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Theorem pgpfaclem2 20048

Description: Lemma for pgpfac 20050. (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)

**Assertion**
Ref	Expression
pgpfaclem2	⊢ (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 Allowed substitution hints: 𝜑(𝑠,𝑟) 𝐵(𝑟) 𝐶(𝑟) 𝑃(𝑡,𝑠,𝑟) ⊕ (𝑡,𝑠,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 pgpfaclem2 Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pgpfac.w	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐻))
2		pgpfac.u	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
3		pgpfac.h	. . . . . . . . 9 ⊢ 𝐻 = (𝐺 ↾_s 𝑈)
4	3	subsubg 19114	. . . . . . . 8 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝑊 ∈ (SubGrp‘𝐻) ↔ (𝑊 ∈ (SubGrp‘𝐺) ∧ 𝑊 ⊆ 𝑈)))
5	2, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑊 ∈ (SubGrp‘𝐻) ↔ (𝑊 ∈ (SubGrp‘𝐺) ∧ 𝑊 ⊆ 𝑈)))
6	1, 5	mpbid 232	. . . . . 6 ⊢ (𝜑 → (𝑊 ∈ (SubGrp‘𝐺) ∧ 𝑊 ⊆ 𝑈))
7	6	simprd 495	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑈)
8		pgpfac.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ Fin)
9		pgpfac.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝐺)
10	9	subgss 19092	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
11	2, 10	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
12	8, 11	ssfid 9170	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ Fin)
13	12, 7	ssfid 9170	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Fin)
14		hashcl 14307	. . . . . . . . 9 ⊢ (𝑊 ∈ Fin → (♯‘𝑊) ∈ ℕ₀)
15	13, 14	syl 17	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ₀)
16	15	nn0red 12488	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑊) ∈ ℝ)
17		pgpfac.0	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝐻)
18	17	fvexi 6846	. . . . . . . . . . 11 ⊢ 0 ∈ V
19		hashsng 14320	. . . . . . . . . . 11 ⊢ ( 0 ∈ V → (♯‘{ 0 }) = 1)
20	18, 19	ax-mp 5	. . . . . . . . . 10 ⊢ (♯‘{ 0 }) = 1
21		subgrcl 19096	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ (SubGrp‘𝐻) → 𝐻 ∈ Grp)
22		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐻) = (Base‘𝐻)
23	22	subgacs 19125	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
24		acsmre 17607	. . . . . . . . . . . . . . . 16 ⊢ ((SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)) → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
25	1, 21, 23, 24	4syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
26		pgpfac.k	. . . . . . . . . . . . . . 15 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐻))
27	25, 26	mrcssvd 17578	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ (Base‘𝐻))
28	3	subgbas 19095	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 = (Base‘𝐻))
29	2, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 = (Base‘𝐻))
30	27, 29	sseqtrrd 3960	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ 𝑈)
31	12, 30	ssfid 9170	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ Fin)
32		pgpfac.x	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
33	32, 29	eleqtrd 2839	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻))
34	26	mrcsncl 17567	. . . . . . . . . . . . . . . 16 ⊢ (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
35	25, 33, 34	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
36	17	subg0cl 19099	. . . . . . . . . . . . . . 15 ⊢ ((𝐾‘{𝑋}) ∈ (SubGrp‘𝐻) → 0 ∈ (𝐾‘{𝑋}))
37	35, 36	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ (𝐾‘{𝑋}))
38	37	snssd 4753	. . . . . . . . . . . . 13 ⊢ (𝜑 → { 0 } ⊆ (𝐾‘{𝑋}))
39	33	snssd 4753	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑋} ⊆ (Base‘𝐻))
40	25, 26, 39	mrcssidd 17580	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑋} ⊆ (𝐾‘{𝑋}))
41		snssg 4728	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ 𝑈 → (𝑋 ∈ (𝐾‘{𝑋}) ↔ {𝑋} ⊆ (𝐾‘{𝑋})))
42	32, 41	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘{𝑋}) ↔ {𝑋} ⊆ (𝐾‘{𝑋})))
43	40, 42	mpbird 257	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ (𝐾‘{𝑋}))
44		pgpfac.oe	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑂‘𝑋) = 𝐸)
45		pgpfac.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 ≠ 1)
46	44, 45	eqnetrd 3000	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘𝑋) ≠ 1)
47		pgpfac.o	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑂 = (od‘𝐻)
48	47, 17	od1 19523	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻 ∈ Grp → (𝑂‘ 0 ) = 1)
49	1, 21, 48	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑂‘ 0 ) = 1)
50		elsni 4585	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ { 0 } → 𝑋 = 0 )
51	50	fveqeq2d 6840	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ { 0 } → ((𝑂‘𝑋) = 1 ↔ (𝑂‘ 0 ) = 1))
52	49, 51	syl5ibrcom 247	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋 ∈ { 0 } → (𝑂‘𝑋) = 1))
53	52	necon3ad 2946	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑂‘𝑋) ≠ 1 → ¬ 𝑋 ∈ { 0 }))
54	46, 53	mpd 15	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑋 ∈ { 0 })
55	38, 43, 54	ssnelpssd 4056	. . . . . . . . . . . 12 ⊢ (𝜑 → { 0 } ⊊ (𝐾‘{𝑋}))
56		php3 9134	. . . . . . . . . . . 12 ⊢ (((𝐾‘{𝑋}) ∈ Fin ∧ { 0 } ⊊ (𝐾‘{𝑋})) → { 0 } ≺ (𝐾‘{𝑋}))
57	31, 55, 56	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → { 0 } ≺ (𝐾‘{𝑋}))
58		snfi 8981	. . . . . . . . . . . 12 ⊢ { 0 } ∈ Fin
59		hashsdom 14332	. . . . . . . . . . . 12 ⊢ (({ 0 } ∈ Fin ∧ (𝐾‘{𝑋}) ∈ Fin) → ((♯‘{ 0 }) < (♯‘(𝐾‘{𝑋})) ↔ { 0 } ≺ (𝐾‘{𝑋})))
60	58, 31, 59	sylancr 588	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘{ 0 }) < (♯‘(𝐾‘{𝑋})) ↔ { 0 } ≺ (𝐾‘{𝑋})))
61	57, 60	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘{ 0 }) < (♯‘(𝐾‘{𝑋})))
62	20, 61	eqbrtrrid 5122	. . . . . . . . 9 ⊢ (𝜑 → 1 < (♯‘(𝐾‘{𝑋})))
63		1red 11134	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
64		hashcl 14307	. . . . . . . . . . . 12 ⊢ ((𝐾‘{𝑋}) ∈ Fin → (♯‘(𝐾‘{𝑋})) ∈ ℕ₀)
65	31, 64	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝐾‘{𝑋})) ∈ ℕ₀)
66	65	nn0red 12488	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(𝐾‘{𝑋})) ∈ ℝ)
67	17	subg0cl 19099	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ (SubGrp‘𝐻) → 0 ∈ 𝑊)
68		ne0i 4282	. . . . . . . . . . . . 13 ⊢ ( 0 ∈ 𝑊 → 𝑊 ≠ ∅)
69	1, 67, 68	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ≠ ∅)
70		hashnncl 14317	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Fin → ((♯‘𝑊) ∈ ℕ ↔ 𝑊 ≠ ∅))
71	13, 70	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑊) ∈ ℕ ↔ 𝑊 ≠ ∅))
72	69, 71	mpbird 257	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ)
73	72	nngt0d 12215	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (♯‘𝑊))
74		ltmul1 11994	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ (♯‘(𝐾‘{𝑋})) ∈ ℝ ∧ ((♯‘𝑊) ∈ ℝ ∧ 0 < (♯‘𝑊))) → (1 < (♯‘(𝐾‘{𝑋})) ↔ (1 · (♯‘𝑊)) < ((♯‘(𝐾‘{𝑋})) · (♯‘𝑊))))
75	63, 66, 16, 73, 74	syl112anc 1377	. . . . . . . . 9 ⊢ (𝜑 → (1 < (♯‘(𝐾‘{𝑋})) ↔ (1 · (♯‘𝑊)) < ((♯‘(𝐾‘{𝑋})) · (♯‘𝑊))))
76	62, 75	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → (1 · (♯‘𝑊)) < ((♯‘(𝐾‘{𝑋})) · (♯‘𝑊)))
77	16	recnd 11162	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝑊) ∈ ℂ)
78	77	mullidd 11152	. . . . . . . 8 ⊢ (𝜑 → (1 · (♯‘𝑊)) = (♯‘𝑊))
79		pgpfac.l	. . . . . . . . . 10 ⊢ ⊕ = (LSSum‘𝐻)
80		eqid 2737	. . . . . . . . . 10 ⊢ (Cntz‘𝐻) = (Cntz‘𝐻)
81		pgpfac.i	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })
82		pgpfac.g	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ Abel)
83	3	subgabl 19800	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)
84	82, 2, 83	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ Abel)
85	80, 84, 35, 1	ablcntzd 19821	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ ((Cntz‘𝐻)‘𝑊))
86	79, 17, 80, 35, 1, 81, 85, 31, 13	lsmhash 19669	. . . . . . . . 9 ⊢ (𝜑 → (♯‘((𝐾‘{𝑋}) ⊕ 𝑊)) = ((♯‘(𝐾‘{𝑋})) · (♯‘𝑊)))
87		pgpfac.s	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈)
88	87	fveq2d 6836	. . . . . . . . 9 ⊢ (𝜑 → (♯‘((𝐾‘{𝑋}) ⊕ 𝑊)) = (♯‘𝑈))
89	86, 88	eqtr3d 2774	. . . . . . . 8 ⊢ (𝜑 → ((♯‘(𝐾‘{𝑋})) · (♯‘𝑊)) = (♯‘𝑈))
90	76, 78, 89	3brtr3d 5117	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑊) < (♯‘𝑈))
91	16, 90	ltned 11271	. . . . . 6 ⊢ (𝜑 → (♯‘𝑊) ≠ (♯‘𝑈))
92		fveq2 6832	. . . . . . 7 ⊢ (𝑊 = 𝑈 → (♯‘𝑊) = (♯‘𝑈))
93	92	necon3i 2965	. . . . . 6 ⊢ ((♯‘𝑊) ≠ (♯‘𝑈) → 𝑊 ≠ 𝑈)
94	91, 93	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ≠ 𝑈)
95		df-pss 3910	. . . . 5 ⊢ (𝑊 ⊊ 𝑈 ↔ (𝑊 ⊆ 𝑈 ∧ 𝑊 ≠ 𝑈))
96	7, 94, 95	sylanbrc 584	. . . 4 ⊢ (𝜑 → 𝑊 ⊊ 𝑈)
97		psseq1 4031	. . . . . 6 ⊢ (𝑡 = 𝑊 → (𝑡 ⊊ 𝑈 ↔ 𝑊 ⊊ 𝑈))
98		eqeq2 2749	. . . . . . . 8 ⊢ (𝑡 = 𝑊 → ((𝐺 DProd 𝑠) = 𝑡 ↔ (𝐺 DProd 𝑠) = 𝑊))
99	98	anbi2d 631	. . . . . . 7 ⊢ (𝑡 = 𝑊 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊)))
100	99	rexbidv 3162	. . . . . 6 ⊢ (𝑡 = 𝑊 → (∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊)))
101	97, 100	imbi12d 344	. . . . 5 ⊢ (𝑡 = 𝑊 → ((𝑡 ⊊ 𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) ↔ (𝑊 ⊊ 𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊))))
102		pgpfac.a	. . . . 5 ⊢ (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))
103	6	simpld 494	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
104	101, 102, 103	rspcdva 3566	. . . 4 ⊢ (𝜑 → (𝑊 ⊊ 𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊)))
105	96, 104	mpd 15	. . 3 ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊))
106		breq2 5090	. . . . 5 ⊢ (𝑠 = 𝑎 → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑎))
107		oveq2 7366	. . . . . 6 ⊢ (𝑠 = 𝑎 → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑎))
108	107	eqeq1d 2739	. . . . 5 ⊢ (𝑠 = 𝑎 → ((𝐺 DProd 𝑠) = 𝑊 ↔ (𝐺 DProd 𝑎) = 𝑊))
109	106, 108	anbi12d 633	. . . 4 ⊢ (𝑠 = 𝑎 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊) ↔ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊)))
110	109	cbvrexvw 3217	. . 3 ⊢ (∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊) ↔ ∃𝑎 ∈ Word 𝐶(𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))
111	105, 110	sylib 218	. 2 ⊢ (𝜑 → ∃𝑎 ∈ Word 𝐶(𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))
112		pgpfac.c	. . 3 ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾_s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
113	82	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝐺 ∈ Abel)
114		pgpfac.p	. . . 4 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
115	114	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑃 pGrp 𝐺)
116	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝐵 ∈ Fin)
117	2	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑈 ∈ (SubGrp‘𝐺))
118	102	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))
119		pgpfac.e	. . 3 ⊢ 𝐸 = (gEx‘𝐻)
120	45	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝐸 ≠ 1)
121	32	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑋 ∈ 𝑈)
122	44	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → (𝑂‘𝑋) = 𝐸)
123	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑊 ∈ (SubGrp‘𝐻))
124	81	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })
125	87	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈)
126		simprl 771	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑎 ∈ Word 𝐶)
127		simprrl 781	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝐺dom DProd 𝑎)
128		simprrr 782	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → (𝐺 DProd 𝑎) = 𝑊)
129		eqid 2737	. . 3 ⊢ (𝑎 ++ ⟨“(𝐾‘{𝑋})”⟩) = (𝑎 ++ ⟨“(𝐾‘{𝑋})”⟩)
130	9, 112, 113, 115, 116, 117, 118, 3, 26, 47, 119, 17, 79, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129	pgpfaclem1 20047	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
131	111, 130	rexlimddv 3145	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 ∩ cin 3889 ⊆ wss 3890 ⊊ wpss 3891 ∅c0 4274 {csn 4568 class class class wbr 5086 dom cdm 5622 ran crn 5623 ‘cfv 6490 (class class class)co 7358 ≺ csdm 8883 Fincfn 8884 ℝcr 11026 0cc0 11027 1c1 11028 · cmul 11032 < clt 11168 ℕcn 12163 ℕ₀cn0 12426 ♯chash 14281 Word cword 14464 ++ cconcat 14521 ⟨“cs1 14547 Basecbs 17168 ↾_s cress 17189 0_gc0g 17391 Moorecmre 17533 mrClscmrc 17534 ACScacs 17536 Grpcgrp 18898 SubGrpcsubg 19085 Cntzccntz 19279 odcod 19488 gExcgex 19489 pGrp cpgp 19490 LSSumclsm 19598 Abelcabl 19745 CycGrpccyg 19841 DProd cdprd 19959

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 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

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 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-tpos 8167 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-oadd 8400 df-er 8634 df-map 8766 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-sup 9346 df-inf 9347 df-oi 9416 df-dju 9814 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12164 df-2 12233 df-n0 12427 df-xnn0 12500 df-z 12514 df-uz 12778 df-fz 13451 df-fzo 13598 df-seq 13953 df-hash 14282 df-word 14465 df-concat 14522 df-s1 14548 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-0g 17393 df-gsum 17394 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18740 df-submnd 18741 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19033 df-subg 19088 df-ghm 19177 df-gim 19223 df-cntz 19281 df-oppg 19310 df-od 19492 df-pgp 19494 df-lsm 19600 df-pj1 19601 df-cmn 19746 df-abl 19747 df-cyg 19842 df-dprd 19961

This theorem is referenced by: pgpfaclem3 20049

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