾ (0..^(#‘𝑆)))) ∈ (SubGrp‘𝐺))
122	86, 121	syl 17	. . . 4 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (0..^(#‘𝑆)))) ∈ (SubGrp‘𝐺))
123		dprdsubg 18246	. . . . 5 ⊢ (𝐺dom DProd (𝑇 ↾ {(#‘𝑆)}) → (𝐺 DProd (𝑇 ↾ {(#‘𝑆)})) ∈ (SubGrp‘𝐺))
124	119, 123	syl 17	. . . 4 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ {(#‘𝑆)})) ∈ (SubGrp‘𝐺))
125	72, 120, 122, 124	ablcntzd 18083	. . 3 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (0..^(#‘𝑆)))) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑇 ↾ {(#‘𝑆)}))))
126		pgpfac.i	. . . 4 ⊢ (𝜑 → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })
127	85	oveq2d 6565	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (0..^(#‘𝑆)))) = (𝐺 DProd 𝑆))
128		pgpfac.5	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd 𝑆) = 𝑊)
129	127, 128	eqtrd 2644	. . . . . 6 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (0..^(#‘𝑆)))) = 𝑊)
130	118	oveq2d 6565	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ {(#‘𝑆)})) = (𝐺 DProd {⟨(#‘𝑆), (𝐾‘{𝑋})⟩}))
131	89	simprd 478	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd {⟨(#‘𝑆), (𝐾‘{𝑋})⟩}) = (𝐾‘{𝑋}))
132	130, 131	eqtrd 2644	. . . . . 6 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ {(#‘𝑆)})) = (𝐾‘{𝑋}))
133	129, 132	ineq12d 3777	. . . . 5 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ (0..^(#‘𝑆)))) ∩ (𝐺 DProd (𝑇 ↾ {(#‘𝑆)}))) = (𝑊 ∩ (𝐾‘{𝑋})))
134		incom 3767	. . . . 5 ⊢ (𝑊 ∩ (𝐾‘{𝑋})) = ((𝐾‘{𝑋}) ∩ 𝑊)
135	133, 134	syl6eq 2660	. . . 4 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ (0..^(#‘𝑆)))) ∩ (𝐺 DProd (𝑇 ↾ {(#‘𝑆)}))) = ((𝐾‘{𝑋}) ∩ 𝑊))
136	4, 73	subg0 17423	. . . . . . 7 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (0_g‘𝐺) = (0_g‘𝐻))
137	3, 136	syl 17	. . . . . 6 ⊢ (𝜑 → (0_g‘𝐺) = (0_g‘𝐻))
138		pgpfac.0	. . . . . 6 ⊢ 0 = (0_g‘𝐻)
139	137, 138	syl6eqr 2662	. . . . 5 ⊢ (𝜑 → (0_g‘𝐺) = 0 )
140	139	sneqd 4137	. . . 4 ⊢ (𝜑 → {(0_g‘𝐺)} = { 0 })
141	126, 135, 140	3eqtr4d 2654	. . 3 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ (0..^(#‘𝑆)))) ∩ (𝐺 DProd (𝑇 ↾ {(#‘𝑆)}))) = {(0_g‘𝐺)})
142	49, 57, 71, 72, 73, 86, 119, 125, 141	dmdprdsplit2 18268	. 2 ⊢ (𝜑 → 𝐺dom DProd 𝑇)
143		eqid 2610	. . . . 5 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
144	49, 57, 71, 143, 142	dprdsplit 18270	. . . 4 ⊢ (𝜑 → (𝐺 DProd 𝑇) = ((𝐺 DProd (𝑇 ↾ (0..^(#‘𝑆))))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ {(#‘𝑆)}))))
145	129, 132	oveq12d 6567	. . . 4 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ (0..^(#‘𝑆))))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ {(#‘𝑆)}))) = (𝑊(LSSum‘𝐺)(𝐾‘{𝑋})))
146	129, 122	eqeltrrd 2689	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
147	143	lsmcom 18084	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑊 ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝑋}) ∈ (SubGrp‘𝐺)) → (𝑊(LSSum‘𝐺)(𝐾‘{𝑋})) = ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊))
148	120, 146, 21, 147	syl3anc 1318	. . . 4 ⊢ (𝜑 → (𝑊(LSSum‘𝐺)(𝐾‘{𝑋})) = ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊))
149	144, 145, 148	3eqtrd 2648	. . 3 ⊢ (𝜑 → (𝐺 DProd 𝑇) = ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊))
150		pgpfac.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐻))
151	7	subgss 17418	. . . . . 6 ⊢ (𝑊 ∈ (SubGrp‘𝐻) → 𝑊 ⊆ (Base‘𝐻))
152	150, 151	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ (Base‘𝐻))
153	152, 13	sseqtr4d 3605	. . . 4 ⊢ (𝜑 → 𝑊 ⊆ 𝑈)
154		pgpfac.l	. . . . 5 ⊢ ⊕ = (LSSum‘𝐻)
155	4, 143, 154	subglsm 17909	. . . 4 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝑋}) ⊆ 𝑈 ∧ 𝑊 ⊆ 𝑈) → ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊) = ((𝐾‘{𝑋}) ⊕ 𝑊))
156	3, 23, 153, 155	syl3anc 1318	. . 3 ⊢ (𝜑 → ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊) = ((𝐾‘{𝑋}) ⊕ 𝑊))
157		pgpfac.s	. . 3 ⊢ (𝜑 → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈)
158	149, 156, 157	3eqtrd 2648	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑇) = 𝑈)
159		breq2 4587	. . . 4 ⊢ (𝑠 = 𝑇 → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑇))
160		oveq2 6557	. . . . 5 ⊢ (𝑠 = 𝑇 → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑇))
161	160	eqeq1d 2612	. . . 4 ⊢ (𝑠 = 𝑇 → ((𝐺 DProd 𝑠) = 𝑈 ↔ (𝐺 DProd 𝑇) = 𝑈))
162	159, 161	anbi12d 743	. . 3 ⊢ (𝑠 = 𝑇 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈) ↔ (𝐺dom DProd 𝑇 ∧ (𝐺 DProd 𝑇) = 𝑈)))
163	162	rspcev 3282	. 2 ⊢ ((𝑇 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑇 ∧ (𝐺 DProd 𝑇) = 𝑈)) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
164	43, 142, 158, 163	syl12anc 1316	1 ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ⊊ wpss 3541 ∅c0 3874 {csn 4125 ⟨cop 4131 class class class wbr 4583 dom cdm 5038 ran crn 5039 ↾ cres 5040 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 0cc0 9815 1c1 9816 + caddc 9818 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ..^cfzo 12334 #chash 12979 Word cword 13146 ++ cconcat 13148 ⟨“cs1 13149 ℙcprime 15223 Basecbs 15695 ↾_s cress 15696 0_gc0g 15923 Moorecmre 16065 mrClscmrc 16066 ACScacs 16068 Grpcgrp 17245 SubGrpcsubg 17411 Cntzccntz 17571 odcod 17767 gExcgex 17768 pGrp cpgp 17769 LSSumclsm 17872 Abelcabl 18017 CycGrpccyg 18102 DProd cdprd 18215

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-gim 17524 df-cntz 17573 df-oppg 17599 df-od 17771 df-pgp 17773 df-lsm 17874 df-cmn 18018 df-abl 18019 df-cyg 18103 df-dprd 18217

This theorem is referenced by: pgpfaclem2 18304

W3C validator