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Theorem pgpfaclem2 18481

Description: Lemma for pgpfac 18483. (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 17617	. . . . . . 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 222	. . . . 5 ⊢ (𝜑 → (𝑊 ∈ (SubGrp‘𝐺) ∧ 𝑊 ⊆ 𝑈))
7	6	simpld 475	. . . 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 479	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑈)
10		pgpfac.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ Fin)
11		pgpfac.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝐺)
12	11	subgss 17595	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
13	2, 12	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
14		ssfi 8180	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ Fin ∧ 𝑈 ⊆ 𝐵) → 𝑈 ∈ Fin)
15	10, 13, 14	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ Fin)
16		ssfi 8180	. . . . . . . . . 10 ⊢ ((𝑈 ∈ Fin ∧ 𝑊 ⊆ 𝑈) → 𝑊 ∈ Fin)
17	15, 9, 16	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Fin)
18		hashcl 13147	. . . . . . . . 9 ⊢ (𝑊 ∈ Fin → (#‘𝑊) ∈ ℕ₀)
19	17, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → (#‘𝑊) ∈ ℕ₀)
20	19	nn0red 11352	. . . . . . 7 ⊢ (𝜑 → (#‘𝑊) ∈ ℝ)
21		pgpfac.0	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝐻)
22		fvex 6201	. . . . . . . . . . . 12 ⊢ (0_g‘𝐻) ∈ V
23	21, 22	eqeltri 2697	. . . . . . . . . . 11 ⊢ 0 ∈ V
24		hashsng 13159	. . . . . . . . . . 11 ⊢ ( 0 ∈ V → (#‘{ 0 }) = 1)
25	23, 24	ax-mp 5	. . . . . . . . . 10 ⊢ (#‘{ 0 }) = 1
26		subgrcl 17599	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ (SubGrp‘𝐻) → 𝐻 ∈ Grp)
27		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐻) = (Base‘𝐻)
28	27	subgacs 17629	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 ∈ Grp → (SubGrp‘𝐻) ∈ (ACS‘(Base‘𝐻)))
29		acsmre 16313	. . . . . . . . . . . . . . . 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 16283	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ (Base‘𝐻))
33	3	subgbas 17598	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 = (Base‘𝐻))
34	2, 33	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 = (Base‘𝐻))
35	32, 34	sseqtr4d 3642	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ 𝑈)
36		ssfi 8180	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Fin ∧ (𝐾‘{𝑋}) ⊆ 𝑈) → (𝐾‘{𝑋}) ∈ Fin)
37	15, 35, 36	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ Fin)
38		pgpfac.x	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
39	38, 34	eleqtrd 2703	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻))
40	31	mrcsncl 16272	. . . . . . . . . . . . . . . 16 ⊢ (((SubGrp‘𝐻) ∈ (Moore‘(Base‘𝐻)) ∧ 𝑋 ∈ (Base‘𝐻)) → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
41	30, 39, 40	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ (SubGrp‘𝐻))
42	21	subg0cl 17602	. . . . . . . . . . . . . . 15 ⊢ ((𝐾‘{𝑋}) ∈ (SubGrp‘𝐻) → 0 ∈ (𝐾‘{𝑋}))
43	41, 42	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ (𝐾‘{𝑋}))
44	43	snssd 4340	. . . . . . . . . . . . 13 ⊢ (𝜑 → { 0 } ⊆ (𝐾‘{𝑋}))
45	39	snssd 4340	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑋} ⊆ (Base‘𝐻))
46	30, 31, 45	mrcssidd 16285	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑋} ⊆ (𝐾‘{𝑋}))
47		snssg 4327	. . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ (𝐾‘{𝑋}))
50		pgpfac.oe	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑂‘𝑋) = 𝐸)
51		pgpfac.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 ≠ 1)
52	50, 51	eqnetrd 2861	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘𝑋) ≠ 1)
53		pgpfac.o	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑂 = (od‘𝐻)
54	53, 21	od1 17976	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻 ∈ Grp → (𝑂‘ 0 ) = 1)
55	1, 26, 54	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑂‘ 0 ) = 1)
56		elsni 4194	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ { 0 } → 𝑋 = 0 )
57	56	fveq2d 6195	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ { 0 } → (𝑂‘𝑋) = (𝑂‘ 0 ))
58	57	eqeq1d 2624	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ { 0 } → ((𝑂‘𝑋) = 1 ↔ (𝑂‘ 0 ) = 1))
59	55, 58	syl5ibrcom 237	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋 ∈ { 0 } → (𝑂‘𝑋) = 1))
60	59	necon3ad 2807	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑂‘𝑋) ≠ 1 → ¬ 𝑋 ∈ { 0 }))
61	52, 60	mpd 15	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑋 ∈ { 0 })
62	44, 49, 61	ssnelpssd 3719	. . . . . . . . . . . 12 ⊢ (𝜑 → { 0 } ⊊ (𝐾‘{𝑋}))
63		php3 8146	. . . . . . . . . . . 12 ⊢ (((𝐾‘{𝑋}) ∈ Fin ∧ { 0 } ⊊ (𝐾‘{𝑋})) → { 0 } ≺ (𝐾‘{𝑋}))
64	37, 62, 63	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → { 0 } ≺ (𝐾‘{𝑋}))
65		snfi 8038	. . . . . . . . . . . 12 ⊢ { 0 } ∈ Fin
66		hashsdom 13170	. . . . . . . . . . . 12 ⊢ (({ 0 } ∈ Fin ∧ (𝐾‘{𝑋}) ∈ Fin) → ((#‘{ 0 }) < (#‘(𝐾‘{𝑋})) ↔ { 0 } ≺ (𝐾‘{𝑋})))
67	65, 37, 66	sylancr 695	. . . . . . . . . . 11 ⊢ (𝜑 → ((#‘{ 0 }) < (#‘(𝐾‘{𝑋})) ↔ { 0 } ≺ (𝐾‘{𝑋})))
68	64, 67	mpbird 247	. . . . . . . . . 10 ⊢ (𝜑 → (#‘{ 0 }) < (#‘(𝐾‘{𝑋})))
69	25, 68	syl5eqbrr 4689	. . . . . . . . 9 ⊢ (𝜑 → 1 < (#‘(𝐾‘{𝑋})))
70		1red 10055	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
71		hashcl 13147	. . . . . . . . . . . 12 ⊢ ((𝐾‘{𝑋}) ∈ Fin → (#‘(𝐾‘{𝑋})) ∈ ℕ₀)
72	37, 71	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘(𝐾‘{𝑋})) ∈ ℕ₀)
73	72	nn0red 11352	. . . . . . . . . 10 ⊢ (𝜑 → (#‘(𝐾‘{𝑋})) ∈ ℝ)
74	21	subg0cl 17602	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ (SubGrp‘𝐻) → 0 ∈ 𝑊)
75		ne0i 3921	. . . . . . . . . . . . 13 ⊢ ( 0 ∈ 𝑊 → 𝑊 ≠ ∅)
76	1, 74, 75	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ≠ ∅)
77		hashnncl 13157	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Fin → ((#‘𝑊) ∈ ℕ ↔ 𝑊 ≠ ∅))
78	17, 77	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((#‘𝑊) ∈ ℕ ↔ 𝑊 ≠ ∅))
79	76, 78	mpbird 247	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘𝑊) ∈ ℕ)
80	79	nngt0d 11064	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (#‘𝑊))
81		ltmul1 10873	. . . . . . . . . 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 1330	. . . . . . . . 9 ⊢ (𝜑 → (1 < (#‘(𝐾‘{𝑋})) ↔ (1 · (#‘𝑊)) < ((#‘(𝐾‘{𝑋})) · (#‘𝑊))))
83	69, 82	mpbid 222	. . . . . . . 8 ⊢ (𝜑 → (1 · (#‘𝑊)) < ((#‘(𝐾‘{𝑋})) · (#‘𝑊)))
84	20	recnd 10068	. . . . . . . . 9 ⊢ (𝜑 → (#‘𝑊) ∈ ℂ)
85	84	mulid2d 10058	. . . . . . . 8 ⊢ (𝜑 → (1 · (#‘𝑊)) = (#‘𝑊))
86		pgpfac.l	. . . . . . . . . 10 ⊢ ⊕ = (LSSum‘𝐻)
87		eqid 2622	. . . . . . . . . 10 ⊢ (Cntz‘𝐻) = (Cntz‘𝐻)
88		pgpfac.i	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })
89		pgpfac.g	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ Abel)
90	3	subgabl 18241	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)
91	89, 2, 90	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ Abel)
92	87, 91, 41, 1	ablcntzd 18260	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾‘{𝑋}) ⊆ ((Cntz‘𝐻)‘𝑊))
93	86, 21, 87, 41, 1, 88, 92, 37, 17	lsmhash 18118	. . . . . . . . 9 ⊢ (𝜑 → (#‘((𝐾‘{𝑋}) ⊕ 𝑊)) = ((#‘(𝐾‘{𝑋})) · (#‘𝑊)))
94		pgpfac.s	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈)
95	94	fveq2d 6195	. . . . . . . . 9 ⊢ (𝜑 → (#‘((𝐾‘{𝑋}) ⊕ 𝑊)) = (#‘𝑈))
96	93, 95	eqtr3d 2658	. . . . . . . 8 ⊢ (𝜑 → ((#‘(𝐾‘{𝑋})) · (#‘𝑊)) = (#‘𝑈))
97	83, 85, 96	3brtr3d 4684	. . . . . . 7 ⊢ (𝜑 → (#‘𝑊) < (#‘𝑈))
98	20, 97	ltned 10173	. . . . . 6 ⊢ (𝜑 → (#‘𝑊) ≠ (#‘𝑈))
99		fveq2 6191	. . . . . . 7 ⊢ (𝑊 = 𝑈 → (#‘𝑊) = (#‘𝑈))
100	99	necon3i 2826	. . . . . 6 ⊢ ((#‘𝑊) ≠ (#‘𝑈) → 𝑊 ≠ 𝑈)
101	98, 100	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ≠ 𝑈)
102		df-pss 3590	. . . . 5 ⊢ (𝑊 ⊊ 𝑈 ↔ (𝑊 ⊆ 𝑈 ∧ 𝑊 ≠ 𝑈))
103	9, 101, 102	sylanbrc 698	. . . 4 ⊢ (𝜑 → 𝑊 ⊊ 𝑈)
104		psseq1 3694	. . . . . 6 ⊢ (𝑡 = 𝑊 → (𝑡 ⊊ 𝑈 ↔ 𝑊 ⊊ 𝑈))
105		eqeq2 2633	. . . . . . . 8 ⊢ (𝑡 = 𝑊 → ((𝐺 DProd 𝑠) = 𝑡 ↔ (𝐺 DProd 𝑠) = 𝑊))
106	105	anbi2d 740	. . . . . . 7 ⊢ (𝑡 = 𝑊 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡) ↔ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊)))
107	106	rexbidv 3052	. . . . . 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 334	. . . . 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 3305	. . . 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 66	. . 3 ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊))
111		breq2 4657	. . . . 5 ⊢ (𝑠 = 𝑎 → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑎))
112		oveq2 6658	. . . . . 6 ⊢ (𝑠 = 𝑎 → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑎))
113	112	eqeq1d 2624	. . . . 5 ⊢ (𝑠 = 𝑎 → ((𝐺 DProd 𝑠) = 𝑊 ↔ (𝐺 DProd 𝑎) = 𝑊))
114	111, 113	anbi12d 747	. . . 4 ⊢ (𝑠 = 𝑎 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑊) ↔ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊)))
115	114	cbvrexv 3172	. . 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 208	. 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 481	. . 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 481	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑃 pGrp 𝐺)
121	10	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝐵 ∈ Fin)
122	2	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑈 ∈ (SubGrp‘𝐺))
123	8	adantr 481	. . 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 481	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝐸 ≠ 1)
126	38	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑋 ∈ 𝑈)
127	50	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → (𝑂‘𝑋) = 𝐸)
128	1	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑊 ∈ (SubGrp‘𝐻))
129	88	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })
130	94	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈)
131		simprl 794	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝑎 ∈ Word 𝐶)
132		simprrl 804	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → 𝐺dom DProd 𝑎)
133		simprrr 805	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑎 ∧ (𝐺 DProd 𝑎) = 𝑊))) → (𝐺 DProd 𝑎) = 𝑊)
134		eqid 2622	. . 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 18480	. 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 3035	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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 ⊊ wpss 3575 ∅c0 3915 {csn 4177 class class class wbr 4653 dom cdm 5114 ran crn 5115 ‘cfv 5888 (class class class)co 6650 ≺ csdm 7954 Fincfn 7955 ℝcr 9935 0cc0 9936 1c1 9937 · cmul 9941 < clt 10074 ℕcn 11020 ℕ₀cn0 11292 #chash 13117 Word cword 13291 ++ cconcat 13293 ⟨“cs1 13294 Basecbs 15857 ↾_s cress 15858 0_gc0g 16100 Moorecmre 16242 mrClscmrc 16243 ACScacs 16245 Grpcgrp 17422 SubGrpcsubg 17588 Cntzccntz 17748 odcod 17944 gExcgex 17945 pGrp cpgp 17946 LSSumclsm 18049 Abelcabl 18194 CycGrpccyg 18279 DProd cdprd 18392

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-gsum 16103 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-gim 17701 df-cntz 17750 df-oppg 17776 df-od 17948 df-pgp 17950 df-lsm 18051 df-pj1 18052 df-cmn 18195 df-abl 18196 df-cyg 18280 df-dprd 18394

This theorem is referenced by: pgpfaclem3 18482

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