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Theorem pgpfaclem2 20107

Description: Lemma for pgpfac 20109. (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 19174	. . . . . . . 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 234	. . . . . 6 ⊢ (𝜑 → (𝑊 ∈ (SubGrp‘𝐺) ∧ 𝑊 ⊆ 𝑈))
7	6	simprd 499	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑈)
8		pgpfac.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ Fin)
9		pgpfac.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝐺)
10	9	subgss 19152	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
11	2, 10	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
12	8, 11	ssfid 9209	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ Fin)
13	12, 7	ssfid 9209	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Fin)
14		hashcl 14366	. . . . . . . . 9 ⊢ (𝑊 ∈ Fin → (♯‘𝑊) ∈ ℕ₀)
15	13, 14	syl 17	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ₀)
16	15	nn0red 12540	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑊) ∈ ℝ)
17		pgpfac.0	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝐻)
18	17	fvexi 6877	. . . . . . . . . . 11 ⊢ 0 ∈ V
19		hashsng 14379	. . . . . . . . . . 11 ⊢ ( 0 ∈ V → (♯‘{ 0 }) = 1)
20	18, 19	ax-mp 5	. . . . . . . . . 10 ⊢ (♯‘{ 0 }) = 1
21		subgrcl 19156	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ (SubGrp‘𝐻) → 𝐻 ∈ Grp)
22		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐻) = (Base‘𝐻)
23	22	subgacs 19185	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
24		acsmre 17667	. . . . . . . . . . . . . . . 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 17638	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ (Base‘𝐻))
28	3	subgbas 19155	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 = (Base‘𝐻))
29	2, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 = (Base‘𝐻))
30	27, 29	sseqtrrd 3973	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ 𝑈)
31	12, 30	ssfid 9209	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ Fin)
32		pgpfac.x	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
33	32, 29	eleqtrd 2863	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻))
34	26	mrcsncl 17627	. . . . . . . . . . . . . . . 16 ⊢ (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
35	25, 33, 34	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
36	17	subg0cl 19159	. . . . . . . . . . . . . . 15 ⊢ ((𝐾‘{𝑋}) ∈ (SubGrp‘𝐻) → 0 ∈ (𝐾‘{𝑋}))
37	35, 36	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ (𝐾‘{𝑋}))
38	37	snssd 4744	. . . . . . . . . . . . 13 ⊢ (𝜑 → { 0 } ⊆ (𝐾‘{𝑋}))
39	33	snssd 4744	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑋} ⊆ (Base‘𝐻))
40	25, 26, 39	mrcssidd 17640	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑋} ⊆ (𝐾‘{𝑋}))
41		snssg 4741	. . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ (𝐾‘{𝑋}))
44		pgpfac.oe	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑂‘𝑋) = 𝐸)
45		pgpfac.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 ≠ 1)
46	44, 45	eqnetrd 3023	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘𝑋) ≠ 1)
47		pgpfac.o	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑂 = (od‘𝐻)
48	47, 17	od1 19582	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻 ∈ Grp → (𝑂‘ 0 ) = 1)
49	1, 21, 48	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑂‘ 0 ) = 1)
50		elsni 4598	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ { 0 } → 𝑋 = 0 )
51	50	fveqeq2d 6871	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ { 0 } → ((𝑂‘𝑋) = 1 ↔ (𝑂‘ 0 ) = 1))
52	49, 51	syl5ibrcom 249	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋 ∈ { 0 } → (𝑂‘𝑋) = 1))
53	52	necon3ad 2969	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑂‘𝑋) ≠ 1 → ¬ 𝑋 ∈ { 0 }))
54	46, 53	mpd 15	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑋 ∈ { 0 })
55	38, 43, 54	ssnelpssd 4069	. . . . . . . . . . . 12 ⊢ (𝜑 → { 0 } ⊊ (𝐾‘{𝑋}))
56		php3 9173	. . . . . . . . . . . 12 ⊢ (((𝐾‘{𝑋}) ∈ Fin ∧ { 0 } ⊊ (𝐾‘{𝑋})) → { 0 } ≺ (𝐾‘{𝑋}))
57	31, 55, 56	syl2anc 593	. . . . . . . . . . 11 ⊢ (𝜑 → { 0 } ≺ (𝐾‘{𝑋}))
58		snfi 9020	. . . . . . . . . . . 12 ⊢ { 0 } ∈ Fin
59		hashsdom 14391	. . . . . . . . . . . 12 ⊢ (({ 0 } ∈ Fin ∧ (𝐾‘{𝑋}) ∈ Fin) → ((♯‘{ 0 }) < (♯‘(𝐾‘{𝑋})) ↔ { 0 } ≺ (𝐾‘{𝑋})))
60	58, 31, 59	sylancr 596	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘{ 0 }) < (♯‘(𝐾‘{𝑋})) ↔ { 0 } ≺ (𝐾‘{𝑋})))
61	57, 60	mpbird 259	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘{ 0 }) < (♯‘(𝐾‘{𝑋})))
62	20, 61	eqbrtrrid 5135	. . . . . . . . 9 ⊢ (𝜑 → 1 < (♯‘(𝐾‘{𝑋})))
63		1red 11179	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
64		hashcl 14366	. . . . . . . . . . . 12 ⊢ ((𝐾‘{𝑋}) ∈ Fin → (♯‘(𝐾‘{𝑋})) ∈ ℕ₀)
65	31, 64	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝐾‘{𝑋})) ∈ ℕ₀)
66	65	nn0red 12540	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(𝐾‘{𝑋})) ∈ ℝ)
67	17	subg0cl 19159	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ (SubGrp‘𝐻) → 0 ∈ 𝑊)
68		ne0i 4293	. . . . . . . . . . . . 13 ⊢ ( 0 ∈ 𝑊 → 𝑊 ≠ ∅)
69	1, 67, 68	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ≠ ∅)
70		hashnncl 14376	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Fin → ((♯‘𝑊) ∈ ℕ ↔ 𝑊 ≠ ∅))
71	13, 70	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑊) ∈ ℕ ↔ 𝑊 ≠ ∅))
72	69, 71	mpbird 259	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ)
73	72	nngt0d 12259	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (♯‘𝑊))
74		ltmul1 12038	. . . . . . . . . 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 1392	. . . . . . . . 9 ⊢ (𝜑 → (1 < (♯‘(𝐾‘{𝑋})) ↔ (1 · (♯‘𝑊)) < ((♯‘(𝐾‘{𝑋})) · (♯‘𝑊))))
76	62, 75	mpbid 234	. . . . . . . 8 ⊢ (𝜑 → (1 · (♯‘𝑊)) < ((♯‘(𝐾‘{𝑋})) · (♯‘𝑊)))
77	16	recnd 11207	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝑊) ∈ ℂ)
78	77	mullidd 11197	. . . . . . . 8 ⊢ (𝜑 → (1 · (♯‘𝑊)) = (♯‘𝑊))
79		pgpfac.l	. . . . . . . . . 10 ⊢ ⊕ = (LSSum‘𝐻)
80		eqid 2761	. . . . . . . . . 10 ⊢ (Cntz‘𝐻) = (Cntz‘𝐻)
81		pgpfac.i	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })
82		pgpfac.g	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ Abel)
83	3	subgabl 19859	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)
84	82, 2, 83	syl2anc 593	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ Abel)
85	80, 84, 35, 1	ablcntzd 19880	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ ((Cntz‘𝐻)‘𝑊))
86	79, 17, 80, 35, 1, 81, 85, 31, 13	lsmhash 19728	. . . . . . . . 9 ⊢ (𝜑 → (♯‘((𝐾‘{𝑋}) ⊕ 𝑊)) = ((♯‘(𝐾‘{𝑋})) · (♯‘𝑊)))
87		pgpfac.s	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈)
88	87	fveq2d 6867	. . . . . . . . 9 ⊢ (𝜑 → (♯‘((𝐾‘{𝑋}) ⊕ 𝑊)) = (♯‘𝑈))
89	86, 88	eqtr3d 2798	. . . . . . . 8 ⊢ (𝜑 → ((♯‘(𝐾‘{𝑋})) · (♯‘𝑊)) = (♯‘𝑈))
90	76, 78, 89	3brtr3d 5130	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑊) < (♯‘𝑈))
91	16, 90	ltned 11316	. . . . . 6 ⊢ (𝜑 → (♯‘𝑊) ≠ (♯‘𝑈))
92		fveq2 6863	. . . . . . 7 ⊢ (𝑊 = 𝑈 → (♯‘𝑊) = (♯‘𝑈))
93	92	necon3i 2988	. . . . . 6 ⊢ ((♯‘𝑊) ≠ (♯‘𝑈) → 𝑊 ≠ 𝑈)
94	91, 93	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ≠ 𝑈)
95		df-pss 3924	. . . . 5 ⊢ (𝑊 ⊊ 𝑈 ↔ (𝑊 ⊆ 𝑈 ∧ 𝑊 ≠ 𝑈))
96	7, 94, 95	sylanbrc 592	. . . 4 ⊢ (𝜑 → 𝑊 ⊊ 𝑈)
97		psseq1 4043	. . . . . 6 ⊢ (𝑡 = 𝑊 → (𝑡 ⊊ 𝑈 ↔ 𝑊 ⊊ 𝑈))
98		eqeq2 2773	. . . . . . . 8 ⊢ (𝑡 = 𝑊 → ((𝐺 DProd 𝑠) = 𝑡 ↔ (𝐺 DProd 𝑠) = 𝑊))
99	98	anbi2d 639	. . . . . . 7 ⊢ (𝑡 = 𝑊 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊)))
100	99	rexbidv 3185	. . . . . 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 346	. . . . 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 498	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
104	101, 102, 103	rspcdva 3582	. . . 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 5103	. . . . 5 ⊢ (𝑠 = 𝑎 → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑎))
107		oveq2 7400	. . . . . 6 ⊢ (𝑠 = 𝑎 → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑎))
108	107	eqeq1d 2763	. . . . 5 ⊢ (𝑠 = 𝑎 → ((𝐺 DProd 𝑠) = 𝑊 ↔ (𝐺 DProd 𝑎) = 𝑊))
109	106, 108	anbi12d 641	. . . 4 ⊢ (𝑠 = 𝑎 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊) ↔ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊)))
110	109	cbvrexvw 3240	. . 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 220	. 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 484	. . 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 484	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑃 pGrp 𝐺)
116	8	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝐵 ∈ Fin)
117	2	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑈 ∈ (SubGrp‘𝐺))
118	102	adantr 484	. . 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 484	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝐸 ≠ 1)
121	32	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑋 ∈ 𝑈)
122	44	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → (𝑂‘𝑋) = 𝐸)
123	1	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑊 ∈ (SubGrp‘𝐻))
124	81	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })
125	87	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈)
126		simprl 780	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑎 ∈ Word 𝐶)
127		simprrl 790	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝐺dom DProd 𝑎)
128		simprrr 791	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → (𝐺 DProd 𝑎) = 𝑊)
129		eqid 2761	. . 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 20106	. 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 3168	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 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 ∃wrex 3085 {crab 3413 Vcvv 3453 ∩ cin 3903 ⊆ wss 3904 ⊊ wpss 3905 ∅c0 4285 {csn 4581 class class class wbr 5099 dom cdm 5645 ran crn 5646 ‘cfv 6517 (class class class)co 7392 ≺ csdm 8922 Fincfn 8923 ℝcr 11069 0cc0 11070 1c1 11071 · cmul 11075 < clt 11213 ℕcn 12207 ℕ₀cn0 12478 ♯chash 14340 Word cword 14523 ++ cconcat 14580 ⟨“cs1 14606 Basecbs 17228 ↾_s cress 17249 0_gc0g 17451 Moorecmre 17593 mrClscmrc 17594 ACScacs 17596 Grpcgrp 18958 SubGrpcsubg 19145 Cntzccntz 19338 odcod 19547 gExcgex 19548 pGrp cpgp 19549 LSSumclsm 19657 Abelcabl 19804 CycGrpccyg 19900 DProd cdprd 20018

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-om 7843 df-1st 7966 df-2nd 7967 df-supp 8136 df-tpos 8201 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-oadd 8436 df-er 8673 df-map 8805 df-ixp 8876 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-fsupp 9305 df-sup 9385 df-inf 9386 df-oi 9455 df-dju 9856 df-card 9894 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-n0 12479 df-xnn0 12552 df-z 12566 df-uz 12837 df-fz 13510 df-fzo 13657 df-seq 14012 df-hash 14341 df-word 14524 df-concat 14581 df-s1 14607 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-0g 17453 df-gsum 17454 df-mre 17597 df-mrc 17598 df-acs 17600 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-mhm 18800 df-submnd 18801 df-grp 18961 df-minusg 18962 df-sbg 18963 df-mulg 19093 df-subg 19148 df-ghm 19237 df-gim 19282 df-cntz 19340 df-oppg 19369 df-od 19551 df-pgp 19553 df-lsm 19659 df-pj1 19660 df-cmn 19805 df-abl 19806 df-cyg 19901 df-dprd 20020

This theorem is referenced by: pgpfaclem3 20108

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