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Theorem pgpfaclem1 20058

Description: Lemma for pgpfac 20061. (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	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
4		pgpfac.h	. . . . . . . . . . 11 ⊢ 𝐻 = (𝐺 ↾_s 𝑈)
5	4	subggrp 19105	. . . . . . . . . 10 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
6	3, 5	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ Grp)
7		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝐻) = (Base‘𝐻)
8	7	subgacs 19136	. . . . . . . . 9 ⊢ (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
9	6, 8	syl 17	. . . . . . . 8 ⊢ (𝜑 → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
10	9	acsmred 17622	. . . . . . 7 ⊢ (𝜑 → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
11		pgpfac.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
12	4	subgbas 19106	. . . . . . . . 9 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 = (Base‘𝐻))
13	3, 12	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 = (Base‘𝐻))
14	11, 13	eleqtrd 2839	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻))
15		pgpfac.k	. . . . . . . 8 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐻))
16	15	mrcsncl 17578	. . . . . . 7 ⊢ (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
17	10, 14, 16	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
18	4	subsubg 19125	. . . . . . 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 232	. . . . 5 ⊢ (𝜑 → ((𝐾‘{𝑋}) ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝑋}) ⊆ 𝑈))
21	20	simpld 494	. . . 4 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐺))
22	4	oveq1i 7377	. . . . . . 7 ⊢ (𝐻 ↾_s (𝐾‘{𝑋})) = ((𝐺 ↾_s 𝑈) ↾_s (𝐾‘{𝑋}))
23	20	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ 𝑈)
24		ressabs 17218	. . . . . . . 8 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝑋}) ⊆ 𝑈) → ((𝐺 ↾_s 𝑈) ↾_s (𝐾‘{𝑋})) = (𝐺 ↾_s (𝐾‘{𝑋})))
25	3, 23, 24	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → ((𝐺 ↾_s 𝑈) ↾_s (𝐾‘{𝑋})) = (𝐺 ↾_s (𝐾‘{𝑋})))
26	22, 25	eqtrid 2784	. . . . . 6 ⊢ (𝜑 → (𝐻 ↾_s (𝐾‘{𝑋})) = (𝐺 ↾_s (𝐾‘{𝑋})))
27	7, 15	cycsubgcyg2 19877	. . . . . . 7 ⊢ ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐻 ↾_s (𝐾‘{𝑋})) ∈ CycGrp)
28	6, 14, 27	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (𝐻 ↾_s (𝐾‘{𝑋})) ∈ CycGrp)
29	26, 28	eqeltrrd 2838	. . . . 5 ⊢ (𝜑 → (𝐺 ↾_s (𝐾‘{𝑋})) ∈ CycGrp)
30		pgpfac.p	. . . . . . 7 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
31		pgpprm 19568	. . . . . . 7 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)
32	30, 31	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℙ)
33		subgpgp 19572	. . . . . . 7 ⊢ ((𝑃 pGrp 𝐺 ∧ (𝐾‘{𝑋}) ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺 ↾_s (𝐾‘{𝑋})))
34	30, 21, 33	syl2anc 585	. . . . . 6 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾_s (𝐾‘{𝑋})))
35		brelrng 5897	. . . . . 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 1374	. . . . 5 ⊢ (𝜑 → (𝐺 ↾_s (𝐾‘{𝑋})) ∈ ran pGrp )
37	29, 36	elind 4141	. . . 4 ⊢ (𝜑 → (𝐺 ↾_s (𝐾‘{𝑋})) ∈ (CycGrp ∩ ran pGrp ))
38		oveq2 7375	. . . . . 6 ⊢ (𝑟 = (𝐾‘{𝑋}) → (𝐺 ↾_s 𝑟) = (𝐺 ↾_s (𝐾‘{𝑋})))
39	38	eleq1d 2822	. . . . 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 3638	. . . 4 ⊢ ((𝐾‘{𝑋}) ∈ 𝐶 ↔ ((𝐾‘{𝑋}) ∈ (SubGrp‘𝐺) ∧ (𝐺 ↾_s (𝐾‘{𝑋})) ∈ (CycGrp ∩ ran pGrp )))
42	21, 37, 41	sylanbrc 584	. . 3 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ 𝐶)
43	1, 2, 42	cats1cld 14817	. 2 ⊢ (𝜑 → 𝑇 ∈ Word 𝐶)
44		wrdf 14480	. . . . 5 ⊢ (𝑇 ∈ Word 𝐶 → 𝑇:(0..^(♯‘𝑇))⟶𝐶)
45	43, 44	syl 17	. . . 4 ⊢ (𝜑 → 𝑇:(0..^(♯‘𝑇))⟶𝐶)
46	40	ssrab3 4023	. . . 4 ⊢ 𝐶 ⊆ (SubGrp‘𝐺)
47		fss 6685	. . . 4 ⊢ ((𝑇:(0..^(♯‘𝑇))⟶𝐶 ∧ 𝐶 ⊆ (SubGrp‘𝐺)) → 𝑇:(0..^(♯‘𝑇))⟶(SubGrp‘𝐺))
48	45, 46, 47	sylancl 587	. . 3 ⊢ (𝜑 → 𝑇:(0..^(♯‘𝑇))⟶(SubGrp‘𝐺))
49		lencl 14495	. . . . . . . 8 ⊢ (𝑆 ∈ Word 𝐶 → (♯‘𝑆) ∈ ℕ₀)
50	2, 49	syl 17	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑆) ∈ ℕ₀)
51	50	nn0zd 12549	. . . . . 6 ⊢ (𝜑 → (♯‘𝑆) ∈ ℤ)
52		fzosn 13691	. . . . . 6 ⊢ ((♯‘𝑆) ∈ ℤ → ((♯‘𝑆)..^((♯‘𝑆) + 1)) = {(♯‘𝑆)})
53	51, 52	syl 17	. . . . 5 ⊢ (𝜑 → ((♯‘𝑆)..^((♯‘𝑆) + 1)) = {(♯‘𝑆)})
54	53	ineq2d 4161	. . . 4 ⊢ (𝜑 → ((0..^(♯‘𝑆)) ∩ ((♯‘𝑆)..^((♯‘𝑆) + 1))) = ((0..^(♯‘𝑆)) ∩ {(♯‘𝑆)}))
55		fzodisj 13648	. . . 4 ⊢ ((0..^(♯‘𝑆)) ∩ ((♯‘𝑆)..^((♯‘𝑆) + 1))) = ∅
56	54, 55	eqtr3di 2787	. . 3 ⊢ (𝜑 → ((0..^(♯‘𝑆)) ∩ {(♯‘𝑆)}) = ∅)
57	1	fveq2i 6844	. . . . . . 7 ⊢ (♯‘𝑇) = (♯‘(𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩))
58	42	s1cld 14566	. . . . . . . 8 ⊢ (𝜑 → ⟨“(𝐾‘{𝑋})”⟩ ∈ Word 𝐶)
59		ccatlen 14537	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐶 ∧ ⟨“(𝐾‘{𝑋})”⟩ ∈ Word 𝐶) → (♯‘(𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)) = ((♯‘𝑆) + (♯‘⟨“(𝐾‘{𝑋})”⟩)))
60	2, 58, 59	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (♯‘(𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)) = ((♯‘𝑆) + (♯‘⟨“(𝐾‘{𝑋})”⟩)))
61	57, 60	eqtrid 2784	. . . . . 6 ⊢ (𝜑 → (♯‘𝑇) = ((♯‘𝑆) + (♯‘⟨“(𝐾‘{𝑋})”⟩)))
62		s1len 14569	. . . . . . 7 ⊢ (♯‘⟨“(𝐾‘{𝑋})”⟩) = 1
63	62	oveq2i 7378	. . . . . 6 ⊢ ((♯‘𝑆) + (♯‘⟨“(𝐾‘{𝑋})”⟩)) = ((♯‘𝑆) + 1)
64	61, 63	eqtrdi 2788	. . . . 5 ⊢ (𝜑 → (♯‘𝑇) = ((♯‘𝑆) + 1))
65	64	oveq2d 7383	. . . 4 ⊢ (𝜑 → (0..^(♯‘𝑇)) = (0..^((♯‘𝑆) + 1)))
66		nn0uz 12826	. . . . . 6 ⊢ ℕ₀ = (ℤ_≥‘0)
67	50, 66	eleqtrdi 2847	. . . . 5 ⊢ (𝜑 → (♯‘𝑆) ∈ (ℤ_≥‘0))
68		fzosplitsn 13731	. . . . 5 ⊢ ((♯‘𝑆) ∈ (ℤ_≥‘0) → (0..^((♯‘𝑆) + 1)) = ((0..^(♯‘𝑆)) ∪ {(♯‘𝑆)}))
69	67, 68	syl 17	. . . 4 ⊢ (𝜑 → (0..^((♯‘𝑆) + 1)) = ((0..^(♯‘𝑆)) ∪ {(♯‘𝑆)}))
70	65, 69	eqtrd 2772	. . 3 ⊢ (𝜑 → (0..^(♯‘𝑇)) = ((0..^(♯‘𝑆)) ∪ {(♯‘𝑆)}))
71		eqid 2737	. . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
72		eqid 2737	. . 3 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
73		pgpfac.4	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
74		cats1un 14683	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐶 ∧ (𝐾‘{𝑋}) ∈ 𝐶) → (𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩) = (𝑆 ∪ {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}))
75	2, 42, 74	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩) = (𝑆 ∪ {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}))
76	1, 75	eqtrid 2784	. . . . . 6 ⊢ (𝜑 → 𝑇 = (𝑆 ∪ {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}))
77	76	reseq1d 5944	. . . . 5 ⊢ (𝜑 → (𝑇 ↾ (0..^(♯‘𝑆))) = ((𝑆 ∪ {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}) ↾ (0..^(♯‘𝑆))))
78		wrdfn 14490	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐶 → 𝑆 Fn (0..^(♯‘𝑆)))
79	2, 78	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 Fn (0..^(♯‘𝑆)))
80		fzonel 13628	. . . . . 6 ⊢ ¬ (♯‘𝑆) ∈ (0..^(♯‘𝑆))
81		fsnunres 7143	. . . . . 6 ⊢ ((𝑆 Fn (0..^(♯‘𝑆)) ∧ ¬ (♯‘𝑆) ∈ (0..^(♯‘𝑆))) → ((𝑆 ∪ {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}) ↾ (0..^(♯‘𝑆))) = 𝑆)
82	79, 80, 81	sylancl 587	. . . . 5 ⊢ (𝜑 → ((𝑆 ∪ {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}) ↾ (0..^(♯‘𝑆))) = 𝑆)
83	77, 82	eqtrd 2772	. . . 4 ⊢ (𝜑 → (𝑇 ↾ (0..^(♯‘𝑆))) = 𝑆)
84	73, 83	breqtrrd 5114	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑇 ↾ (0..^(♯‘𝑆))))
85		fvex 6854	. . . . . 6 ⊢ (♯‘𝑆) ∈ V
86		dprdsn 20013	. . . . . 6 ⊢ (((♯‘𝑆) ∈ V ∧ (𝐾‘{𝑋}) ∈ (SubGrp‘𝐺)) → (𝐺dom DProd {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩} ∧ (𝐺 DProd {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}) = (𝐾‘{𝑋})))
87	85, 21, 86	sylancr 588	. . . . 5 ⊢ (𝜑 → (𝐺dom DProd {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩} ∧ (𝐺 DProd {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}) = (𝐾‘{𝑋})))
88	87	simpld 494	. . . 4 ⊢ (𝜑 → 𝐺dom DProd {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩})
89		wrdfn 14490	. . . . . . 7 ⊢ (𝑇 ∈ Word 𝐶 → 𝑇 Fn (0..^(♯‘𝑇)))
90	43, 89	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑇 Fn (0..^(♯‘𝑇)))
91		ssun2 4120	. . . . . . . 8 ⊢ {(♯‘𝑆)} ⊆ ((0..^(♯‘𝑆)) ∪ {(♯‘𝑆)})
92	85	snss 4729	. . . . . . . 8 ⊢ ((♯‘𝑆) ∈ ((0..^(♯‘𝑆)) ∪ {(♯‘𝑆)}) ↔ {(♯‘𝑆)} ⊆ ((0..^(♯‘𝑆)) ∪ {(♯‘𝑆)}))
93	91, 92	mpbir 231	. . . . . . 7 ⊢ (♯‘𝑆) ∈ ((0..^(♯‘𝑆)) ∪ {(♯‘𝑆)})
94	93, 70	eleqtrrid 2844	. . . . . 6 ⊢ (𝜑 → (♯‘𝑆) ∈ (0..^(♯‘𝑇)))
95		fnressn 7112	. . . . . 6 ⊢ ((𝑇 Fn (0..^(♯‘𝑇)) ∧ (♯‘𝑆) ∈ (0..^(♯‘𝑇))) → (𝑇 ↾ {(♯‘𝑆)}) = {⟨(♯‘𝑆), (𝑇‘(♯‘𝑆))⟩})
96	90, 94, 95	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝑇 ↾ {(♯‘𝑆)}) = {⟨(♯‘𝑆), (𝑇‘(♯‘𝑆))⟩})
97	1	fveq1i 6842	. . . . . . . . 9 ⊢ (𝑇‘(♯‘𝑆)) = ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(♯‘𝑆))
98	50	nn0cnd 12500	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑆) ∈ ℂ)
99	98	addlidd 11347	. . . . . . . . . 10 ⊢ (𝜑 → (0 + (♯‘𝑆)) = (♯‘𝑆))
100	99	fveq2d 6845	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(0 + (♯‘𝑆))) = ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(♯‘𝑆)))
101	97, 100	eqtr4id 2791	. . . . . . . 8 ⊢ (𝜑 → (𝑇‘(♯‘𝑆)) = ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(0 + (♯‘𝑆))))
102		1nn 12185	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
103	62, 102	eqeltri 2833	. . . . . . . . . . 11 ⊢ (♯‘⟨“(𝐾‘{𝑋})”⟩) ∈ ℕ
104		lbfzo0 13654	. . . . . . . . . . 11 ⊢ (0 ∈ (0..^(♯‘⟨“(𝐾‘{𝑋})”⟩)) ↔ (♯‘⟨“(𝐾‘{𝑋})”⟩) ∈ ℕ)
105	103, 104	mpbir 231	. . . . . . . . . 10 ⊢ 0 ∈ (0..^(♯‘⟨“(𝐾‘{𝑋})”⟩))
106	105	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (0..^(♯‘⟨“(𝐾‘{𝑋})”⟩)))
107		ccatval3 14541	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐶 ∧ ⟨“(𝐾‘{𝑋})”⟩ ∈ Word 𝐶 ∧ 0 ∈ (0..^(♯‘⟨“(𝐾‘{𝑋})”⟩))) → ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(0 + (♯‘𝑆))) = (⟨“(𝐾‘{𝑋})”⟩‘0))
108	2, 58, 106, 107	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(0 + (♯‘𝑆))) = (⟨“(𝐾‘{𝑋})”⟩‘0))
109		fvex 6854	. . . . . . . . 9 ⊢ (𝐾‘{𝑋}) ∈ V
110		s1fv 14573	. . . . . . . . 9 ⊢ ((𝐾‘{𝑋}) ∈ V → (⟨“(𝐾‘{𝑋})”⟩‘0) = (𝐾‘{𝑋}))
111	109, 110	mp1i 13	. . . . . . . 8 ⊢ (𝜑 → (⟨“(𝐾‘{𝑋})”⟩‘0) = (𝐾‘{𝑋}))
112	101, 108, 111	3eqtrd 2776	. . . . . . 7 ⊢ (𝜑 → (𝑇‘(♯‘𝑆)) = (𝐾‘{𝑋}))
113	112	opeq2d 4824	. . . . . 6 ⊢ (𝜑 → ⟨(♯‘𝑆), (𝑇‘(♯‘𝑆))⟩ = ⟨(♯‘𝑆), (𝐾‘{𝑋})⟩)
114	113	sneqd 4580	. . . . 5 ⊢ (𝜑 → {⟨(♯‘𝑆), (𝑇‘(♯‘𝑆))⟩} = {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩})
115	96, 114	eqtrd 2772	. . . 4 ⊢ (𝜑 → (𝑇 ↾ {(♯‘𝑆)}) = {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩})
116	88, 115	breqtrrd 5114	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑇 ↾ {(♯‘𝑆)}))
117		pgpfac.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Abel)
118		dprdsubg 20001	. . . . 5 ⊢ (𝐺dom DProd (𝑇 ↾ (0..^(♯‘𝑆))) → (𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆)))) ∈ (SubGrp‘𝐺))
119	84, 118	syl 17	. . . 4 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆)))) ∈ (SubGrp‘𝐺))
120		dprdsubg 20001	. . . . 5 ⊢ (𝐺dom DProd (𝑇 ↾ {(♯‘𝑆)}) → (𝐺 DProd (𝑇 ↾ {(♯‘𝑆)})) ∈ (SubGrp‘𝐺))
121	116, 120	syl 17	. . . 4 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ {(♯‘𝑆)})) ∈ (SubGrp‘𝐺))
122	71, 117, 119, 121	ablcntzd 19832	. . 3 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆)))) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑇 ↾ {(♯‘𝑆)}))))
123		pgpfac.i	. . . 4 ⊢ (𝜑 → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })
124	83	oveq2d 7383	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆)))) = (𝐺 DProd 𝑆))
125		pgpfac.5	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd 𝑆) = 𝑊)
126	124, 125	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆)))) = 𝑊)
127	115	oveq2d 7383	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ {(♯‘𝑆)})) = (𝐺 DProd {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}))
128	87	simprd 495	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}) = (𝐾‘{𝑋}))
129	127, 128	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ {(♯‘𝑆)})) = (𝐾‘{𝑋}))
130	126, 129	ineq12d 4162	. . . . 5 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆)))) ∩ (𝐺 DProd (𝑇 ↾ {(♯‘𝑆)}))) = (𝑊 ∩ (𝐾‘{𝑋})))
131		incom 4150	. . . . 5 ⊢ (𝑊 ∩ (𝐾‘{𝑋})) = ((𝐾‘{𝑋}) ∩ 𝑊)
132	130, 131	eqtrdi 2788	. . . 4 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆)))) ∩ (𝐺 DProd (𝑇 ↾ {(♯‘𝑆)}))) = ((𝐾‘{𝑋}) ∩ 𝑊))
133	4, 72	subg0 19108	. . . . . . 7 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (0_g‘𝐺) = (0_g‘𝐻))
134	3, 133	syl 17	. . . . . 6 ⊢ (𝜑 → (0_g‘𝐺) = (0_g‘𝐻))
135		pgpfac.0	. . . . . 6 ⊢ 0 = (0_g‘𝐻)
136	134, 135	eqtr4di 2790	. . . . 5 ⊢ (𝜑 → (0_g‘𝐺) = 0 )
137	136	sneqd 4580	. . . 4 ⊢ (𝜑 → {(0_g‘𝐺)} = { 0 })
138	123, 132, 137	3eqtr4d 2782	. . 3 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆)))) ∩ (𝐺 DProd (𝑇 ↾ {(♯‘𝑆)}))) = {(0_g‘𝐺)})
139	48, 56, 70, 71, 72, 84, 116, 122, 138	dmdprdsplit2 20023	. 2 ⊢ (𝜑 → 𝐺dom DProd 𝑇)
140		eqid 2737	. . . . 5 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
141	48, 56, 70, 140, 139	dprdsplit 20025	. . . 4 ⊢ (𝜑 → (𝐺 DProd 𝑇) = ((𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆))))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ {(♯‘𝑆)}))))
142	126, 129	oveq12d 7385	. . . 4 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆))))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ {(♯‘𝑆)}))) = (𝑊(LSSum‘𝐺)(𝐾‘{𝑋})))
143	126, 119	eqeltrrd 2838	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
144	140	lsmcom 19833	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑊 ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝑋}) ∈ (SubGrp‘𝐺)) → (𝑊(LSSum‘𝐺)(𝐾‘{𝑋})) = ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊))
145	117, 143, 21, 144	syl3anc 1374	. . . 4 ⊢ (𝜑 → (𝑊(LSSum‘𝐺)(𝐾‘{𝑋})) = ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊))
146	141, 142, 145	3eqtrd 2776	. . 3 ⊢ (𝜑 → (𝐺 DProd 𝑇) = ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊))
147		pgpfac.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐻))
148	7	subgss 19103	. . . . . 6 ⊢ (𝑊 ∈ (SubGrp‘𝐻) → 𝑊 ⊆ (Base‘𝐻))
149	147, 148	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ (Base‘𝐻))
150	149, 13	sseqtrrd 3960	. . . 4 ⊢ (𝜑 → 𝑊 ⊆ 𝑈)
151		pgpfac.l	. . . . 5 ⊢ ⊕ = (LSSum‘𝐻)
152	4, 140, 151	subglsm 19648	. . . 4 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝑋}) ⊆ 𝑈 ∧ 𝑊 ⊆ 𝑈) → ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊) = ((𝐾‘{𝑋}) ⊕ 𝑊))
153	3, 23, 150, 152	syl3anc 1374	. . 3 ⊢ (𝜑 → ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊) = ((𝐾‘{𝑋}) ⊕ 𝑊))
154		pgpfac.s	. . 3 ⊢ (𝜑 → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈)
155	146, 153, 154	3eqtrd 2776	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑇) = 𝑈)
156		breq2 5090	. . . 4 ⊢ (𝑠 = 𝑇 → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑇))
157		oveq2 7375	. . . . 5 ⊢ (𝑠 = 𝑇 → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑇))
158	157	eqeq1d 2739	. . . 4 ⊢ (𝑠 = 𝑇 → ((𝐺 DProd 𝑠) = 𝑈 ↔ (𝐺 DProd 𝑇) = 𝑈))
159	156, 158	anbi12d 633	. . 3 ⊢ (𝑠 = 𝑇 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈) ↔ (𝐺dom DProd 𝑇 ∧ (𝐺 DProd 𝑇) = 𝑈)))
160	159	rspcev 3565	. 2 ⊢ ((𝑇 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑇 ∧ (𝐺 DProd 𝑇) = 𝑈)) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
161	43, 139, 155, 160	syl12anc 837	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 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 {crab 3390 Vcvv 3430 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 ⊊ wpss 3891 ∅c0 4274 {csn 4568 ⟨cop 4574 class class class wbr 5086 dom cdm 5631 ran crn 5632 ↾ cres 5633 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 (class class class)co 7367 Fincfn 8893 0cc0 11038 1c1 11039 + caddc 11041 ℕcn 12174 ℕ₀cn0 12437 ℤcz 12524 ℤ_≥cuz 12788 ..^cfzo 13608 ♯chash 14292 Word cword 14475 ++ cconcat 14532 ⟨“cs1 14558 ℙcprime 16640 Basecbs 17179 ↾_s cress 17200 0_gc0g 17402 Moorecmre 17544 mrClscmrc 17545 ACScacs 17547 Grpcgrp 18909 SubGrpcsubg 19096 Cntzccntz 19290 odcod 19499 gExcgex 19500 pGrp cpgp 19501 LSSumclsm 19609 Abelcabl 19756 CycGrpccyg 19852 DProd cdprd 19970

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-fzo 13609 df-seq 13964 df-hash 14293 df-word 14476 df-concat 14533 df-s1 14559 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-0g 17404 df-gsum 17405 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-ghm 19188 df-gim 19234 df-cntz 19292 df-oppg 19321 df-od 19503 df-pgp 19505 df-lsm 19611 df-cmn 19757 df-abl 19758 df-cyg 19853 df-dprd 19972

This theorem is referenced by: pgpfaclem2 20059

W3C validator