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Theorem pgpfaclem2 20021

Description: Lemma for pgpfac 20023. (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 19088	. . . . . . . 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 19066	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
11	2, 10	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
12	8, 11	ssfid 9219	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ Fin)
13	12, 7	ssfid 9219	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Fin)
14		hashcl 14328	. . . . . . . . 9 ⊢ (𝑊 ∈ Fin → (♯‘𝑊) ∈ ℕ₀)
15	13, 14	syl 17	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ₀)
16	15	nn0red 12511	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑊) ∈ ℝ)
17		pgpfac.0	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝐻)
18	17	fvexi 6875	. . . . . . . . . . 11 ⊢ 0 ∈ V
19		hashsng 14341	. . . . . . . . . . 11 ⊢ ( 0 ∈ V → (♯‘{ 0 }) = 1)
20	18, 19	ax-mp 5	. . . . . . . . . 10 ⊢ (♯‘{ 0 }) = 1
21		subgrcl 19070	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ (SubGrp‘𝐻) → 𝐻 ∈ Grp)
22		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐻) = (Base‘𝐻)
23	22	subgacs 19100	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
24		acsmre 17620	. . . . . . . . . . . . . . . 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 17591	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ (Base‘𝐻))
28	3	subgbas 19069	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 = (Base‘𝐻))
29	2, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 = (Base‘𝐻))
30	27, 29	sseqtrrd 3987	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ 𝑈)
31	12, 30	ssfid 9219	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ Fin)
32		pgpfac.x	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
33	32, 29	eleqtrd 2831	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻))
34	26	mrcsncl 17580	. . . . . . . . . . . . . . . 16 ⊢ (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
35	25, 33, 34	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
36	17	subg0cl 19073	. . . . . . . . . . . . . . 15 ⊢ ((𝐾‘{𝑋}) ∈ (SubGrp‘𝐻) → 0 ∈ (𝐾‘{𝑋}))
37	35, 36	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ (𝐾‘{𝑋}))
38	37	snssd 4776	. . . . . . . . . . . . 13 ⊢ (𝜑 → { 0 } ⊆ (𝐾‘{𝑋}))
39	33	snssd 4776	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑋} ⊆ (Base‘𝐻))
40	25, 26, 39	mrcssidd 17593	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑋} ⊆ (𝐾‘{𝑋}))
41		snssg 4750	. . . . . . . . . . . . . . 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 2993	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘𝑋) ≠ 1)
47		pgpfac.o	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑂 = (od‘𝐻)
48	47, 17	od1 19496	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻 ∈ Grp → (𝑂‘ 0 ) = 1)
49	1, 21, 48	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑂‘ 0 ) = 1)
50		elsni 4609	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ { 0 } → 𝑋 = 0 )
51	50	fveqeq2d 6869	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ { 0 } → ((𝑂‘𝑋) = 1 ↔ (𝑂‘ 0 ) = 1))
52	49, 51	syl5ibrcom 247	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋 ∈ { 0 } → (𝑂‘𝑋) = 1))
53	52	necon3ad 2939	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑂‘𝑋) ≠ 1 → ¬ 𝑋 ∈ { 0 }))
54	46, 53	mpd 15	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑋 ∈ { 0 })
55	38, 43, 54	ssnelpssd 4081	. . . . . . . . . . . 12 ⊢ (𝜑 → { 0 } ⊊ (𝐾‘{𝑋}))
56		php3 9179	. . . . . . . . . . . 12 ⊢ (((𝐾‘{𝑋}) ∈ Fin ∧ { 0 } ⊊ (𝐾‘{𝑋})) → { 0 } ≺ (𝐾‘{𝑋}))
57	31, 55, 56	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → { 0 } ≺ (𝐾‘{𝑋}))
58		snfi 9017	. . . . . . . . . . . 12 ⊢ { 0 } ∈ Fin
59		hashsdom 14353	. . . . . . . . . . . 12 ⊢ (({ 0 } ∈ Fin ∧ (𝐾‘{𝑋}) ∈ Fin) → ((♯‘{ 0 }) < (♯‘(𝐾‘{𝑋})) ↔ { 0 } ≺ (𝐾‘{𝑋})))
60	58, 31, 59	sylancr 587	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘{ 0 }) < (♯‘(𝐾‘{𝑋})) ↔ { 0 } ≺ (𝐾‘{𝑋})))
61	57, 60	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘{ 0 }) < (♯‘(𝐾‘{𝑋})))
62	20, 61	eqbrtrrid 5146	. . . . . . . . 9 ⊢ (𝜑 → 1 < (♯‘(𝐾‘{𝑋})))
63		1red 11182	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
64		hashcl 14328	. . . . . . . . . . . 12 ⊢ ((𝐾‘{𝑋}) ∈ Fin → (♯‘(𝐾‘{𝑋})) ∈ ℕ₀)
65	31, 64	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝐾‘{𝑋})) ∈ ℕ₀)
66	65	nn0red 12511	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(𝐾‘{𝑋})) ∈ ℝ)
67	17	subg0cl 19073	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ (SubGrp‘𝐻) → 0 ∈ 𝑊)
68		ne0i 4307	. . . . . . . . . . . . 13 ⊢ ( 0 ∈ 𝑊 → 𝑊 ≠ ∅)
69	1, 67, 68	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ≠ ∅)
70		hashnncl 14338	. . . . . . . . . . . . 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 12242	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (♯‘𝑊))
74		ltmul1 12039	. . . . . . . . . 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 1376	. . . . . . . . 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 11209	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝑊) ∈ ℂ)
78	77	mullidd 11199	. . . . . . . 8 ⊢ (𝜑 → (1 · (♯‘𝑊)) = (♯‘𝑊))
79		pgpfac.l	. . . . . . . . . 10 ⊢ ⊕ = (LSSum‘𝐻)
80		eqid 2730	. . . . . . . . . 10 ⊢ (Cntz‘𝐻) = (Cntz‘𝐻)
81		pgpfac.i	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })
82		pgpfac.g	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ Abel)
83	3	subgabl 19773	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)
84	82, 2, 83	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ Abel)
85	80, 84, 35, 1	ablcntzd 19794	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ ((Cntz‘𝐻)‘𝑊))
86	79, 17, 80, 35, 1, 81, 85, 31, 13	lsmhash 19642	. . . . . . . . 9 ⊢ (𝜑 → (♯‘((𝐾‘{𝑋}) ⊕ 𝑊)) = ((♯‘(𝐾‘{𝑋})) · (♯‘𝑊)))
87		pgpfac.s	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈)
88	87	fveq2d 6865	. . . . . . . . 9 ⊢ (𝜑 → (♯‘((𝐾‘{𝑋}) ⊕ 𝑊)) = (♯‘𝑈))
89	86, 88	eqtr3d 2767	. . . . . . . 8 ⊢ (𝜑 → ((♯‘(𝐾‘{𝑋})) · (♯‘𝑊)) = (♯‘𝑈))
90	76, 78, 89	3brtr3d 5141	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑊) < (♯‘𝑈))
91	16, 90	ltned 11317	. . . . . 6 ⊢ (𝜑 → (♯‘𝑊) ≠ (♯‘𝑈))
92		fveq2 6861	. . . . . . 7 ⊢ (𝑊 = 𝑈 → (♯‘𝑊) = (♯‘𝑈))
93	92	necon3i 2958	. . . . . 6 ⊢ ((♯‘𝑊) ≠ (♯‘𝑈) → 𝑊 ≠ 𝑈)
94	91, 93	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ≠ 𝑈)
95		df-pss 3937	. . . . 5 ⊢ (𝑊 ⊊ 𝑈 ↔ (𝑊 ⊆ 𝑈 ∧ 𝑊 ≠ 𝑈))
96	7, 94, 95	sylanbrc 583	. . . 4 ⊢ (𝜑 → 𝑊 ⊊ 𝑈)
97		psseq1 4056	. . . . . 6 ⊢ (𝑡 = 𝑊 → (𝑡 ⊊ 𝑈 ↔ 𝑊 ⊊ 𝑈))
98		eqeq2 2742	. . . . . . . 8 ⊢ (𝑡 = 𝑊 → ((𝐺 DProd 𝑠) = 𝑡 ↔ (𝐺 DProd 𝑠) = 𝑊))
99	98	anbi2d 630	. . . . . . 7 ⊢ (𝑡 = 𝑊 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊)))
100	99	rexbidv 3158	. . . . . 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 3592	. . . 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 5114	. . . . 5 ⊢ (𝑠 = 𝑎 → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑎))
107		oveq2 7398	. . . . . 6 ⊢ (𝑠 = 𝑎 → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑎))
108	107	eqeq1d 2732	. . . . 5 ⊢ (𝑠 = 𝑎 → ((𝐺 DProd 𝑠) = 𝑊 ↔ (𝐺 DProd 𝑎) = 𝑊))
109	106, 108	anbi12d 632	. . . 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 770	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑎 ∈ Word 𝐶)
127		simprrl 780	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝐺dom DProd 𝑎)
128		simprrr 781	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → (𝐺 DProd 𝑎) = 𝑊)
129		eqid 2730	. . 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 20020	. 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 3141	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 2926 ∀wral 3045 ∃wrex 3054 {crab 3408 Vcvv 3450 ∩ cin 3916 ⊆ wss 3917 ⊊ wpss 3918 ∅c0 4299 {csn 4592 class class class wbr 5110 dom cdm 5641 ran crn 5642 ‘cfv 6514 (class class class)co 7390 ≺ csdm 8920 Fincfn 8921 ℝcr 11074 0cc0 11075 1c1 11076 · cmul 11080 < clt 11215 ℕcn 12193 ℕ₀cn0 12449 ♯chash 14302 Word cword 14485 ++ cconcat 14542 ⟨“cs1 14567 Basecbs 17186 ↾_s cress 17207 0_gc0g 17409 Moorecmre 17550 mrClscmrc 17551 ACScacs 17553 Grpcgrp 18872 SubGrpcsubg 19059 Cntzccntz 19254 odcod 19461 gExcgex 19462 pGrp cpgp 19463 LSSumclsm 19571 Abelcabl 19718 CycGrpccyg 19814 DProd cdprd 19932

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

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 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 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-er 8674 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-sup 9400 df-inf 9401 df-oi 9470 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-n0 12450 df-xnn0 12523 df-z 12537 df-uz 12801 df-fz 13476 df-fzo 13623 df-seq 13974 df-hash 14303 df-word 14486 df-concat 14543 df-s1 14568 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-0g 17411 df-gsum 17412 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-mhm 18717 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-mulg 19007 df-subg 19062 df-ghm 19152 df-gim 19198 df-cntz 19256 df-oppg 19285 df-od 19465 df-pgp 19467 df-lsm 19573 df-pj1 19574 df-cmn 19719 df-abl 19720 df-cyg 19815 df-dprd 19934

This theorem is referenced by: pgpfaclem3 20022

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