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Theorem pgpfaclem1 19988

Description: Lemma for pgpfac 19991. (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 19034	. . . . . . . . . 10 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
6	3, 5	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ Grp)
7		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝐻) = (Base‘𝐻)
8	7	subgacs 19066	. . . . . . . . 9 ⊢ (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
9	6, 8	syl 17	. . . . . . . 8 ⊢ (𝜑 → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
10	9	acsmred 17554	. . . . . . 7 ⊢ (𝜑 → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
11		pgpfac.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
12	4	subgbas 19035	. . . . . . . . 9 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 = (Base‘𝐻))
13	3, 12	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 = (Base‘𝐻))
14	11, 13	eleqtrd 2831	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻))
15		pgpfac.k	. . . . . . . 8 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐻))
16	15	mrcsncl 17510	. . . . . . 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 19054	. . . . . . 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 7351	. . . . . . 7 ⊢ (𝐻 ↾_s (𝐾‘{𝑋})) = ((𝐺 ↾_s 𝑈) ↾_s (𝐾‘{𝑋}))
23	20	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ 𝑈)
24		ressabs 17151	. . . . . . . 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 2777	. . . . . 6 ⊢ (𝜑 → (𝐻 ↾_s (𝐾‘{𝑋})) = (𝐺 ↾_s (𝐾‘{𝑋})))
27	7, 15	cycsubgcyg2 19807	. . . . . . 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 2830	. . . . 5 ⊢ (𝜑 → (𝐺 ↾_s (𝐾‘{𝑋})) ∈ CycGrp)
30		pgpfac.p	. . . . . . 7 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
31		pgpprm 19498	. . . . . . 7 ⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)
32	30, 31	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℙ)
33		subgpgp 19502	. . . . . . 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 5878	. . . . . 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 4148	. . . 4 ⊢ (𝜑 → (𝐺 ↾_s (𝐾‘{𝑋})) ∈ (CycGrp ∩ ran pGrp ))
38		oveq2 7349	. . . . . 6 ⊢ (𝑟 = (𝐾‘{𝑋}) → (𝐺 ↾_s 𝑟) = (𝐺 ↾_s (𝐾‘{𝑋})))
39	38	eleq1d 2814	. . . . 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 3648	. . . 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 14754	. 2 ⊢ (𝜑 → 𝑇 ∈ Word 𝐶)
44		wrdf 14417	. . . . 5 ⊢ (𝑇 ∈ Word 𝐶 → 𝑇:(0..^(♯‘𝑇))⟶𝐶)
45	43, 44	syl 17	. . . 4 ⊢ (𝜑 → 𝑇:(0..^(♯‘𝑇))⟶𝐶)
46	40	ssrab3 4030	. . . 4 ⊢ 𝐶 ⊆ (SubGrp‘𝐺)
47		fss 6663	. . . 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 14432	. . . . . . . 8 ⊢ (𝑆 ∈ Word 𝐶 → (♯‘𝑆) ∈ ℕ₀)
50	2, 49	syl 17	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑆) ∈ ℕ₀)
51	50	nn0zd 12486	. . . . . 6 ⊢ (𝜑 → (♯‘𝑆) ∈ ℤ)
52		fzosn 13628	. . . . . 6 ⊢ ((♯‘𝑆) ∈ ℤ → ((♯‘𝑆)..^((♯‘𝑆) + 1)) = {(♯‘𝑆)})
53	51, 52	syl 17	. . . . 5 ⊢ (𝜑 → ((♯‘𝑆)..^((♯‘𝑆) + 1)) = {(♯‘𝑆)})
54	53	ineq2d 4168	. . . 4 ⊢ (𝜑 → ((0..^(♯‘𝑆)) ∩ ((♯‘𝑆)..^((♯‘𝑆) + 1))) = ((0..^(♯‘𝑆)) ∩ {(♯‘𝑆)}))
55		fzodisj 13585	. . . 4 ⊢ ((0..^(♯‘𝑆)) ∩ ((♯‘𝑆)..^((♯‘𝑆) + 1))) = ∅
56	54, 55	eqtr3di 2780	. . 3 ⊢ (𝜑 → ((0..^(♯‘𝑆)) ∩ {(♯‘𝑆)}) = ∅)
57	1	fveq2i 6820	. . . . . . 7 ⊢ (♯‘𝑇) = (♯‘(𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩))
58	42	s1cld 14503	. . . . . . . 8 ⊢ (𝜑 → ⟨“(𝐾‘{𝑋})”⟩ ∈ Word 𝐶)
59		ccatlen 14474	. . . . . . . 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 2777	. . . . . 6 ⊢ (𝜑 → (♯‘𝑇) = ((♯‘𝑆) + (♯‘⟨“(𝐾‘{𝑋})”⟩)))
62		s1len 14506	. . . . . . 7 ⊢ (♯‘⟨“(𝐾‘{𝑋})”⟩) = 1
63	62	oveq2i 7352	. . . . . 6 ⊢ ((♯‘𝑆) + (♯‘⟨“(𝐾‘{𝑋})”⟩)) = ((♯‘𝑆) + 1)
64	61, 63	eqtrdi 2781	. . . . 5 ⊢ (𝜑 → (♯‘𝑇) = ((♯‘𝑆) + 1))
65	64	oveq2d 7357	. . . 4 ⊢ (𝜑 → (0..^(♯‘𝑇)) = (0..^((♯‘𝑆) + 1)))
66		nn0uz 12766	. . . . . 6 ⊢ ℕ₀ = (ℤ_≥‘0)
67	50, 66	eleqtrdi 2839	. . . . 5 ⊢ (𝜑 → (♯‘𝑆) ∈ (ℤ_≥‘0))
68		fzosplitsn 13668	. . . . 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 2765	. . 3 ⊢ (𝜑 → (0..^(♯‘𝑇)) = ((0..^(♯‘𝑆)) ∪ {(♯‘𝑆)}))
71		eqid 2730	. . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
72		eqid 2730	. . 3 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
73		pgpfac.4	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
74		cats1un 14620	. . . . . . . 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 2777	. . . . . 6 ⊢ (𝜑 → 𝑇 = (𝑆 ∪ {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}))
77	76	reseq1d 5924	. . . . 5 ⊢ (𝜑 → (𝑇 ↾ (0..^(♯‘𝑆))) = ((𝑆 ∪ {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩}) ↾ (0..^(♯‘𝑆))))
78		wrdfn 14427	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐶 → 𝑆 Fn (0..^(♯‘𝑆)))
79	2, 78	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 Fn (0..^(♯‘𝑆)))
80		fzonel 13565	. . . . . 6 ⊢ ¬ (♯‘𝑆) ∈ (0..^(♯‘𝑆))
81		fsnunres 7117	. . . . . 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 2765	. . . 4 ⊢ (𝜑 → (𝑇 ↾ (0..^(♯‘𝑆))) = 𝑆)
84	73, 83	breqtrrd 5117	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑇 ↾ (0..^(♯‘𝑆))))
85		fvex 6830	. . . . . 6 ⊢ (♯‘𝑆) ∈ V
86		dprdsn 19943	. . . . . 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 14427	. . . . . . 7 ⊢ (𝑇 ∈ Word 𝐶 → 𝑇 Fn (0..^(♯‘𝑇)))
90	43, 89	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑇 Fn (0..^(♯‘𝑇)))
91		ssun2 4127	. . . . . . . 8 ⊢ {(♯‘𝑆)} ⊆ ((0..^(♯‘𝑆)) ∪ {(♯‘𝑆)})
92	85	snss 4735	. . . . . . . 8 ⊢ ((♯‘𝑆) ∈ ((0..^(♯‘𝑆)) ∪ {(♯‘𝑆)}) ↔ {(♯‘𝑆)} ⊆ ((0..^(♯‘𝑆)) ∪ {(♯‘𝑆)}))
93	91, 92	mpbir 231	. . . . . . 7 ⊢ (♯‘𝑆) ∈ ((0..^(♯‘𝑆)) ∪ {(♯‘𝑆)})
94	93, 70	eleqtrrid 2836	. . . . . 6 ⊢ (𝜑 → (♯‘𝑆) ∈ (0..^(♯‘𝑇)))
95		fnressn 7086	. . . . . 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 6818	. . . . . . . . 9 ⊢ (𝑇‘(♯‘𝑆)) = ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(♯‘𝑆))
98	50	nn0cnd 12436	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑆) ∈ ℂ)
99	98	addlidd 11306	. . . . . . . . . 10 ⊢ (𝜑 → (0 + (♯‘𝑆)) = (♯‘𝑆))
100	99	fveq2d 6821	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(0 + (♯‘𝑆))) = ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(♯‘𝑆)))
101	97, 100	eqtr4id 2784	. . . . . . . 8 ⊢ (𝜑 → (𝑇‘(♯‘𝑆)) = ((𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)‘(0 + (♯‘𝑆))))
102		1nn 12128	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
103	62, 102	eqeltri 2825	. . . . . . . . . . 11 ⊢ (♯‘⟨“(𝐾‘{𝑋})”⟩) ∈ ℕ
104		lbfzo0 13591	. . . . . . . . . . 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 14478	. . . . . . . . 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 6830	. . . . . . . . 9 ⊢ (𝐾‘{𝑋}) ∈ V
110		s1fv 14510	. . . . . . . . 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 2769	. . . . . . 7 ⊢ (𝜑 → (𝑇‘(♯‘𝑆)) = (𝐾‘{𝑋}))
113	112	opeq2d 4830	. . . . . 6 ⊢ (𝜑 → ⟨(♯‘𝑆), (𝑇‘(♯‘𝑆))⟩ = ⟨(♯‘𝑆), (𝐾‘{𝑋})⟩)
114	113	sneqd 4586	. . . . 5 ⊢ (𝜑 → {⟨(♯‘𝑆), (𝑇‘(♯‘𝑆))⟩} = {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩})
115	96, 114	eqtrd 2765	. . . 4 ⊢ (𝜑 → (𝑇 ↾ {(♯‘𝑆)}) = {⟨(♯‘𝑆), (𝐾‘{𝑋})⟩})
116	88, 115	breqtrrd 5117	. . 3 ⊢ (𝜑 → 𝐺dom DProd (𝑇 ↾ {(♯‘𝑆)}))
117		pgpfac.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Abel)
118		dprdsubg 19931	. . . . 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 19931	. . . . 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 19762	. . 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 7357	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆)))) = (𝐺 DProd 𝑆))
125		pgpfac.5	. . . . . . 7 ⊢ (𝜑 → (𝐺 DProd 𝑆) = 𝑊)
126	124, 125	eqtrd 2765	. . . . . 6 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆)))) = 𝑊)
127	115	oveq2d 7357	. . . . . . 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 2765	. . . . . 6 ⊢ (𝜑 → (𝐺 DProd (𝑇 ↾ {(♯‘𝑆)})) = (𝐾‘{𝑋}))
130	126, 129	ineq12d 4169	. . . . 5 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆)))) ∩ (𝐺 DProd (𝑇 ↾ {(♯‘𝑆)}))) = (𝑊 ∩ (𝐾‘{𝑋})))
131		incom 4157	. . . . 5 ⊢ (𝑊 ∩ (𝐾‘{𝑋})) = ((𝐾‘{𝑋}) ∩ 𝑊)
132	130, 131	eqtrdi 2781	. . . 4 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆)))) ∩ (𝐺 DProd (𝑇 ↾ {(♯‘𝑆)}))) = ((𝐾‘{𝑋}) ∩ 𝑊))
133	4, 72	subg0 19037	. . . . . . 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 2783	. . . . 5 ⊢ (𝜑 → (0_g‘𝐺) = 0 )
137	136	sneqd 4586	. . . 4 ⊢ (𝜑 → {(0_g‘𝐺)} = { 0 })
138	123, 132, 137	3eqtr4d 2775	. . 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 19953	. 2 ⊢ (𝜑 → 𝐺dom DProd 𝑇)
140		eqid 2730	. . . . 5 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
141	48, 56, 70, 140, 139	dprdsplit 19955	. . . 4 ⊢ (𝜑 → (𝐺 DProd 𝑇) = ((𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆))))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ {(♯‘𝑆)}))))
142	126, 129	oveq12d 7359	. . . 4 ⊢ (𝜑 → ((𝐺 DProd (𝑇 ↾ (0..^(♯‘𝑆))))(LSSum‘𝐺)(𝐺 DProd (𝑇 ↾ {(♯‘𝑆)}))) = (𝑊(LSSum‘𝐺)(𝐾‘{𝑋})))
143	126, 119	eqeltrrd 2830	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
144	140	lsmcom 19763	. . . . 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 2769	. . 3 ⊢ (𝜑 → (𝐺 DProd 𝑇) = ((𝐾‘{𝑋})(LSSum‘𝐺)𝑊))
147		pgpfac.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐻))
148	7	subgss 19032	. . . . . 6 ⊢ (𝑊 ∈ (SubGrp‘𝐻) → 𝑊 ⊆ (Base‘𝐻))
149	147, 148	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ (Base‘𝐻))
150	149, 13	sseqtrrd 3970	. . . 4 ⊢ (𝜑 → 𝑊 ⊆ 𝑈)
151		pgpfac.l	. . . . 5 ⊢ ⊕ = (LSSum‘𝐻)
152	4, 140, 151	subglsm 19578	. . . 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 2769	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑇) = 𝑈)
156		breq2 5093	. . . 4 ⊢ (𝑠 = 𝑇 → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑇))
157		oveq2 7349	. . . . 5 ⊢ (𝑠 = 𝑇 → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑇))
158	157	eqeq1d 2732	. . . 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 3575	. 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 1541 ∈ wcel 2110 ≠ wne 2926 ∀wral 3045 ∃wrex 3054 {crab 3393 Vcvv 3434 ∪ cun 3898 ∩ cin 3899 ⊆ wss 3900 ⊊ wpss 3901 ∅c0 4281 {csn 4574 ⟨cop 4580 class class class wbr 5089 dom cdm 5614 ran crn 5615 ↾ cres 5616 Fn wfn 6472 ⟶wf 6473 ‘cfv 6477 (class class class)co 7341 Fincfn 8864 0cc0 10998 1c1 10999 + caddc 11001 ℕcn 12117 ℕ₀cn0 12373 ℤcz 12460 ℤ_≥cuz 12724 ..^cfzo 13546 ♯chash 14229 Word cword 14412 ++ cconcat 14469 ⟨“cs1 14495 ℙcprime 16574 Basecbs 17112 ↾_s cress 17133 0_gc0g 17335 Moorecmre 17476 mrClscmrc 17477 ACScacs 17479 Grpcgrp 18838 SubGrpcsubg 19025 Cntzccntz 19220 odcod 19429 gExcgex 19430 pGrp cpgp 19431 LSSumclsm 19539 Abelcabl 19686 CycGrpccyg 19782 DProd cdprd 19900

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-isom 6486 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-tpos 8151 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-map 8747 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-sup 9321 df-inf 9322 df-oi 9391 df-card 9824 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-nn 12118 df-2 12180 df-n0 12374 df-z 12461 df-uz 12725 df-fz 13400 df-fzo 13547 df-seq 13901 df-hash 14230 df-word 14413 df-concat 14470 df-s1 14496 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-ress 17134 df-plusg 17166 df-0g 17337 df-gsum 17338 df-mre 17480 df-mrc 17481 df-acs 17483 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-mhm 18683 df-submnd 18684 df-grp 18841 df-minusg 18842 df-sbg 18843 df-mulg 18973 df-subg 19028 df-ghm 19118 df-gim 19164 df-cntz 19222 df-oppg 19251 df-od 19433 df-pgp 19435 df-lsm 19541 df-cmn 19687 df-abl 19688 df-cyg 19783 df-dprd 19902

This theorem is referenced by: pgpfaclem2 19989

W3C validator