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 e. (SubGrp ` G ) | ( G ↾s r ) e. (CycGrp i^i ran pGrp ) }
pgpfac.g	|- ( ph -> G e. Abel )
pgpfac.p	|- ( ph -> P pGrp G )
pgpfac.f	|- ( ph -> B e. Fin )
pgpfac.u	|- ( ph -> U e. (SubGrp ` G ) )
pgpfac.a	|- ( ph -> A. t e. (SubGrp ` G ) ( t C. U -> E. s e. Word C ( G dom 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. = ( 0g ` H )
pgpfac.l	|- .(+) = ( LSSum ` H )
pgpfac.1	|- ( ph -> E =/= 1 )
pgpfac.x	|- ( ph -> X e. U )
pgpfac.oe	|- ( ph -> ( O ` X ) = E )
pgpfac.w	|- ( ph -> W e. (SubGrp ` H ) )
pgpfac.i	|- ( ph -> ( ( K ` { X } ) i^i W ) = { .0. } )
pgpfac.s	|- ( ph -> ( ( K ` { X } ) .(+) W ) = U )

Assertion

Ref	Expression
pgpfaclem2	|- ( ph -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = U ) )

Distinct variable groups:   t, s, C   s, r, t, G   K, r, s   ph, t   B, s, t   U, r, s, t   W, s, t   X, r, s

Allowed substitution hints:   ph( 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 |- ( ph -> W e. (SubGrp ` H ) )
2		pgpfac.u	. . . . . . 7 |- ( ph -> U e. (SubGrp ` G ) )
3		pgpfac.h	. . . . . . . 8 |- H = ( G ↾s U )
4	3	subsubg 17617	. . . . . . 7 |- ( U e. (SubGrp ` G ) -> ( W e. (SubGrp ` H ) <-> ( W e. (SubGrp ` G ) /\ W C_ U ) ) )
5	2, 4	syl 17	. . . . . 6 |- ( ph -> ( W e. (SubGrp ` H ) <-> ( W e. (SubGrp ` G ) /\ W C_ U ) ) )
6	1, 5	mpbid 222	. . . . 5 |- ( ph -> ( W e. (SubGrp ` G ) /\ W C_ U ) )
7	6	simpld 475	. . . 4 |- ( ph -> W e. (SubGrp ` G ) )
8		pgpfac.a	. . . 4 |- ( ph -> A. t e. (SubGrp ` G ) ( t C. U -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) )
9	6	simprd 479	. . . . 5 |- ( ph -> W C_ U )
10		pgpfac.f	. . . . . . . . . . 11 |- ( ph -> B e. Fin )
11		pgpfac.b	. . . . . . . . . . . . 13 |- B = ( Base ` G )
12	11	subgss 17595	. . . . . . . . . . . 12 |- ( U e. (SubGrp ` G ) -> U C_ B )
13	2, 12	syl 17	. . . . . . . . . . 11 |- ( ph -> U C_ B )
14		ssfi 8180	. . . . . . . . . . 11 |- ( ( B e. Fin /\ U C_ B ) -> U e. Fin )
15	10, 13, 14	syl2anc 693	. . . . . . . . . 10 |- ( ph -> U e. Fin )
16		ssfi 8180	. . . . . . . . . 10 |- ( ( U e. Fin /\ W C_ U ) -> W e. Fin )
17	15, 9, 16	syl2anc 693	. . . . . . . . 9 |- ( ph -> W e. Fin )
18		hashcl 13147	. . . . . . . . 9 |- ( W e. Fin -> ( # ` W ) e. NN0 )
19	17, 18	syl 17	. . . . . . . 8 |- ( ph -> ( # ` W ) e. NN0 )
20	19	nn0red 11352	. . . . . . 7 |- ( ph -> ( # ` W ) e. RR )
21		pgpfac.0	. . . . . . . . . . . 12 |- .0. = ( 0g ` H )
22		fvex 6201	. . . . . . . . . . . 12 |- ( 0g ` H ) e. _V
23	21, 22	eqeltri 2697	. . . . . . . . . . 11 |- .0. e. _V
24		hashsng 13159	. . . . . . . . . . 11 |- ( .0. e. _V -> ( # ` { .0. } ) = 1 )
25	23, 24	ax-mp 5	. . . . . . . . . 10 |- ( # ` { .0. } ) = 1
26		subgrcl 17599	. . . . . . . . . . . . . . . 16 |- ( W e. (SubGrp ` H ) -> H e. Grp )
27		eqid 2622	. . . . . . . . . . . . . . . . 17 |- ( Base ` H ) = ( Base ` H )
28	27	subgacs 17629	. . . . . . . . . . . . . . . 16 |- ( H e. Grp -> (SubGrp ` H ) e. (ACS ` ( Base ` H ) ) )
29		acsmre 16313	. . . . . . . . . . . . . . . 16 |- ( (SubGrp ` H ) e. (ACS ` ( Base ` H ) ) -> (SubGrp ` H ) e. (Moore ` ( Base ` H ) ) )
30	1, 26, 28, 29	4syl 19	. . . . . . . . . . . . . . 15 |- ( ph -> (SubGrp ` H ) e. (Moore ` ( Base ` H ) ) )
31		pgpfac.k	. . . . . . . . . . . . . . 15 |- K = (mrCls ` (SubGrp ` H ) )
32	30, 31	mrcssvd 16283	. . . . . . . . . . . . . 14 |- ( ph -> ( K ` { X } ) C_ ( Base ` H ) )
33	3	subgbas 17598	. . . . . . . . . . . . . . 15 |- ( U e. (SubGrp ` G ) -> U = ( Base ` H ) )
34	2, 33	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> U = ( Base ` H ) )
35	32, 34	sseqtr4d 3642	. . . . . . . . . . . . 13 |- ( ph -> ( K ` { X } ) C_ U )
36		ssfi 8180	. . . . . . . . . . . . 13 |- ( ( U e. Fin /\ ( K ` { X } ) C_ U ) -> ( K ` { X } ) e. Fin )
37	15, 35, 36	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> ( K ` { X } ) e. Fin )
38		pgpfac.x	. . . . . . . . . . . . . . . . 17 |- ( ph -> X e. U )
39	38, 34	eleqtrd 2703	. . . . . . . . . . . . . . . 16 |- ( ph -> X e. ( Base ` H ) )
40	31	mrcsncl 16272	. . . . . . . . . . . . . . . 16 |- ( ( (SubGrp ` H ) e. (Moore ` ( Base ` H ) ) /\ X e. ( Base ` H ) ) -> ( K ` { X } ) e. (SubGrp ` H ) )
41	30, 39, 40	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ph -> ( K ` { X } ) e. (SubGrp ` H ) )
42	21	subg0cl 17602	. . . . . . . . . . . . . . 15 |- ( ( K ` { X } ) e. (SubGrp ` H ) -> .0. e. ( K ` { X } ) )
43	41, 42	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> .0. e. ( K ` { X } ) )
44	43	snssd 4340	. . . . . . . . . . . . 13 |- ( ph -> { .0. } C_ ( K ` { X } ) )
45	39	snssd 4340	. . . . . . . . . . . . . . 15 |- ( ph -> { X } C_ ( Base ` H ) )
46	30, 31, 45	mrcssidd 16285	. . . . . . . . . . . . . 14 |- ( ph -> { X } C_ ( K ` { X } ) )
47		snssg 4327	. . . . . . . . . . . . . . 15 |- ( X e. U -> ( X e. ( K ` { X } ) <-> { X } C_ ( K ` { X } ) ) )
48	38, 47	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> ( X e. ( K ` { X } ) <-> { X } C_ ( K ` { X } ) ) )
49	46, 48	mpbird 247	. . . . . . . . . . . . 13 |- ( ph -> X e. ( K ` { X } ) )
50		pgpfac.oe	. . . . . . . . . . . . . . 15 |- ( ph -> ( O ` X ) = E )
51		pgpfac.1	. . . . . . . . . . . . . . 15 |- ( ph -> E =/= 1 )
52	50, 51	eqnetrd 2861	. . . . . . . . . . . . . 14 |- ( ph -> ( O ` X ) =/= 1 )
53		pgpfac.o	. . . . . . . . . . . . . . . . . 18 |- O = ( od ` H )
54	53, 21	od1 17976	. . . . . . . . . . . . . . . . 17 |- ( H e. Grp -> ( O ` .0. ) = 1 )
55	1, 26, 54	3syl 18	. . . . . . . . . . . . . . . 16 |- ( ph -> ( O ` .0. ) = 1 )
56		elsni 4194	. . . . . . . . . . . . . . . . . 18 |- ( X e. { .0. } -> X = .0. )
57	56	fveq2d 6195	. . . . . . . . . . . . . . . . 17 |- ( X e. { .0. } -> ( O ` X ) = ( O ` .0. ) )
58	57	eqeq1d 2624	. . . . . . . . . . . . . . . 16 |- ( X e. { .0. } -> ( ( O ` X ) = 1 <-> ( O ` .0. ) = 1 ) )
59	55, 58	syl5ibrcom 237	. . . . . . . . . . . . . . 15 |- ( ph -> ( X e. { .0. } -> ( O ` X ) = 1 ) )
60	59	necon3ad 2807	. . . . . . . . . . . . . 14 |- ( ph -> ( ( O ` X ) =/= 1 -> -. X e. { .0. } ) )
61	52, 60	mpd 15	. . . . . . . . . . . . 13 |- ( ph -> -. X e. { .0. } )
62	44, 49, 61	ssnelpssd 3719	. . . . . . . . . . . 12 |- ( ph -> { .0. } C. ( K ` { X } ) )
63		php3 8146	. . . . . . . . . . . 12 |- ( ( ( K ` { X } ) e. Fin /\ { .0. } C. ( K ` { X } ) ) -> { .0. } ~< ( K ` { X } ) )
64	37, 62, 63	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> { .0. } ~< ( K ` { X } ) )
65		snfi 8038	. . . . . . . . . . . 12 |- { .0. } e. Fin
66		hashsdom 13170	. . . . . . . . . . . 12 |- ( ( { .0. } e. Fin /\ ( K ` { X } ) e. Fin ) -> ( ( # ` { .0. } ) < ( # ` ( K ` { X } ) ) <-> { .0. } ~< ( K ` { X } ) ) )
67	65, 37, 66	sylancr 695	. . . . . . . . . . 11 |- ( ph -> ( ( # ` { .0. } ) < ( # ` ( K ` { X } ) ) <-> { .0. } ~< ( K ` { X } ) ) )
68	64, 67	mpbird 247	. . . . . . . . . 10 |- ( ph -> ( # ` { .0. } ) < ( # ` ( K ` { X } ) ) )
69	25, 68	syl5eqbrr 4689	. . . . . . . . 9 |- ( ph -> 1 < ( # ` ( K ` { X } ) ) )
70		1red 10055	. . . . . . . . . 10 |- ( ph -> 1 e. RR )
71		hashcl 13147	. . . . . . . . . . . 12 |- ( ( K ` { X } ) e. Fin -> ( # ` ( K ` { X } ) ) e. NN0 )
72	37, 71	syl 17	. . . . . . . . . . 11 |- ( ph -> ( # ` ( K ` { X } ) ) e. NN0 )
73	72	nn0red 11352	. . . . . . . . . 10 |- ( ph -> ( # ` ( K ` { X } ) ) e. RR )
74	21	subg0cl 17602	. . . . . . . . . . . . 13 |- ( W e. (SubGrp ` H ) -> .0. e. W )
75		ne0i 3921	. . . . . . . . . . . . 13 |- ( .0. e. W -> W =/= (/) )
76	1, 74, 75	3syl 18	. . . . . . . . . . . 12 |- ( ph -> W =/= (/) )
77		hashnncl 13157	. . . . . . . . . . . . 13 |- ( W e. Fin -> ( ( # ` W ) e. NN <-> W =/= (/) ) )
78	17, 77	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( ( # ` W ) e. NN <-> W =/= (/) ) )
79	76, 78	mpbird 247	. . . . . . . . . . 11 |- ( ph -> ( # ` W ) e. NN )
80	79	nngt0d 11064	. . . . . . . . . 10 |- ( ph -> 0 < ( # ` W ) )
81		ltmul1 10873	. . . . . . . . . 10 |- ( ( 1 e. RR /\ ( # ` ( K ` { X } ) ) e. RR /\ ( ( # ` W ) e. RR /\ 0 < ( # ` W ) ) ) -> ( 1 < ( # ` ( K ` { X } ) ) <-> ( 1 x. ( # ` W ) ) < ( ( # ` ( K ` { X } ) ) x. ( # ` W ) ) ) )
82	70, 73, 20, 80, 81	syl112anc 1330	. . . . . . . . 9 |- ( ph -> ( 1 < ( # ` ( K ` { X } ) ) <-> ( 1 x. ( # ` W ) ) < ( ( # ` ( K ` { X } ) ) x. ( # ` W ) ) ) )
83	69, 82	mpbid 222	. . . . . . . 8 |- ( ph -> ( 1 x. ( # ` W ) ) < ( ( # ` ( K ` { X } ) ) x. ( # ` W ) ) )
84	20	recnd 10068	. . . . . . . . 9 |- ( ph -> ( # ` W ) e. CC )
85	84	mulid2d 10058	. . . . . . . 8 |- ( ph -> ( 1 x. ( # ` W ) ) = ( # ` W ) )
86		pgpfac.l	. . . . . . . . . 10 |- .(+) = ( LSSum ` H )
87		eqid 2622	. . . . . . . . . 10 |- (Cntz ` H ) = (Cntz ` H )
88		pgpfac.i	. . . . . . . . . 10 |- ( ph -> ( ( K ` { X } ) i^i W ) = { .0. } )
89		pgpfac.g	. . . . . . . . . . . 12 |- ( ph -> G e. Abel )
90	3	subgabl 18241	. . . . . . . . . . . 12 |- ( ( G e. Abel /\ U e. (SubGrp ` G ) ) -> H e. Abel )
91	89, 2, 90	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> H e. Abel )
92	87, 91, 41, 1	ablcntzd 18260	. . . . . . . . . 10 |- ( ph -> ( K ` { X } ) C_ ( (Cntz ` H ) ` W ) )
93	86, 21, 87, 41, 1, 88, 92, 37, 17	lsmhash 18118	. . . . . . . . 9 |- ( ph -> ( # ` ( ( K ` { X } ) .(+) W ) ) = ( ( # ` ( K ` { X } ) ) x. ( # ` W ) ) )
94		pgpfac.s	. . . . . . . . . 10 |- ( ph -> ( ( K ` { X } ) .(+) W ) = U )
95	94	fveq2d 6195	. . . . . . . . 9 |- ( ph -> ( # ` ( ( K ` { X } ) .(+) W ) ) = ( # ` U ) )
96	93, 95	eqtr3d 2658	. . . . . . . 8 |- ( ph -> ( ( # ` ( K ` { X } ) ) x. ( # ` W ) ) = ( # ` U ) )
97	83, 85, 96	3brtr3d 4684	. . . . . . 7 |- ( ph -> ( # ` W ) < ( # ` U ) )
98	20, 97	ltned 10173	. . . . . 6 |- ( ph -> ( # ` 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 |- ( ph -> W =/= U )
102		df-pss 3590	. . . . 5 |- ( W C. U <-> ( W C_ U /\ W =/= U ) )
103	9, 101, 102	sylanbrc 698	. . . 4 |- ( ph -> W C. U )
104		psseq1 3694	. . . . . 6 |- ( t = W -> ( t C. U <-> W C. U ) )
105		eqeq2 2633	. . . . . . . 8 |- ( t = W -> ( ( G DProd s ) = t <-> ( G DProd s ) = W ) )
106	105	anbi2d 740	. . . . . . 7 |- ( t = W -> ( ( G dom DProd s /\ ( G DProd s ) = t ) <-> ( G dom DProd s /\ ( G DProd s ) = W ) ) )
107	106	rexbidv 3052	. . . . . 6 |- ( t = W -> ( E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) <-> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = W ) ) )
108	104, 107	imbi12d 334	. . . . 5 |- ( t = W -> ( ( t C. U -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) <-> ( W C. U -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = W ) ) ) )
109	108	rspcv 3305	. . . 4 |- ( W e. (SubGrp ` G ) -> ( A. t e. (SubGrp ` G ) ( t C. U -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) -> ( W C. U -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = W ) ) ) )
110	7, 8, 103, 109	syl3c 66	. . 3 |- ( ph -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = W ) )
111		breq2 4657	. . . . 5 |- ( s = a -> ( G dom DProd s <-> G dom 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 -> ( ( G dom DProd s /\ ( G DProd s ) = W ) <-> ( G dom DProd a /\ ( G DProd a ) = W ) ) )
115	114	cbvrexv 3172	. . 3 |- ( E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = W ) <-> E. a e. Word C ( G dom DProd a /\ ( G DProd a ) = W ) )
116	110, 115	sylib 208	. 2 |- ( ph -> E. a e. Word C ( G dom DProd a /\ ( G DProd a ) = W ) )
117		pgpfac.c	. . 3 |- C = { r e. (SubGrp ` G ) | ( G ↾s r ) e. (CycGrp i^i ran pGrp ) }
118	89	adantr 481	. . 3 |- ( ( ph /\ ( a e. Word C /\ ( G dom DProd a /\ ( G DProd a ) = W ) ) ) -> G e. Abel )
119		pgpfac.p	. . . 4 |- ( ph -> P pGrp G )
120	119	adantr 481	. . 3 |- ( ( ph /\ ( a e. Word C /\ ( G dom DProd a /\ ( G DProd a ) = W ) ) ) -> P pGrp G )
121	10	adantr 481	. . 3 |- ( ( ph /\ ( a e. Word C /\ ( G dom DProd a /\ ( G DProd a ) = W ) ) ) -> B e. Fin )
122	2	adantr 481	. . 3 |- ( ( ph /\ ( a e. Word C /\ ( G dom DProd a /\ ( G DProd a ) = W ) ) ) -> U e. (SubGrp ` G ) )
123	8	adantr 481	. . 3 |- ( ( ph /\ ( a e. Word C /\ ( G dom DProd a /\ ( G DProd a ) = W ) ) ) -> A. t e. (SubGrp ` G ) ( t C. U -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) )
124		pgpfac.e	. . 3 |- E = (gEx ` H )
125	51	adantr 481	. . 3 |- ( ( ph /\ ( a e. Word C /\ ( G dom DProd a /\ ( G DProd a ) = W ) ) ) -> E =/= 1 )
126	38	adantr 481	. . 3 |- ( ( ph /\ ( a e. Word C /\ ( G dom DProd a /\ ( G DProd a ) = W ) ) ) -> X e. U )
127	50	adantr 481	. . 3 |- ( ( ph /\ ( a e. Word C /\ ( G dom DProd a /\ ( G DProd a ) = W ) ) ) -> ( O ` X ) = E )
128	1	adantr 481	. . 3 |- ( ( ph /\ ( a e. Word C /\ ( G dom DProd a /\ ( G DProd a ) = W ) ) ) -> W e. (SubGrp ` H ) )
129	88	adantr 481	. . 3 |- ( ( ph /\ ( a e. Word C /\ ( G dom DProd a /\ ( G DProd a ) = W ) ) ) -> ( ( K ` { X } ) i^i W ) = { .0. } )
130	94	adantr 481	. . 3 |- ( ( ph /\ ( a e. Word C /\ ( G dom DProd a /\ ( G DProd a ) = W ) ) ) -> ( ( K ` { X } ) .(+) W ) = U )
131		simprl 794	. . 3 |- ( ( ph /\ ( a e. Word C /\ ( G dom DProd a /\ ( G DProd a ) = W ) ) ) -> a e. Word C )
132		simprrl 804	. . 3 |- ( ( ph /\ ( a e. Word C /\ ( G dom DProd a /\ ( G DProd a ) = W ) ) ) -> G dom DProd a )
133		simprrr 805	. . 3 |- ( ( ph /\ ( a e. Word C /\ ( G dom 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 |- ( ( ph /\ ( a e. Word C /\ ( G dom DProd a /\ ( G DProd a ) = W ) ) ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = U ) )
136	116, 135	rexlimddv 3035	1 |- ( ph -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = U ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   C. 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   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   NNcn 11020   NN0cn0 11292   #chash 13117  Word cword 13291   ++ cconcat 13293   <"cs1 13294   Basecbs 15857   ↾_s cress 15858   0gc0g 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