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Theorem pgpfaclem1 20013

Description: Lemma for pgpfac 20016. (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 19061	. . . . . . . . . 10 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
6	3, 5	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ Grp)
7		eqid 2729	. . . . . . . . . 10 ⊢ (Base‘𝐻) = (Base‘𝐻)
8	7	subgacs 19093	. . . . . . . . 9 ⊢ (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
9	6, 8	syl 17	. . . . . . . 8 ⊢ (𝜑 → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
10	9	acsmred 17617	. . . . . . 7 ⊢ (𝜑 → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
11		pgpfac.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
12	4	subgbas 19062	. . . . . . . . 9 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 = (Base‘𝐻))
13	3, 12	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 = (Base‘𝐻))
14	11, 13	eleqtrd 2830	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻))
15		pgpfac.k	. . . . . . . 8 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐻))
16	15	mrcsncl 17573	. . . . . . 7 ⊢ (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
17	10, 14, 16	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
18	4	subsubg 19081	. . . . . . 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 7397	. . . . . . 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 584	. . . . . . 7 ⊢ (𝜑 → ((𝐺 ↾_s 𝑈) ↾_s (𝐾‘{𝑋})) = (𝐺 ↾_s (𝐾‘{𝑋})))
26	22, 25	eqtrid 2776	. . . . . 6 ⊢ (𝜑 → (𝐻 ↾_s (𝐾‘{𝑋})) = (𝐺 ↾_s (𝐾‘{𝑋})))
27	7, 15	cycsubgcyg2 19832	. . . . . . 7 ⊢ ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐻 ↾_s (𝐾‘{𝑋})) ∈ CycGrp)
28	6, 14, 27	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐻 ↾_s (𝐾‘{𝑋})) ∈ CycGrp)
29	26, 28	eqeltrrd 2829	. . . . 5 ⊢ (𝜑 → (𝐺 ↾_s (𝐾‘{𝑋})) ∈ CycGrp)
30		pgpfac.p	. . . . . . 7 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
31		pgpprm 19523	. . . . . . 7 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)
32	30, 31	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℙ)
33		subgpgp 19527	. . . . . . 7 ⊢ ((𝑃 pGrp 𝐺 ∧ (𝐾‘{𝑋}) ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺 ↾_s (𝐾‘{𝑋})))
34	30, 21, 33	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾_s (𝐾‘{𝑋})))
35		brelrng 5905	. . . . . 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 1373	. . . . 5 ⊢ (𝜑 → (𝐺 ↾_s (𝐾‘{𝑋})) ∈ ran pGrp )
37	29, 36	elind 4163	. . . 4 ⊢ (𝜑 → (𝐺 ↾_s (𝐾‘{𝑋})) ∈ (CycGrp ∩ ran pGrp ))
38		oveq2 7395	. . . . . 6 ⊢ (𝑟 = (𝐾‘{𝑋}) → (𝐺 ↾_s 𝑟) = (𝐺 ↾_s (𝐾‘{𝑋})))
39	38	eleq1d 2813	. . . . 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 3662	. . . 4 ⊢ ((𝐾‘{𝑋}) ∈ 𝐶 ↔ ((𝐾‘{𝑋}) ∈ (SubGrp‘𝐺) ∧ (𝐺 ↾_s (𝐾‘{𝑋})) ∈ (CycGrp ∩ ran pGrp )))
42	21, 37, 41	sylanbrc 583	. . 3 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ 𝐶)
43	1, 2, 42	cats1cld 14821	. 2 ⊢ (𝜑 → 𝑇 ∈ Word 𝐶)
44		wrdf 14483	. . . . 5 ⊢ (𝑇 ∈ Word 𝐶 → 𝑇:(0..^(♯‘𝑇))⟶𝐶)
45	43, 44	syl 17	. . . 4 ⊢ (𝜑 → 𝑇:(0..^(♯‘𝑇))⟶𝐶)
46	40	ssrab3 4045	. . . 4 ⊢ 𝐶 ⊆ (SubGrp‘𝐺)
47		fss 6704	. . . 4 ⊢ ((𝑇:(0..^(♯‘𝑇))⟶𝐶 ∧ 𝐶 ⊆ (SubGrp‘𝐺)) → 𝑇:(0..^(♯‘𝑇))⟶(SubGrp‘𝐺))
48	45, 46, 47	sylancl 586	. . 3 ⊢ (𝜑 → 𝑇:(0..^(♯‘𝑇))⟶(SubGrp‘𝐺))
49		lencl 14498	. . . . . . . 8 ⊢ (𝑆 ∈ Word 𝐶 → (♯‘𝑆) ∈ ℕ₀)
50	2, 49	syl 17	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑆) ∈ ℕ₀)
51	50	nn0zd 12555	. . . . . 6 ⊢ (𝜑 → (♯‘𝑆) ∈ ℤ)
52		fzosn 13697	. . . . . 6 ⊢ ((♯‘𝑆) ∈ ℤ → ((♯‘𝑆)..^((♯‘𝑆) + 1)) = {(♯‘𝑆)})
53	51, 52	syl 17	. . . . 5 ⊢ (𝜑 → ((♯‘𝑆)..^((♯‘𝑆) + 1)) = {(♯‘𝑆)})
54	53	ineq2d 4183	. . . 4 ⊢ (𝜑 → ((0..^(♯‘𝑆)) ∩ ((♯‘𝑆)..^((♯‘𝑆) + 1))) = ((0..^(♯‘𝑆)) ∩ {(♯‘𝑆)}))
55		fzodisj 13654	. . . 4 ⊢ ((0..^(♯‘𝑆)) ∩ ((♯‘𝑆)..^((♯‘𝑆) + 1))) = ∅
56	54, 55	eqtr3di 2779	. . 3 ⊢ (𝜑 → ((0..^(♯‘𝑆)) ∩ {(♯‘𝑆)}) = ∅)
57	1	fveq2i 6861	. . . . . . 7 ⊢ (♯‘𝑇) = (♯‘(𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩))
58	42	s1cld 14568	. . . . . . . 8 ⊢ (𝜑 → ⟨“(𝐾‘{𝑋})”⟩ ∈ Word 𝐶)
59		ccatlen 14540	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐶 ∧ ⟨“(𝐾‘{𝑋})”⟩ ∈ Word 𝐶) → (♯‘(𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)) = ((♯‘𝑆) + (♯‘⟨“(𝐾‘{𝑋})”⟩)))
60	2, 58, 59	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (♯‘(𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)) = ((♯‘𝑆) + (♯‘⟨“(𝐾‘{𝑋})”⟩)))
61	57, 60	eqtrid 2776	. . . . . 6 ⊢ (𝜑 → (♯‘𝑇) = ((♯‘𝑆) + (♯‘⟨“(𝐾‘{𝑋})”⟩)))
62		s1len 14571	. . . . . . 7 ⊢ (♯‘⟨“(𝐾‘{𝑋})”⟩) = 1
63	62	oveq2i 7398	. . . . . 6 ⊢ ((♯‘𝑆) + (♯‘⟨“(𝐾‘{𝑋})”⟩)) = ((♯‘𝑆) + 1)
64	61, 63	eqtrdi 2780	. . . . 5 ⊢ (𝜑 → (♯‘𝑇) = ((♯‘𝑆) + 1))
65	64	oveq2d 7403	. . . 4 ⊢ (𝜑 → (0..^(♯‘𝑇)) = (0..^((♯‘𝑆) + 1)))
66		nn0uz 12835	. . . . . 6 ⊢ ℕ₀ = (ℤ_≥‘0)
67	50, 66	eleqtrdi 2838	. . . . 5 ⊢ (𝜑 → (♯‘𝑆) ∈ (ℤ_≥‘0))
68		fzosplitsn 13736	. . . . 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 2764	. . 3 ⊢ (𝜑 → (0..^(♯‘𝑇)) = ((0..^(♯‘𝑆)) ∪ {(♯‘𝑆)}))
71		eqid 2729	. . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
72		eqid 2729	. . 3 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
73		pgpfac.4	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
74		cats1un 14686	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐶 ∧ (𝐾‘{𝑋}) ∈ 𝐶) → (𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩) = (𝑆 ∪ {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}))
75	2, 42, 74	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩) = (𝑆 ∪ {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}))
76	1, 75	eqtrid 2776	. . . . . 6 ⊢ (𝜑 → 𝑇 = (𝑆 ∪ {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}))
77	76	reseq1d 5949	. . . . 5 ⊢ (𝜑 → (𝑇 ↾ (0..^(♯‘𝑆))) = ((𝑆 ∪ {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}) ↾ (0..^(♯‘𝑆))))
78		wrdfn 14493	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐶 → 𝑆 Fn (0..^(♯‘𝑆)))
79	2, 78	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 Fn (0..^(♯‘𝑆)))
80		fzonel 13634	. . . . . 6 ⊢ ¬ (♯‘𝑆) ∈ (0..^(♯‘𝑆))
81		fsnunres 7162	. . . . . 6 ⊢ ((𝑆 Fn (0..^(♯‘𝑆)) ∧ ¬ (♯‘𝑆) ∈ (0..^(♯‘𝑆))) → ((𝑆 ∪ {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}) ↾ (0..^(♯‘𝑆))) = 𝑆)
82	79, 80, 81	sylancl 586	. . . . 5 ⊢ (𝜑 → ((𝑆 ∪ {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}) ↾ (0..^(♯‘𝑆))) = 𝑆)
83	77, 82	eqtrd 2764	. . . 4 ⊢ (𝜑 → (𝑇 ↾ (0..^(♯‘𝑆))) = 𝑆)
84	73, 83	breqtrrd 5135	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑇 ↾ (0..^(♯‘𝑆))))
85		fvex 6871	. . . . . 6 ⊢ (♯‘𝑆) ∈ V
86		dprdsn 19968	. . . . . 6 ⊢ (((♯‘𝑆) ∈ V ∧ (𝐾‘{𝑋}) ∈ (SubGrp‘𝐺)) → (𝐺dom DProd {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩} ∧ (𝐺 DProd {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}) = (𝐾‘{𝑋})))
87	85, 21, 86	sylancr 587	. . . . 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 14493	. . . . . . 7 ⊢ (𝑇 ∈ Word 𝐶 → 𝑇 Fn (0..^(♯‘𝑇)))
90	43, 89	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑇 Fn (0..^(♯‘𝑇)))
91		ssun2 4142	. . . . . . . 8 ⊢ {(♯‘𝑆)} ⊆ ((0..^(♯‘𝑆)) ∪ {(♯‘𝑆)})
92	85	snss 4749	. . . . . . . 8 ⊢ ((♯‘𝑆) ∈ ((0..^(♯‘𝑆)) ∪ {(♯‘𝑆)}) ↔ {(♯‘𝑆)} ⊆ ((0..^(♯‘𝑆)) ∪ {(♯‘𝑆)}))
93	91, 92	mpbir 231	. . . . . . 7 ⊢ (♯‘𝑆) ∈ ((0..^(♯‘𝑆)) ∪ {(♯‘𝑆)})
94	93, 70	eleqtrrid 2835	. . . . . 6 ⊢ (𝜑 → (♯‘𝑆) ∈ (0..^(♯‘𝑇)))
95		fnressn 7130	. . . . . 6 ⊢ ((𝑇 Fn (0..^(♯‘𝑇)) ∧ (♯‘𝑆) ∈ (0..^(♯‘𝑇))) → (𝑇 ↾ {(♯‘𝑆)}) = {⟨(♯‘𝑆), (𝑇‘(♯‘𝑆))⟩})
96	90, 94, 95	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑇 ↾ {(♯‘𝑆)}) = {⟨(♯‘𝑆), (𝑇‘(♯‘𝑆))⟩})
97	1	fveq1i 6859	. . . . . . . . 9 ⊢ (𝑇‘(♯‘𝑆)) = ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(♯‘𝑆))
98	50	nn0cnd 12505	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑆) ∈ ℂ)
99	98	addlidd 11375	. . . . . . . . . 10 ⊢ (𝜑 → (0 + (♯‘𝑆)) = (♯‘𝑆))
100	99	fveq2d 6862	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(0 + (♯‘𝑆))) = ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(♯‘𝑆)))
101	97, 100	eqtr4id 2783	. . . . . . . 8 ⊢ (𝜑 → (𝑇‘(♯‘𝑆)) = ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(0 + (♯‘𝑆))))
102		1nn 12197	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
103	62, 102	eqeltri 2824	. . . . . . . . . . 11 ⊢ (♯‘⟨“(𝐾‘{𝑋})”⟩) ∈ ℕ
104		lbfzo0 13660	. . . . . . . . . . 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 14544	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(0 + (♯‘𝑆))) = (⟨“(𝐾‘{𝑋})”⟩‘0))
109		fvex 6871	. . . . . . . . 9 ⊢ (𝐾‘{𝑋}) ∈ V
110		s1fv 14575	. . . . . . . . 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 2768	. . . . . . 7 ⊢ (𝜑 → (𝑇‘(♯‘𝑆)) = (𝐾‘{𝑋}))
113	112	opeq2d 4844	. . . . . 6 ⊢ (𝜑 → ⟨(♯‘𝑆), (𝑇‘(♯‘𝑆))⟩ = ⟨(♯‘𝑆), (𝐾‘{𝑋})⟩)
114	113	sneqd 4601	. . . . 5 ⊢ (𝜑 → {⟨(♯‘𝑆), (𝑇‘(♯‘𝑆))⟩} = {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩})
115	96, 114	eqtrd 2764	. . . 4 ⊢ (𝜑 → (𝑇 ↾ {(♯‘𝑆)}) = {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩})
116	88, 115	breqtrrd 5135	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑇 ↾ {(♯‘𝑆)}))
117		pgpfac.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Abel)
118		dprdsubg 19956	. . . . 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 19956	. . . . 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 19787	. . 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 7403	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆)))) = (𝐺 DProd 𝑆))
125		pgpfac.5	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd 𝑆) = 𝑊)
126	124, 125	eqtrd 2764	. . . . . 6 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆)))) = 𝑊)
127	115	oveq2d 7403	. . . . . . 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 2764	. . . . . 6 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ {(♯‘𝑆)})) = (𝐾‘{𝑋}))
130	126, 129	ineq12d 4184	. . . . 5 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆)))) ∩ (𝐺 DProd (𝑇 ↾ {(♯‘𝑆)}))) = (𝑊 ∩ (𝐾‘{𝑋})))
131		incom 4172	. . . . 5 ⊢ (𝑊 ∩ (𝐾‘{𝑋})) = ((𝐾‘{𝑋}) ∩ 𝑊)
132	130, 131	eqtrdi 2780	. . . 4 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆)))) ∩ (𝐺 DProd (𝑇 ↾ {(♯‘𝑆)}))) = ((𝐾‘{𝑋}) ∩ 𝑊))
133	4, 72	subg0 19064	. . . . . . 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 2782	. . . . 5 ⊢ (𝜑 → (0_g‘𝐺) = 0 )
137	136	sneqd 4601	. . . 4 ⊢ (𝜑 → {(0_g‘𝐺)} = { 0 })
138	123, 132, 137	3eqtr4d 2774	. . 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 19978	. 2 ⊢ (𝜑 → 𝐺dom DProd 𝑇)
140		eqid 2729	. . . . 5 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
141	48, 56, 70, 140, 139	dprdsplit 19980	. . . 4 ⊢ (𝜑 → (𝐺 DProd 𝑇) = ((𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆))))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ {(♯‘𝑆)}))))
142	126, 129	oveq12d 7405	. . . 4 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆))))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ {(♯‘𝑆)}))) = (𝑊(LSSum‘𝐺)(𝐾‘{𝑋})))
143	126, 119	eqeltrrd 2829	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
144	140	lsmcom 19788	. . . . 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 1373	. . . 4 ⊢ (𝜑 → (𝑊(LSSum‘𝐺)(𝐾‘{𝑋})) = ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊))
146	141, 142, 145	3eqtrd 2768	. . 3 ⊢ (𝜑 → (𝐺 DProd 𝑇) = ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊))
147		pgpfac.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐻))
148	7	subgss 19059	. . . . . 6 ⊢ (𝑊 ∈ (SubGrp‘𝐻) → 𝑊 ⊆ (Base‘𝐻))
149	147, 148	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ (Base‘𝐻))
150	149, 13	sseqtrrd 3984	. . . 4 ⊢ (𝜑 → 𝑊 ⊆ 𝑈)
151		pgpfac.l	. . . . 5 ⊢ ⊕ = (LSSum‘𝐻)
152	4, 140, 151	subglsm 19603	. . . 4 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝑋}) ⊆ 𝑈 ∧ 𝑊 ⊆ 𝑈) → ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊) = ((𝐾‘{𝑋}) ⊕ 𝑊))
153	3, 23, 150, 152	syl3anc 1373	. . 3 ⊢ (𝜑 → ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊) = ((𝐾‘{𝑋}) ⊕ 𝑊))
154		pgpfac.s	. . 3 ⊢ (𝜑 → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈)
155	146, 153, 154	3eqtrd 2768	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑇) = 𝑈)
156		breq2 5111	. . . 4 ⊢ (𝑠 = 𝑇 → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑇))
157		oveq2 7395	. . . . 5 ⊢ (𝑠 = 𝑇 → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑇))
158	157	eqeq1d 2731	. . . 4 ⊢ (𝑠 = 𝑇 → ((𝐺 DProd 𝑠) = 𝑈 ↔ (𝐺 DProd 𝑇) = 𝑈))
159	156, 158	anbi12d 632	. . 3 ⊢ (𝑠 = 𝑇 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈) ↔ (𝐺dom DProd 𝑇 ∧ (𝐺 DProd 𝑇) = 𝑈)))
160	159	rspcev 3588	. 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 836	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 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 {crab 3405 Vcvv 3447 ∪ cun 3912 ∩ cin 3913 ⊆ wss 3914 ⊊ wpss 3915 ∅c0 4296 {csn 4589 ⟨cop 4595 class class class wbr 5107 dom cdm 5638 ran crn 5639 ↾ cres 5640 Fn wfn 6506 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 Fincfn 8918 0cc0 11068 1c1 11069 + caddc 11071 ℕcn 12186 ℕ₀cn0 12442 ℤcz 12529 ℤ_≥cuz 12793 ..^cfzo 13615 ♯chash 14295 Word cword 14478 ++ cconcat 14535 ⟨“cs1 14560 ℙcprime 16641 Basecbs 17179 ↾_s cress 17200 0_gc0g 17402 Moorecmre 17543 mrClscmrc 17544 ACScacs 17546 Grpcgrp 18865 SubGrpcsubg 19052 Cntzccntz 19247 odcod 19454 gExcgex 19455 pGrp cpgp 19456 LSSumclsm 19564 Abelcabl 19711 CycGrpccyg 19807 DProd cdprd 19925

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 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-fzo 13616 df-seq 13967 df-hash 14296 df-word 14479 df-concat 14536 df-s1 14561 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 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-ghm 19145 df-gim 19191 df-cntz 19249 df-oppg 19278 df-od 19458 df-pgp 19460 df-lsm 19566 df-cmn 19712 df-abl 19713 df-cyg 19808 df-dprd 19927

This theorem is referenced by: pgpfaclem2 20014

W3C validator