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Theorem pgpfaclem1 18526

Description: Lemma for pgpfac 18529. (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 17644	. . . . . . . . 9 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
6	3, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ Grp)
7		eqid 2651	. . . . . . . . 9 ⊢ (Base‘𝐻) = (Base‘𝐻)
8	7	subgacs 17676	. . . . . . . 8 ⊢ (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
9		acsmre 16360	. . . . . . . 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 17645	. . . . . . . . 9 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 = (Base‘𝐻))
13	3, 12	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 = (Base‘𝐻))
14	11, 13	eleqtrd 2732	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻))
15		pgpfac.k	. . . . . . . 8 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐻))
16	15	mrcsncl 16319	. . . . . . 7 ⊢ (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
17	10, 14, 16	syl2anc 694	. . . . . 6 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
18	4	subsubg 17664	. . . . . . 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 222	. . . . 5 ⊢ (𝜑 → ((𝐾‘{𝑋}) ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝑋}) ⊆ 𝑈))
21	20	simpld 474	. . . 4 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐺))
22	4	oveq1i 6700	. . . . . . 7 ⊢ (𝐻 ↾_s (𝐾‘{𝑋})) = ((𝐺 ↾_s 𝑈) ↾_s (𝐾‘{𝑋}))
23	20	simprd 478	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ 𝑈)
24		ressabs 15986	. . . . . . . 8 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝑋}) ⊆ 𝑈) → ((𝐺 ↾_s 𝑈) ↾_s (𝐾‘{𝑋})) = (𝐺 ↾_s (𝐾‘{𝑋})))
25	3, 23, 24	syl2anc 694	. . . . . . 7 ⊢ (𝜑 → ((𝐺 ↾_s 𝑈) ↾_s (𝐾‘{𝑋})) = (𝐺 ↾_s (𝐾‘{𝑋})))
26	22, 25	syl5eq 2697	. . . . . 6 ⊢ (𝜑 → (𝐻 ↾_s (𝐾‘{𝑋})) = (𝐺 ↾_s (𝐾‘{𝑋})))
27	7, 15	cycsubgcyg2 18349	. . . . . . 7 ⊢ ((𝐻 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐻 ↾_s (𝐾‘{𝑋})) ∈ CycGrp)
28	6, 14, 27	syl2anc 694	. . . . . 6 ⊢ (𝜑 → (𝐻 ↾_s (𝐾‘{𝑋})) ∈ CycGrp)
29	26, 28	eqeltrrd 2731	. . . . 5 ⊢ (𝜑 → (𝐺 ↾_s (𝐾‘{𝑋})) ∈ CycGrp)
30		pgpfac.p	. . . . . . 7 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
31		pgpprm 18054	. . . . . . 7 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)
32	30, 31	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℙ)
33		subgpgp 18058	. . . . . . 7 ⊢ ((𝑃 pGrp 𝐺 ∧ (𝐾‘{𝑋}) ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺 ↾_s (𝐾‘{𝑋})))
34	30, 21, 33	syl2anc 694	. . . . . 6 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾_s (𝐾‘{𝑋})))
35		brelrng 5387	. . . . . 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 1366	. . . . 5 ⊢ (𝜑 → (𝐺 ↾_s (𝐾‘{𝑋})) ∈ ran pGrp )
37	29, 36	elind 3831	. . . 4 ⊢ (𝜑 → (𝐺 ↾_s (𝐾‘{𝑋})) ∈ (CycGrp ∩ ran pGrp ))
38		oveq2 6698	. . . . . 6 ⊢ (𝑟 = (𝐾‘{𝑋}) → (𝐺 ↾_s 𝑟) = (𝐺 ↾_s (𝐾‘{𝑋})))
39	38	eleq1d 2715	. . . . 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 3399	. . . 4 ⊢ ((𝐾‘{𝑋}) ∈ 𝐶 ↔ ((𝐾‘{𝑋}) ∈ (SubGrp‘𝐺) ∧ (𝐺 ↾_s (𝐾‘{𝑋})) ∈ (CycGrp ∩ ran pGrp )))
42	21, 37, 41	sylanbrc 699	. . 3 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ 𝐶)
43	1, 2, 42	cats1cld 13646	. 2 ⊢ (𝜑 → 𝑇 ∈ Word 𝐶)
44		wrdf 13342	. . . . 5 ⊢ (𝑇 ∈ Word 𝐶 → 𝑇:(0..^(#‘𝑇))⟶𝐶)
45	43, 44	syl 17	. . . 4 ⊢ (𝜑 → 𝑇:(0..^(#‘𝑇))⟶𝐶)
46		ssrab2 3720	. . . . 5 ⊢ {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾_s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ (SubGrp‘𝐺)
47	40, 46	eqsstri 3668	. . . 4 ⊢ 𝐶 ⊆ (SubGrp‘𝐺)
48		fss 6094	. . . 4 ⊢ ((𝑇:(0..^(#‘𝑇))⟶𝐶 ∧ 𝐶 ⊆ (SubGrp‘𝐺)) → 𝑇:(0..^(#‘𝑇))⟶(SubGrp‘𝐺))
49	45, 47, 48	sylancl 695	. . 3 ⊢ (𝜑 → 𝑇:(0..^(#‘𝑇))⟶(SubGrp‘𝐺))
50		fzodisj 12541	. . . 4 ⊢ ((0..^(#‘𝑆)) ∩ ((#‘𝑆)..^((#‘𝑆) + 1))) = ∅
51		lencl 13356	. . . . . . . 8 ⊢ (𝑆 ∈ Word 𝐶 → (#‘𝑆) ∈ ℕ₀)
52	2, 51	syl 17	. . . . . . 7 ⊢ (𝜑 → (#‘𝑆) ∈ ℕ₀)
53	52	nn0zd 11518	. . . . . 6 ⊢ (𝜑 → (#‘𝑆) ∈ ℤ)
54		fzosn 12578	. . . . . 6 ⊢ ((#‘𝑆) ∈ ℤ → ((#‘𝑆)..^((#‘𝑆) + 1)) = {(#‘𝑆)})
55	53, 54	syl 17	. . . . 5 ⊢ (𝜑 → ((#‘𝑆)..^((#‘𝑆) + 1)) = {(#‘𝑆)})
56	55	ineq2d 3847	. . . 4 ⊢ (𝜑 → ((0..^(#‘𝑆)) ∩ ((#‘𝑆)..^((#‘𝑆) + 1))) = ((0..^(#‘𝑆)) ∩ {(#‘𝑆)}))
57	50, 56	syl5reqr 2700	. . 3 ⊢ (𝜑 → ((0..^(#‘𝑆)) ∩ {(#‘𝑆)}) = ∅)
58	1	fveq2i 6232	. . . . . . 7 ⊢ (#‘𝑇) = (#‘(𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩))
59	42	s1cld 13419	. . . . . . . 8 ⊢ (𝜑 → ⟨“(𝐾‘{𝑋})”⟩ ∈ Word 𝐶)
60		ccatlen 13393	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐶 ∧ ⟨“(𝐾‘{𝑋})”⟩ ∈ Word 𝐶) → (#‘(𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)) = ((#‘𝑆) + (#‘⟨“(𝐾‘{𝑋})”⟩)))
61	2, 59, 60	syl2anc 694	. . . . . . 7 ⊢ (𝜑 → (#‘(𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)) = ((#‘𝑆) + (#‘⟨“(𝐾‘{𝑋})”⟩)))
62	58, 61	syl5eq 2697	. . . . . 6 ⊢ (𝜑 → (#‘𝑇) = ((#‘𝑆) + (#‘⟨“(𝐾‘{𝑋})”⟩)))
63		s1len 13422	. . . . . . 7 ⊢ (#‘⟨“(𝐾‘{𝑋})”⟩) = 1
64	63	oveq2i 6701	. . . . . 6 ⊢ ((#‘𝑆) + (#‘⟨“(𝐾‘{𝑋})”⟩)) = ((#‘𝑆) + 1)
65	62, 64	syl6eq 2701	. . . . 5 ⊢ (𝜑 → (#‘𝑇) = ((#‘𝑆) + 1))
66	65	oveq2d 6706	. . . 4 ⊢ (𝜑 → (0..^(#‘𝑇)) = (0..^((#‘𝑆) + 1)))
67		nn0uz 11760	. . . . . 6 ⊢ ℕ₀ = (ℤ_≥‘0)
68	52, 67	syl6eleq 2740	. . . . 5 ⊢ (𝜑 → (#‘𝑆) ∈ (ℤ_≥‘0))
69		fzosplitsn 12616	. . . . 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 2685	. . 3 ⊢ (𝜑 → (0..^(#‘𝑇)) = ((0..^(#‘𝑆)) ∪ {(#‘𝑆)}))
72		eqid 2651	. . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
73		eqid 2651	. . 3 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
74		pgpfac.4	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
75		cats1un 13521	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐶 ∧ (𝐾‘{𝑋}) ∈ 𝐶) → (𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩) = (𝑆 ∪ {⟨(#‘𝑆), (𝐾‘{𝑋})⟩}))
76	2, 42, 75	syl2anc 694	. . . . . . 7 ⊢ (𝜑 → (𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩) = (𝑆 ∪ {⟨(#‘𝑆), (𝐾‘{𝑋})⟩}))
77	1, 76	syl5eq 2697	. . . . . 6 ⊢ (𝜑 → 𝑇 = (𝑆 ∪ {⟨(#‘𝑆), (𝐾‘{𝑋})⟩}))
78	77	reseq1d 5427	. . . . 5 ⊢ (𝜑 → (𝑇 ↾ (0..^(#‘𝑆))) = ((𝑆 ∪ {⟨(#‘𝑆), (𝐾‘{𝑋})⟩}) ↾ (0..^(#‘𝑆))))
79		wrdf 13342	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐶 → 𝑆:(0..^(#‘𝑆))⟶𝐶)
80		ffn 6083	. . . . . . 7 ⊢ (𝑆:(0..^(#‘𝑆))⟶𝐶 → 𝑆 Fn (0..^(#‘𝑆)))
81	2, 79, 80	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝑆 Fn (0..^(#‘𝑆)))
82		fzonel 12522	. . . . . 6 ⊢ ¬ (#‘𝑆) ∈ (0..^(#‘𝑆))
83		fsnunres 6495	. . . . . 6 ⊢ ((𝑆 Fn (0..^(#‘𝑆)) ∧ ¬ (#‘𝑆) ∈ (0..^(#‘𝑆))) → ((𝑆 ∪ {⟨(#‘𝑆), (𝐾‘{𝑋})⟩}) ↾ (0..^(#‘𝑆))) = 𝑆)
84	81, 82, 83	sylancl 695	. . . . 5 ⊢ (𝜑 → ((𝑆 ∪ {⟨(#‘𝑆), (𝐾‘{𝑋})⟩}) ↾ (0..^(#‘𝑆))) = 𝑆)
85	78, 84	eqtrd 2685	. . . 4 ⊢ (𝜑 → (𝑇 ↾ (0..^(#‘𝑆))) = 𝑆)
86	74, 85	breqtrrd 4713	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑇 ↾ (0..^(#‘𝑆))))
87		fvex 6239	. . . . . 6 ⊢ (#‘𝑆) ∈ V
88		dprdsn 18481	. . . . . 6 ⊢ (((#‘𝑆) ∈ V ∧ (𝐾‘{𝑋}) ∈ (SubGrp‘𝐺)) → (𝐺dom DProd {⟨(#‘𝑆), (𝐾‘{𝑋})⟩} ∧ (𝐺 DProd {⟨(#‘𝑆), (𝐾‘{𝑋})⟩}) = (𝐾‘{𝑋})))
89	87, 21, 88	sylancr 696	. . . . 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 6083	. . . . . . 7 ⊢ (𝑇:(0..^(#‘𝑇))⟶𝐶 → 𝑇 Fn (0..^(#‘𝑇)))
92	43, 44, 91	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝑇 Fn (0..^(#‘𝑇)))
93		ssun2 3810	. . . . . . . 8 ⊢ {(#‘𝑆)} ⊆ ((0..^(#‘𝑆)) ∪ {(#‘𝑆)})
94	87	snss 4348	. . . . . . . 8 ⊢ ((#‘𝑆) ∈ ((0..^(#‘𝑆)) ∪ {(#‘𝑆)}) ↔ {(#‘𝑆)} ⊆ ((0..^(#‘𝑆)) ∪ {(#‘𝑆)}))
95	93, 94	mpbir 221	. . . . . . 7 ⊢ (#‘𝑆) ∈ ((0..^(#‘𝑆)) ∪ {(#‘𝑆)})
96	95, 71	syl5eleqr 2737	. . . . . 6 ⊢ (𝜑 → (#‘𝑆) ∈ (0..^(#‘𝑇)))
97		fnressn 6465	. . . . . 6 ⊢ ((𝑇 Fn (0..^(#‘𝑇)) ∧ (#‘𝑆) ∈ (0..^(#‘𝑇))) → (𝑇 ↾ {(#‘𝑆)}) = {⟨(#‘𝑆), (𝑇‘(#‘𝑆))⟩})
98	92, 96, 97	syl2anc 694	. . . . 5 ⊢ (𝜑 → (𝑇 ↾ {(#‘𝑆)}) = {⟨(#‘𝑆), (𝑇‘(#‘𝑆))⟩})
99	1	fveq1i 6230	. . . . . . . . 9 ⊢ (𝑇‘(#‘𝑆)) = ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(#‘𝑆))
100	52	nn0cnd 11391	. . . . . . . . . . . 12 ⊢ (𝜑 → (#‘𝑆) ∈ ℂ)
101	100	addid2d 10275	. . . . . . . . . . 11 ⊢ (𝜑 → (0 + (#‘𝑆)) = (#‘𝑆))
102	101	eqcomd 2657	. . . . . . . . . 10 ⊢ (𝜑 → (#‘𝑆) = (0 + (#‘𝑆)))
103	102	fveq2d 6233	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(#‘𝑆)) = ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(0 + (#‘𝑆))))
104	99, 103	syl5eq 2697	. . . . . . . 8 ⊢ (𝜑 → (𝑇‘(#‘𝑆)) = ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(0 + (#‘𝑆))))
105		1nn 11069	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
106	105	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℕ)
107	63, 106	syl5eqel 2734	. . . . . . . . . 10 ⊢ (𝜑 → (#‘⟨“(𝐾‘{𝑋})”⟩) ∈ ℕ)
108		lbfzo0 12547	. . . . . . . . . 10 ⊢ (0 ∈ (0..^(#‘⟨“(𝐾‘{𝑋})”⟩)) ↔ (#‘⟨“(𝐾‘{𝑋})”⟩) ∈ ℕ)
109	107, 108	sylibr 224	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (0..^(#‘⟨“(𝐾‘{𝑋})”⟩)))
110		ccatval3 13397	. . . . . . . . 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 1366	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(0 + (#‘𝑆))) = (⟨“(𝐾‘{𝑋})”⟩‘0))
112		fvex 6239	. . . . . . . . 9 ⊢ (𝐾‘{𝑋}) ∈ V
113		s1fv 13427	. . . . . . . . 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 2689	. . . . . . 7 ⊢ (𝜑 → (𝑇‘(#‘𝑆)) = (𝐾‘{𝑋}))
116	115	opeq2d 4440	. . . . . 6 ⊢ (𝜑 → ⟨(#‘𝑆), (𝑇‘(#‘𝑆))⟩ = ⟨(#‘𝑆), (𝐾‘{𝑋})⟩)
117	116	sneqd 4222	. . . . 5 ⊢ (𝜑 → {⟨(#‘𝑆), (𝑇‘(#‘𝑆))⟩} = {⟨(#‘𝑆), (𝐾‘{𝑋})⟩})
118	98, 117	eqtrd 2685	. . . 4 ⊢ (𝜑 → (𝑇 ↾ {(#‘𝑆)}) = {⟨(#‘𝑆), (𝐾‘{𝑋})⟩})
119	90, 118	breqtrrd 4713	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑇 ↾ {(#‘𝑆)}))
120		pgpfac.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Abel)
121		dprdsubg 18469	. . . . 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 18469	. . . . 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 18306	. . 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 6706	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (0..^(#‘𝑆)))) = (𝐺 DProd 𝑆))
128		pgpfac.5	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd 𝑆) = 𝑊)
129	127, 128	eqtrd 2685	. . . . . 6 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (0..^(#‘𝑆)))) = 𝑊)
130	118	oveq2d 6706	. . . . . . 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 2685	. . . . . 6 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ {(#‘𝑆)})) = (𝐾‘{𝑋}))
133	129, 132	ineq12d 3848	. . . . 5 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ (0..^(#‘𝑆)))) ∩ (𝐺 DProd (𝑇 ↾ {(#‘𝑆)}))) = (𝑊 ∩ (𝐾‘{𝑋})))
134		incom 3838	. . . . 5 ⊢ (𝑊 ∩ (𝐾‘{𝑋})) = ((𝐾‘{𝑋}) ∩ 𝑊)
135	133, 134	syl6eq 2701	. . . 4 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ (0..^(#‘𝑆)))) ∩ (𝐺 DProd (𝑇 ↾ {(#‘𝑆)}))) = ((𝐾‘{𝑋}) ∩ 𝑊))
136	4, 73	subg0 17647	. . . . . . 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 2703	. . . . 5 ⊢ (𝜑 → (0_g‘𝐺) = 0 )
140	139	sneqd 4222	. . . 4 ⊢ (𝜑 → {(0_g‘𝐺)} = { 0 })
141	126, 135, 140	3eqtr4d 2695	. . 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 18491	. 2 ⊢ (𝜑 → 𝐺dom DProd 𝑇)
143		eqid 2651	. . . . 5 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
144	49, 57, 71, 143, 142	dprdsplit 18493	. . . 4 ⊢ (𝜑 → (𝐺 DProd 𝑇) = ((𝐺 DProd (𝑇 ↾ (0..^(#‘𝑆))))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ {(#‘𝑆)}))))
145	129, 132	oveq12d 6708	. . . 4 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ (0..^(#‘𝑆))))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ {(#‘𝑆)}))) = (𝑊(LSSum‘𝐺)(𝐾‘{𝑋})))
146	129, 122	eqeltrrd 2731	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
147	143	lsmcom 18307	. . . . 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 1366	. . . 4 ⊢ (𝜑 → (𝑊(LSSum‘𝐺)(𝐾‘{𝑋})) = ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊))
149	144, 145, 148	3eqtrd 2689	. . 3 ⊢ (𝜑 → (𝐺 DProd 𝑇) = ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊))
150		pgpfac.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐻))
151	7	subgss 17642	. . . . . 6 ⊢ (𝑊 ∈ (SubGrp‘𝐻) → 𝑊 ⊆ (Base‘𝐻))
152	150, 151	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ (Base‘𝐻))
153	152, 13	sseqtr4d 3675	. . . 4 ⊢ (𝜑 → 𝑊 ⊆ 𝑈)
154		pgpfac.l	. . . . 5 ⊢ ⊕ = (LSSum‘𝐻)
155	4, 143, 154	subglsm 18132	. . . 4 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝑋}) ⊆ 𝑈 ∧ 𝑊 ⊆ 𝑈) → ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊) = ((𝐾‘{𝑋}) ⊕ 𝑊))
156	3, 23, 153, 155	syl3anc 1366	. . 3 ⊢ (𝜑 → ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊) = ((𝐾‘{𝑋}) ⊕ 𝑊))
157		pgpfac.s	. . 3 ⊢ (𝜑 → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈)
158	149, 156, 157	3eqtrd 2689	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑇) = 𝑈)
159		breq2 4689	. . . 4 ⊢ (𝑠 = 𝑇 → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑇))
160		oveq2 6698	. . . . 5 ⊢ (𝑠 = 𝑇 → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑇))
161	160	eqeq1d 2653	. . . 4 ⊢ (𝑠 = 𝑇 → ((𝐺 DProd 𝑠) = 𝑈 ↔ (𝐺 DProd 𝑇) = 𝑈))
162	159, 161	anbi12d 747	. . 3 ⊢ (𝑠 = 𝑇 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈) ↔ (𝐺dom DProd 𝑇 ∧ (𝐺 DProd 𝑇) = 𝑈)))
163	162	rspcev 3340	. 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 1364	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 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∀wral 2941 ∃wrex 2942 {crab 2945 Vcvv 3231 ∪ cun 3605 ∩ cin 3606 ⊆ wss 3607 ⊊ wpss 3608 ∅c0 3948 {csn 4210 ⟨cop 4216 class class class wbr 4685 dom cdm 5143 ran crn 5144 ↾ cres 5145 Fn wfn 5921 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 Fincfn 7997 0cc0 9974 1c1 9975 + caddc 9977 ℕcn 11058 ℕ₀cn0 11330 ℤcz 11415 ℤ_≥cuz 11725 ..^cfzo 12504 #chash 13157 Word cword 13323 ++ cconcat 13325 ⟨“cs1 13326 ℙcprime 15432 Basecbs 15904 ↾_s cress 15905 0_gc0g 16147 Moorecmre 16289 mrClscmrc 16290 ACScacs 16292 Grpcgrp 17469 SubGrpcsubg 17635 Cntzccntz 17794 odcod 17990 gExcgex 17991 pGrp cpgp 17992 LSSumclsm 18095 Abelcabl 18240 CycGrpccyg 18325 DProd cdprd 18438

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-tpos 7397 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-fzo 12505 df-seq 12842 df-hash 13158 df-word 13331 df-concat 13333 df-s1 13334 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-0g 16149 df-gsum 16150 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-mhm 17382 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-mulg 17588 df-subg 17638 df-ghm 17705 df-gim 17748 df-cntz 17796 df-oppg 17822 df-od 17994 df-pgp 17996 df-lsm 18097 df-cmn 18241 df-abl 18242 df-cyg 18326 df-dprd 18440

This theorem is referenced by: pgpfaclem2 18527

W3C validator