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Theorem pgpfaclem2 18304

Description: Lemma for pgpfac 18306. (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	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐻))
2		pgpfac.u	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
3		pgpfac.h	. . . . . . . 8 ⊢ 𝐻 = (𝐺 ↾_s 𝑈)
4	3	subsubg 17440	. . . . . . 7 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝑊 ∈ (SubGrp‘𝐻) ↔ (𝑊 ∈ (SubGrp‘𝐺) ∧ 𝑊 ⊆ 𝑈)))
5	2, 4	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑊 ∈ (SubGrp‘𝐻) ↔ (𝑊 ∈ (SubGrp‘𝐺) ∧ 𝑊 ⊆ 𝑈)))
6	1, 5	mpbid 221	. . . . 5 ⊢ (𝜑 → (𝑊 ∈ (SubGrp‘𝐺) ∧ 𝑊 ⊆ 𝑈))
7	6	simpld 474	. . . 4 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
8		pgpfac.a	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))
9	6	simprd 478	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑈)
10		pgpfac.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ Fin)
11		pgpfac.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝐺)
12	11	subgss 17418	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
13	2, 12	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
14		ssfi 8065	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ Fin ∧ 𝑈 ⊆ 𝐵) → 𝑈 ∈ Fin)
15	10, 13, 14	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ Fin)
16		ssfi 8065	. . . . . . . . . 10 ⊢ ((𝑈 ∈ Fin ∧ 𝑊 ⊆ 𝑈) → 𝑊 ∈ Fin)
17	15, 9, 16	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Fin)
18		hashcl 13009	. . . . . . . . 9 ⊢ (𝑊 ∈ Fin → (#‘𝑊) ∈ ℕ₀)
19	17, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → (#‘𝑊) ∈ ℕ₀)
20	19	nn0red 11229	. . . . . . 7 ⊢ (𝜑 → (#‘𝑊) ∈ ℝ)
21		pgpfac.0	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝐻)
22		fvex 6113	. . . . . . . . . . . 12 ⊢ (0_g‘𝐻) ∈ V
23	21, 22	eqeltri 2684	. . . . . . . . . . 11 ⊢ 0 ∈ V
24		hashsng 13020	. . . . . . . . . . 11 ⊢ ( 0 ∈ V → (#‘{ 0 }) = 1)
25	23, 24	ax-mp 5	. . . . . . . . . 10 ⊢ (#‘{ 0 }) = 1
26		subgrcl 17422	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ (SubGrp‘𝐻) → 𝐻 ∈ Grp)
27		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐻) = (Base‘𝐻)
28	27	subgacs 17452	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
29		acsmre 16136	. . . . . . . . . . . . . . . 16 ⊢ ((SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)) → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
30	1, 26, 28, 29	4syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)))
31		pgpfac.k	. . . . . . . . . . . . . . 15 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐻))
32	30, 31	mrcssvd 16106	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ (Base‘𝐻))
33	3	subgbas 17421	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 = (Base‘𝐻))
34	2, 33	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 = (Base‘𝐻))
35	32, 34	sseqtr4d 3605	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ 𝑈)
36		ssfi 8065	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Fin ∧ (𝐾‘{𝑋}) ⊆ 𝑈) → (𝐾‘{𝑋}) ∈ Fin)
37	15, 35, 36	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ Fin)
38		pgpfac.x	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
39	38, 34	eleqtrd 2690	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻))
40	31	mrcsncl 16095	. . . . . . . . . . . . . . . 16 ⊢ (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
41	30, 39, 40	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
42	21	subg0cl 17425	. . . . . . . . . . . . . . 15 ⊢ ((𝐾‘{𝑋}) ∈ (SubGrp‘𝐻) → 0 ∈ (𝐾‘{𝑋}))
43	41, 42	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ (𝐾‘{𝑋}))
44	43	snssd 4281	. . . . . . . . . . . . 13 ⊢ (𝜑 → { 0 } ⊆ (𝐾‘{𝑋}))
45	39	snssd 4281	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑋} ⊆ (Base‘𝐻))
46	30, 31, 45	mrcssidd 16108	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑋} ⊆ (𝐾‘{𝑋}))
47		snssg 4268	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ 𝑈 → (𝑋 ∈ (𝐾‘{𝑋}) ↔ {𝑋} ⊆ (𝐾‘{𝑋})))
48	38, 47	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘{𝑋}) ↔ {𝑋} ⊆ (𝐾‘{𝑋})))
49	46, 48	mpbird 246	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ (𝐾‘{𝑋}))
50		pgpfac.oe	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑂‘𝑋) = 𝐸)
51		pgpfac.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 ≠ 1)
52	50, 51	eqnetrd 2849	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘𝑋) ≠ 1)
53		pgpfac.o	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑂 = (od‘𝐻)
54	53, 21	od1 17799	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻 ∈ Grp → (𝑂‘ 0 ) = 1)
55	1, 26, 54	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑂‘ 0 ) = 1)
56		elsni 4142	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ { 0 } → 𝑋 = 0 )
57	56	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ { 0 } → (𝑂‘𝑋) = (𝑂‘ 0 ))
58	57	eqeq1d 2612	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ { 0 } → ((𝑂‘𝑋) = 1 ↔ (𝑂‘ 0 ) = 1))
59	55, 58	syl5ibrcom 236	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋 ∈ { 0 } → (𝑂‘𝑋) = 1))
60	59	necon3ad 2795	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑂‘𝑋) ≠ 1 → ¬ 𝑋 ∈ { 0 }))
61	52, 60	mpd 15	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑋 ∈ { 0 })
62	44, 49, 61	ssnelpssd 3681	. . . . . . . . . . . 12 ⊢ (𝜑 → { 0 } ⊊ (𝐾‘{𝑋}))
63		php3 8031	. . . . . . . . . . . 12 ⊢ (((𝐾‘{𝑋}) ∈ Fin ∧ { 0 } ⊊ (𝐾‘{𝑋})) → { 0 } ≺ (𝐾‘{𝑋}))
64	37, 62, 63	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → { 0 } ≺ (𝐾‘{𝑋}))
65		snfi 7923	. . . . . . . . . . . 12 ⊢ { 0 } ∈ Fin
66		hashsdom 13031	. . . . . . . . . . . 12 ⊢ (({ 0 } ∈ Fin ∧ (𝐾‘{𝑋}) ∈ Fin) → ((#‘{ 0 }) < (#‘(𝐾‘{𝑋})) ↔ { 0 } ≺ (𝐾‘{𝑋})))
67	65, 37, 66	sylancr 694	. . . . . . . . . . 11 ⊢ (𝜑 → ((#‘{ 0 }) < (#‘(𝐾‘{𝑋})) ↔ { 0 } ≺ (𝐾‘{𝑋})))
68	64, 67	mpbird 246	. . . . . . . . . 10 ⊢ (𝜑 → (#‘{ 0 }) < (#‘(𝐾‘{𝑋})))
69	25, 68	syl5eqbrr 4619	. . . . . . . . 9 ⊢ (𝜑 → 1 < (#‘(𝐾‘{𝑋})))
70		1red 9934	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
71		hashcl 13009	. . . . . . . . . . . 12 ⊢ ((𝐾‘{𝑋}) ∈ Fin → (#‘(𝐾‘{𝑋})) ∈ ℕ₀)
72	37, 71	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘(𝐾‘{𝑋})) ∈ ℕ₀)
73	72	nn0red 11229	. . . . . . . . . 10 ⊢ (𝜑 → (#‘(𝐾‘{𝑋})) ∈ ℝ)
74	21	subg0cl 17425	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ (SubGrp‘𝐻) → 0 ∈ 𝑊)
75		ne0i 3880	. . . . . . . . . . . . 13 ⊢ ( 0 ∈ 𝑊 → 𝑊 ≠ ∅)
76	1, 74, 75	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ≠ ∅)
77		hashnncl 13018	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Fin → ((#‘𝑊) ∈ ℕ ↔ 𝑊 ≠ ∅))
78	17, 77	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((#‘𝑊) ∈ ℕ ↔ 𝑊 ≠ ∅))
79	76, 78	mpbird 246	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘𝑊) ∈ ℕ)
80	79	nngt0d 10941	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (#‘𝑊))
81		ltmul1 10752	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ (#‘(𝐾‘{𝑋})) ∈ ℝ ∧ ((#‘𝑊) ∈ ℝ ∧ 0 < (#‘𝑊))) → (1 < (#‘(𝐾‘{𝑋})) ↔ (1 · (#‘𝑊)) < ((#‘(𝐾‘{𝑋})) · (#‘𝑊))))
82	70, 73, 20, 80, 81	syl112anc 1322	. . . . . . . . 9 ⊢ (𝜑 → (1 < (#‘(𝐾‘{𝑋})) ↔ (1 · (#‘𝑊)) < ((#‘(𝐾‘{𝑋})) · (#‘𝑊))))
83	69, 82	mpbid 221	. . . . . . . 8 ⊢ (𝜑 → (1 · (#‘𝑊)) < ((#‘(𝐾‘{𝑋})) · (#‘𝑊)))
84	20	recnd 9947	. . . . . . . . 9 ⊢ (𝜑 → (#‘𝑊) ∈ ℂ)
85	84	mulid2d 9937	. . . . . . . 8 ⊢ (𝜑 → (1 · (#‘𝑊)) = (#‘𝑊))
86		pgpfac.l	. . . . . . . . . 10 ⊢ ⊕ = (LSSum‘𝐻)
87		eqid 2610	. . . . . . . . . 10 ⊢ (Cntz‘𝐻) = (Cntz‘𝐻)
88		pgpfac.i	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })
89		pgpfac.g	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ Abel)
90	3	subgabl 18064	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)
91	89, 2, 90	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ Abel)
92	87, 91, 41, 1	ablcntzd 18083	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ ((Cntz‘𝐻)‘𝑊))
93	86, 21, 87, 41, 1, 88, 92, 37, 17	lsmhash 17941	. . . . . . . . 9 ⊢ (𝜑 → (#‘((𝐾‘{𝑋}) ⊕ 𝑊)) = ((#‘(𝐾‘{𝑋})) · (#‘𝑊)))
94		pgpfac.s	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈)
95	94	fveq2d 6107	. . . . . . . . 9 ⊢ (𝜑 → (#‘((𝐾‘{𝑋}) ⊕ 𝑊)) = (#‘𝑈))
96	93, 95	eqtr3d 2646	. . . . . . . 8 ⊢ (𝜑 → ((#‘(𝐾‘{𝑋})) · (#‘𝑊)) = (#‘𝑈))
97	83, 85, 96	3brtr3d 4614	. . . . . . 7 ⊢ (𝜑 → (#‘𝑊) < (#‘𝑈))
98	20, 97	ltned 10052	. . . . . 6 ⊢ (𝜑 → (#‘𝑊) ≠ (#‘𝑈))
99		fveq2 6103	. . . . . . 7 ⊢ (𝑊 = 𝑈 → (#‘𝑊) = (#‘𝑈))
100	99	necon3i 2814	. . . . . 6 ⊢ ((#‘𝑊) ≠ (#‘𝑈) → 𝑊 ≠ 𝑈)
101	98, 100	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ≠ 𝑈)
102		df-pss 3556	. . . . 5 ⊢ (𝑊 ⊊ 𝑈 ↔ (𝑊 ⊆ 𝑈 ∧ 𝑊 ≠ 𝑈))
103	9, 101, 102	sylanbrc 695	. . . 4 ⊢ (𝜑 → 𝑊 ⊊ 𝑈)
104		psseq1 3656	. . . . . 6 ⊢ (𝑡 = 𝑊 → (𝑡 ⊊ 𝑈 ↔ 𝑊 ⊊ 𝑈))
105		eqeq2 2621	. . . . . . . 8 ⊢ (𝑡 = 𝑊 → ((𝐺 DProd 𝑠) = 𝑡 ↔ (𝐺 DProd 𝑠) = 𝑊))
106	105	anbi2d 736	. . . . . . 7 ⊢ (𝑡 = 𝑊 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊)))
107	106	rexbidv 3034	. . . . . 6 ⊢ (𝑡 = 𝑊 → (∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊)))
108	104, 107	imbi12d 333	. . . . 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))))
109	108	rspcv 3278	. . . 4 ⊢ (𝑊 ∈ (SubGrp‘𝐺) → (∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)) → (𝑊 ⊊ 𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊))))
110	7, 8, 103, 109	syl3c 64	. . 3 ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊))
111		breq2 4587	. . . . 5 ⊢ (𝑠 = 𝑎 → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑎))
112		oveq2 6557	. . . . . 6 ⊢ (𝑠 = 𝑎 → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑎))
113	112	eqeq1d 2612	. . . . 5 ⊢ (𝑠 = 𝑎 → ((𝐺 DProd 𝑠) = 𝑊 ↔ (𝐺 DProd 𝑎) = 𝑊))
114	111, 113	anbi12d 743	. . . 4 ⊢ (𝑠 = 𝑎 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊) ↔ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊)))
115	114	cbvrexv 3148	. . 3 ⊢ (∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊) ↔ ∃𝑎 ∈ Word 𝐶(𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))
116	110, 115	sylib 207	. 2 ⊢ (𝜑 → ∃𝑎 ∈ Word 𝐶(𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))
117		pgpfac.c	. . 3 ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾_s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
118	89	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝐺 ∈ Abel)
119		pgpfac.p	. . . 4 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
120	119	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑃 pGrp 𝐺)
121	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝐵 ∈ Fin)
122	2	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑈 ∈ (SubGrp‘𝐺))
123	8	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)))
124		pgpfac.e	. . 3 ⊢ 𝐸 = (gEx‘𝐻)
125	51	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝐸 ≠ 1)
126	38	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑋 ∈ 𝑈)
127	50	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → (𝑂‘𝑋) = 𝐸)
128	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑊 ∈ (SubGrp‘𝐻))
129	88	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })
130	94	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈)
131		simprl 790	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑎 ∈ Word 𝐶)
132		simprrl 800	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝐺dom DProd 𝑎)
133		simprrr 801	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → (𝐺 DProd 𝑎) = 𝑊)
134		eqid 2610	. . 3 ⊢ (𝑎 ++ ⟨“(𝐾‘{𝑋})”⟩) = (𝑎 ++ ⟨“(𝐾‘{𝑋})”⟩)
135	11, 117, 118, 120, 121, 122, 123, 3, 31, 53, 124, 21, 86, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134	pgpfaclem1 18303	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
136	116, 135	rexlimddv 3017	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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ⊊ wpss 3541 ∅c0 3874 {csn 4125 class class class wbr 4583 dom cdm 5038 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ≺ csdm 7840 Fincfn 7841 ℝcr 9814 0cc0 9815 1c1 9816 · cmul 9820 < clt 9953 ℕcn 10897 ℕ₀cn0 11169 #chash 12979 Word cword 13146 ++ cconcat 13148 ⟨“cs1 13149 Basecbs 15695 ↾_s cress 15696 0_gc0g 15923 Moorecmre 16065 mrClscmrc 16066 ACScacs 16068 Grpcgrp 17245 SubGrpcsubg 17411 Cntzccntz 17571 odcod 17767 gExcgex 17768 pGrp cpgp 17769 LSSumclsm 17872 Abelcabl 18017 CycGrpccyg 18102 DProd cdprd 18215

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-gim 17524 df-cntz 17573 df-oppg 17599 df-od 17771 df-pgp 17773 df-lsm 17874 df-pj1 17875 df-cmn 18018 df-abl 18019 df-cyg 18103 df-dprd 18217

This theorem is referenced by: pgpfaclem3 18305

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