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Theorem pgpfaclem1 20016

Description: Lemma for pgpfac 20019. (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 19063	. . . . . . . . . 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 19094	. . . . . . . . 9 ⊢ (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
9	6, 8	syl 17	. . . . . . . 8 ⊢ (𝜑 → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
10	9	acsmred 17583	. . . . . . 7 ⊢ (𝜑 → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
11		pgpfac.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
12	4	subgbas 19064	. . . . . . . . 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 17539	. . . . . . 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 19083	. . . . . . 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 7370	. . . . . . 7 ⊢ (𝐻 ↾_s (𝐾‘{𝑋})) = ((𝐺 ↾_s 𝑈) ↾_s (𝐾‘{𝑋}))
23	20	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ 𝑈)
24		ressabs 17179	. . . . . . . 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 19835	. . . . . . 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 19526	. . . . . . 7 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)
32	30, 31	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℙ)
33		subgpgp 19530	. . . . . . 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 5891	. . . . . 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 4153	. . . 4 ⊢ (𝜑 → (𝐺 ↾_s (𝐾‘{𝑋})) ∈ (CycGrp ∩ ran pGrp ))
38		oveq2 7368	. . . . . 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 3650	. . . 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 14782	. 2 ⊢ (𝜑 → 𝑇 ∈ Word 𝐶)
44		wrdf 14445	. . . . 5 ⊢ (𝑇 ∈ Word 𝐶 → 𝑇:(0..^(♯‘𝑇))⟶𝐶)
45	43, 44	syl 17	. . . 4 ⊢ (𝜑 → 𝑇:(0..^(♯‘𝑇))⟶𝐶)
46	40	ssrab3 4035	. . . 4 ⊢ 𝐶 ⊆ (SubGrp‘𝐺)
47		fss 6679	. . . 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 14460	. . . . . . . 8 ⊢ (𝑆 ∈ Word 𝐶 → (♯‘𝑆) ∈ ℕ₀)
50	2, 49	syl 17	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑆) ∈ ℕ₀)
51	50	nn0zd 12517	. . . . . 6 ⊢ (𝜑 → (♯‘𝑆) ∈ ℤ)
52		fzosn 13656	. . . . . 6 ⊢ ((♯‘𝑆) ∈ ℤ → ((♯‘𝑆)..^((♯‘𝑆) + 1)) = {(♯‘𝑆)})
53	51, 52	syl 17	. . . . 5 ⊢ (𝜑 → ((♯‘𝑆)..^((♯‘𝑆) + 1)) = {(♯‘𝑆)})
54	53	ineq2d 4173	. . . 4 ⊢ (𝜑 → ((0..^(♯‘𝑆)) ∩ ((♯‘𝑆)..^((♯‘𝑆) + 1))) = ((0..^(♯‘𝑆)) ∩ {(♯‘𝑆)}))
55		fzodisj 13613	. . . 4 ⊢ ((0..^(♯‘𝑆)) ∩ ((♯‘𝑆)..^((♯‘𝑆) + 1))) = ∅
56	54, 55	eqtr3di 2787	. . 3 ⊢ (𝜑 → ((0..^(♯‘𝑆)) ∩ {(♯‘𝑆)}) = ∅)
57	1	fveq2i 6838	. . . . . . 7 ⊢ (♯‘𝑇) = (♯‘(𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩))
58	42	s1cld 14531	. . . . . . . 8 ⊢ (𝜑 → ⟨“(𝐾‘{𝑋})”⟩ ∈ Word 𝐶)
59		ccatlen 14502	. . . . . . . 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 14534	. . . . . . 7 ⊢ (♯‘⟨“(𝐾‘{𝑋})”⟩) = 1
63	62	oveq2i 7371	. . . . . 6 ⊢ ((♯‘𝑆) + (♯‘⟨“(𝐾‘{𝑋})”⟩)) = ((♯‘𝑆) + 1)
64	61, 63	eqtrdi 2788	. . . . 5 ⊢ (𝜑 → (♯‘𝑇) = ((♯‘𝑆) + 1))
65	64	oveq2d 7376	. . . 4 ⊢ (𝜑 → (0..^(♯‘𝑇)) = (0..^((♯‘𝑆) + 1)))
66		nn0uz 12793	. . . . . 6 ⊢ ℕ₀ = (ℤ_≥‘0)
67	50, 66	eleqtrdi 2847	. . . . 5 ⊢ (𝜑 → (♯‘𝑆) ∈ (ℤ_≥‘0))
68		fzosplitsn 13696	. . . . 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 14648	. . . . . . . 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 5938	. . . . 5 ⊢ (𝜑 → (𝑇 ↾ (0..^(♯‘𝑆))) = ((𝑆 ∪ {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}) ↾ (0..^(♯‘𝑆))))
78		wrdfn 14455	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐶 → 𝑆 Fn (0..^(♯‘𝑆)))
79	2, 78	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 Fn (0..^(♯‘𝑆)))
80		fzonel 13593	. . . . . 6 ⊢ ¬ (♯‘𝑆) ∈ (0..^(♯‘𝑆))
81		fsnunres 7136	. . . . . 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 5127	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑇 ↾ (0..^(♯‘𝑆))))
85		fvex 6848	. . . . . 6 ⊢ (♯‘𝑆) ∈ V
86		dprdsn 19971	. . . . . 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 14455	. . . . . . 7 ⊢ (𝑇 ∈ Word 𝐶 → 𝑇 Fn (0..^(♯‘𝑇)))
90	43, 89	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑇 Fn (0..^(♯‘𝑇)))
91		ssun2 4132	. . . . . . . 8 ⊢ {(♯‘𝑆)} ⊆ ((0..^(♯‘𝑆)) ∪ {(♯‘𝑆)})
92	85	snss 4742	. . . . . . . 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 7105	. . . . . 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 6836	. . . . . . . . 9 ⊢ (𝑇‘(♯‘𝑆)) = ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(♯‘𝑆))
98	50	nn0cnd 12468	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑆) ∈ ℂ)
99	98	addlidd 11338	. . . . . . . . . 10 ⊢ (𝜑 → (0 + (♯‘𝑆)) = (♯‘𝑆))
100	99	fveq2d 6839	. . . . . . . . 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 12160	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
103	62, 102	eqeltri 2833	. . . . . . . . . . 11 ⊢ (♯‘⟨“(𝐾‘{𝑋})”⟩) ∈ ℕ
104		lbfzo0 13619	. . . . . . . . . . 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 14506	. . . . . . . . 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 6848	. . . . . . . . 9 ⊢ (𝐾‘{𝑋}) ∈ V
110		s1fv 14538	. . . . . . . . 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 4837	. . . . . 6 ⊢ (𝜑 → ⟨(♯‘𝑆), (𝑇‘(♯‘𝑆))⟩ = ⟨(♯‘𝑆), (𝐾‘{𝑋})⟩)
114	113	sneqd 4593	. . . . 5 ⊢ (𝜑 → {⟨(♯‘𝑆), (𝑇‘(♯‘𝑆))⟩} = {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩})
115	96, 114	eqtrd 2772	. . . 4 ⊢ (𝜑 → (𝑇 ↾ {(♯‘𝑆)}) = {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩})
116	88, 115	breqtrrd 5127	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑇 ↾ {(♯‘𝑆)}))
117		pgpfac.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Abel)
118		dprdsubg 19959	. . . . 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 19959	. . . . 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 19790	. . 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 7376	. . . . . . 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 7376	. . . . . . 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 4174	. . . . 5 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆)))) ∩ (𝐺 DProd (𝑇 ↾ {(♯‘𝑆)}))) = (𝑊 ∩ (𝐾‘{𝑋})))
131		incom 4162	. . . . 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 19066	. . . . . . 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 4593	. . . 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 19981	. 2 ⊢ (𝜑 → 𝐺dom DProd 𝑇)
140		eqid 2737	. . . . 5 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
141	48, 56, 70, 140, 139	dprdsplit 19983	. . . 4 ⊢ (𝜑 → (𝐺 DProd 𝑇) = ((𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆))))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ {(♯‘𝑆)}))))
142	126, 129	oveq12d 7378	. . . 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 19791	. . . . 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 19061	. . . . . 6 ⊢ (𝑊 ∈ (SubGrp‘𝐻) → 𝑊 ⊆ (Base‘𝐻))
149	147, 148	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ (Base‘𝐻))
150	149, 13	sseqtrrd 3972	. . . 4 ⊢ (𝜑 → 𝑊 ⊆ 𝑈)
151		pgpfac.l	. . . . 5 ⊢ ⊕ = (LSSum‘𝐻)
152	4, 140, 151	subglsm 19606	. . . 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 5103	. . . 4 ⊢ (𝑠 = 𝑇 → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑇))
157		oveq2 7368	. . . . 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 3577	. 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 3061 {crab 3400 Vcvv 3441 ∪ cun 3900 ∩ cin 3901 ⊆ wss 3902 ⊊ wpss 3903 ∅c0 4286 {csn 4581 ⟨cop 4587 class class class wbr 5099 dom cdm 5625 ran crn 5626 ↾ cres 5627 Fn wfn 6488 ⟶wf 6489 ‘cfv 6493 (class class class)co 7360 Fincfn 8887 0cc0 11030 1c1 11031 + caddc 11033 ℕcn 12149 ℕ₀cn0 12405 ℤcz 12492 ℤ_≥cuz 12755 ..^cfzo 13574 ♯chash 14257 Word cword 14440 ++ cconcat 14497 ⟨“cs1 14523 ℙcprime 16602 Basecbs 17140 ↾_s cress 17161 0_gc0g 17363 Moorecmre 17505 mrClscmrc 17506 ACScacs 17508 Grpcgrp 18867 SubGrpcsubg 19054 Cntzccntz 19248 odcod 19457 gExcgex 19458 pGrp cpgp 19459 LSSumclsm 19567 Abelcabl 19714 CycGrpccyg 19810 DProd cdprd 19928

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 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

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 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-2 12212 df-n0 12406 df-z 12493 df-uz 12756 df-fz 13428 df-fzo 13575 df-seq 13929 df-hash 14258 df-word 14441 df-concat 14498 df-s1 14524 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-0g 17365 df-gsum 17366 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18712 df-submnd 18713 df-grp 18870 df-minusg 18871 df-sbg 18872 df-mulg 19002 df-subg 19057 df-ghm 19146 df-gim 19192 df-cntz 19250 df-oppg 19279 df-od 19461 df-pgp 19463 df-lsm 19569 df-cmn 19715 df-abl 19716 df-cyg 19811 df-dprd 19930

This theorem is referenced by: pgpfaclem2 20017

W3C validator