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Theorem pgpfaclem2 20126

Description: Lemma for pgpfac 20128. (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 19189	. . . . . . . 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 19167	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
11	2, 10	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
12	8, 11	ssfid 9329	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ Fin)
13	12, 7	ssfid 9329	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Fin)
14		hashcl 14405	. . . . . . . . 9 ⊢ (𝑊 ∈ Fin → (♯‘𝑊) ∈ ℕ₀)
15	13, 14	syl 17	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ₀)
16	15	nn0red 12614	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑊) ∈ ℝ)
17		pgpfac.0	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝐻)
18	17	fvexi 6934	. . . . . . . . . . 11 ⊢ 0 ∈ V
19		hashsng 14418	. . . . . . . . . . 11 ⊢ ( 0 ∈ V → (♯‘{ 0 }) = 1)
20	18, 19	ax-mp 5	. . . . . . . . . 10 ⊢ (♯‘{ 0 }) = 1
21		subgrcl 19171	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ (SubGrp‘𝐻) → 𝐻 ∈ Grp)
22		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐻) = (Base‘𝐻)
23	22	subgacs 19201	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
24		acsmre 17710	. . . . . . . . . . . . . . . 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 17681	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ (Base‘𝐻))
28	3	subgbas 19170	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 = (Base‘𝐻))
29	2, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 = (Base‘𝐻))
30	27, 29	sseqtrrd 4050	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ 𝑈)
31	12, 30	ssfid 9329	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ Fin)
32		pgpfac.x	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
33	32, 29	eleqtrd 2846	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻))
34	26	mrcsncl 17670	. . . . . . . . . . . . . . . 16 ⊢ (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
35	25, 33, 34	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
36	17	subg0cl 19174	. . . . . . . . . . . . . . 15 ⊢ ((𝐾‘{𝑋}) ∈ (SubGrp‘𝐻) → 0 ∈ (𝐾‘{𝑋}))
37	35, 36	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ (𝐾‘{𝑋}))
38	37	snssd 4834	. . . . . . . . . . . . 13 ⊢ (𝜑 → { 0 } ⊆ (𝐾‘{𝑋}))
39	33	snssd 4834	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑋} ⊆ (Base‘𝐻))
40	25, 26, 39	mrcssidd 17683	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑋} ⊆ (𝐾‘{𝑋}))
41		snssg 4808	. . . . . . . . . . . . . . 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 3014	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘𝑋) ≠ 1)
47		pgpfac.o	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑂 = (od‘𝐻)
48	47, 17	od1 19601	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻 ∈ Grp → (𝑂‘ 0 ) = 1)
49	1, 21, 48	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑂‘ 0 ) = 1)
50		elsni 4665	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ { 0 } → 𝑋 = 0 )
51	50	fveqeq2d 6928	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ { 0 } → ((𝑂‘𝑋) = 1 ↔ (𝑂‘ 0 ) = 1))
52	49, 51	syl5ibrcom 247	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋 ∈ { 0 } → (𝑂‘𝑋) = 1))
53	52	necon3ad 2959	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑂‘𝑋) ≠ 1 → ¬ 𝑋 ∈ { 0 }))
54	46, 53	mpd 15	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑋 ∈ { 0 })
55	38, 43, 54	ssnelpssd 4138	. . . . . . . . . . . 12 ⊢ (𝜑 → { 0 } ⊊ (𝐾‘{𝑋}))
56		php3 9275	. . . . . . . . . . . 12 ⊢ (((𝐾‘{𝑋}) ∈ Fin ∧ { 0 } ⊊ (𝐾‘{𝑋})) → { 0 } ≺ (𝐾‘{𝑋}))
57	31, 55, 56	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → { 0 } ≺ (𝐾‘{𝑋}))
58		snfi 9109	. . . . . . . . . . . 12 ⊢ { 0 } ∈ Fin
59		hashsdom 14430	. . . . . . . . . . . 12 ⊢ (({ 0 } ∈ Fin ∧ (𝐾‘{𝑋}) ∈ Fin) → ((♯‘{ 0 }) < (♯‘(𝐾‘{𝑋})) ↔ { 0 } ≺ (𝐾‘{𝑋})))
60	58, 31, 59	sylancr 586	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘{ 0 }) < (♯‘(𝐾‘{𝑋})) ↔ { 0 } ≺ (𝐾‘{𝑋})))
61	57, 60	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘{ 0 }) < (♯‘(𝐾‘{𝑋})))
62	20, 61	eqbrtrrid 5202	. . . . . . . . 9 ⊢ (𝜑 → 1 < (♯‘(𝐾‘{𝑋})))
63		1red 11291	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
64		hashcl 14405	. . . . . . . . . . . 12 ⊢ ((𝐾‘{𝑋}) ∈ Fin → (♯‘(𝐾‘{𝑋})) ∈ ℕ₀)
65	31, 64	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝐾‘{𝑋})) ∈ ℕ₀)
66	65	nn0red 12614	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(𝐾‘{𝑋})) ∈ ℝ)
67	17	subg0cl 19174	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ (SubGrp‘𝐻) → 0 ∈ 𝑊)
68		ne0i 4364	. . . . . . . . . . . . 13 ⊢ ( 0 ∈ 𝑊 → 𝑊 ≠ ∅)
69	1, 67, 68	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ≠ ∅)
70		hashnncl 14415	. . . . . . . . . . . . 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 12342	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (♯‘𝑊))
74		ltmul1 12144	. . . . . . . . . 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 1374	. . . . . . . . 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 11318	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝑊) ∈ ℂ)
78	77	mullidd 11308	. . . . . . . 8 ⊢ (𝜑 → (1 · (♯‘𝑊)) = (♯‘𝑊))
79		pgpfac.l	. . . . . . . . . 10 ⊢ ⊕ = (LSSum‘𝐻)
80		eqid 2740	. . . . . . . . . 10 ⊢ (Cntz‘𝐻) = (Cntz‘𝐻)
81		pgpfac.i	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })
82		pgpfac.g	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ Abel)
83	3	subgabl 19878	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)
84	82, 2, 83	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ Abel)
85	80, 84, 35, 1	ablcntzd 19899	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ ((Cntz‘𝐻)‘𝑊))
86	79, 17, 80, 35, 1, 81, 85, 31, 13	lsmhash 19747	. . . . . . . . 9 ⊢ (𝜑 → (♯‘((𝐾‘{𝑋}) ⊕ 𝑊)) = ((♯‘(𝐾‘{𝑋})) · (♯‘𝑊)))
87		pgpfac.s	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈)
88	87	fveq2d 6924	. . . . . . . . 9 ⊢ (𝜑 → (♯‘((𝐾‘{𝑋}) ⊕ 𝑊)) = (♯‘𝑈))
89	86, 88	eqtr3d 2782	. . . . . . . 8 ⊢ (𝜑 → ((♯‘(𝐾‘{𝑋})) · (♯‘𝑊)) = (♯‘𝑈))
90	76, 78, 89	3brtr3d 5197	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑊) < (♯‘𝑈))
91	16, 90	ltned 11426	. . . . . 6 ⊢ (𝜑 → (♯‘𝑊) ≠ (♯‘𝑈))
92		fveq2 6920	. . . . . . 7 ⊢ (𝑊 = 𝑈 → (♯‘𝑊) = (♯‘𝑈))
93	92	necon3i 2979	. . . . . 6 ⊢ ((♯‘𝑊) ≠ (♯‘𝑈) → 𝑊 ≠ 𝑈)
94	91, 93	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ≠ 𝑈)
95		df-pss 3996	. . . . 5 ⊢ (𝑊 ⊊ 𝑈 ↔ (𝑊 ⊆ 𝑈 ∧ 𝑊 ≠ 𝑈))
96	7, 94, 95	sylanbrc 582	. . . 4 ⊢ (𝜑 → 𝑊 ⊊ 𝑈)
97		psseq1 4113	. . . . . 6 ⊢ (𝑡 = 𝑊 → (𝑡 ⊊ 𝑈 ↔ 𝑊 ⊊ 𝑈))
98		eqeq2 2752	. . . . . . . 8 ⊢ (𝑡 = 𝑊 → ((𝐺 DProd 𝑠) = 𝑡 ↔ (𝐺 DProd 𝑠) = 𝑊))
99	98	anbi2d 629	. . . . . . 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 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 3636	. . . 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 5170	. . . . 5 ⊢ (𝑠 = 𝑎 → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑎))
107		oveq2 7456	. . . . . 6 ⊢ (𝑠 = 𝑎 → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑎))
108	107	eqeq1d 2742	. . . . 5 ⊢ (𝑠 = 𝑎 → ((𝐺 DProd 𝑠) = 𝑊 ↔ (𝐺 DProd 𝑎) = 𝑊))
109	106, 108	anbi12d 631	. . . 4 ⊢ (𝑠 = 𝑎 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊) ↔ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊)))
110	109	cbvrexvw 3244	. . 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 2740	. . 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 20125	. 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 3167	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 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 {crab 3443 Vcvv 3488 ∩ cin 3975 ⊆ wss 3976 ⊊ wpss 3977 ∅c0 4352 {csn 4648 class class class wbr 5166 dom cdm 5700 ran crn 5701 ‘cfv 6573 (class class class)co 7448 ≺ csdm 9002 Fincfn 9003 ℝcr 11183 0cc0 11184 1c1 11185 · cmul 11189 < clt 11324 ℕcn 12293 ℕ₀cn0 12553 ♯chash 14379 Word cword 14562 ++ cconcat 14618 ⟨“cs1 14643 Basecbs 17258 ↾_s cress 17287 0_gc0g 17499 Moorecmre 17640 mrClscmrc 17641 ACScacs 17643 Grpcgrp 18973 SubGrpcsubg 19160 Cntzccntz 19355 odcod 19566 gExcgex 19567 pGrp cpgp 19568 LSSumclsm 19676 Abelcabl 19823 CycGrpccyg 19919 DProd cdprd 20037

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-n0 12554 df-xnn0 12626 df-z 12640 df-uz 12904 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-word 14563 df-concat 14619 df-s1 14644 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-0g 17501 df-gsum 17502 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-ghm 19253 df-gim 19299 df-cntz 19357 df-oppg 19386 df-od 19570 df-pgp 19572 df-lsm 19678 df-pj1 19679 df-cmn 19824 df-abl 19825 df-cyg 19920 df-dprd 20039

This theorem is referenced by: pgpfaclem3 20127

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