Theorem ablfac1c 18470

Description: The factors of ablfac1b 18469 cover the entire group. (Contributed by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
ablfac1.b	|- B = ( Base ` G )
ablfac1.o	|- O = ( od ` G )
ablfac1.s	|- S = ( p e. A |-> { x e. B | ( O ` x ) || ( p ^ ( p pCnt ( # ` B ) ) ) } )
ablfac1.g	|- ( ph -> G e. Abel )
ablfac1.f	|- ( ph -> B e. Fin )
ablfac1.1	|- ( ph -> A C_ Prime )
ablfac1c.d	|- D = { w e. Prime | w || ( # ` B ) }
ablfac1.2	|- ( ph -> D C_ A )

Assertion

Ref	Expression
ablfac1c	|- ( ph -> ( G DProd S ) = B )

Distinct variable groups:   w, p, x, B   D, p, x   ph, p, w, x   A, p, x   O, p, x   G, p, x

Allowed substitution hints:   A( w)   D( w)   S( x, w, p)   G( w)   O( w)

Proof of Theorem ablfac1c

Dummy variable q is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ablfac1.f	. 2 |- ( ph -> B e. Fin )
2		ablfac1.b	. . . 4 |- B = ( Base ` G )
3	2	dprdssv 18415	. . 3 |- ( G DProd S ) C_ B
4	3	a1i 11	. 2 |- ( ph -> ( G DProd S ) C_ B )
5		ssfi 8180	. . . . . 6 |- ( ( B e. Fin /\ ( G DProd S ) C_ B ) -> ( G DProd S ) e. Fin )
6	1, 3, 5	sylancl 694	. . . . 5 |- ( ph -> ( G DProd S ) e. Fin )
7		hashcl 13147	. . . . 5 |- ( ( G DProd S ) e. Fin -> ( # ` ( G DProd S ) ) e. NN0 )
8	6, 7	syl 17	. . . 4 |- ( ph -> ( # ` ( G DProd S ) ) e. NN0 )
9		hashcl 13147	. . . . 5 |- ( B e. Fin -> ( # ` B ) e. NN0 )
10	1, 9	syl 17	. . . 4 |- ( ph -> ( # ` B ) e. NN0 )
11		ablfac1.o	. . . . . . 7 |- O = ( od ` G )
12		ablfac1.s	. . . . . . 7 |- S = ( p e. A |-> { x e. B | ( O ` x ) || ( p ^ ( p pCnt ( # ` B ) ) ) } )
13		ablfac1.g	. . . . . . 7 |- ( ph -> G e. Abel )
14		ablfac1.1	. . . . . . 7 |- ( ph -> A C_ Prime )
15	2, 11, 12, 13, 1, 14	ablfac1b 18469	. . . . . 6 |- ( ph -> G dom DProd S )
16		dprdsubg 18423	. . . . . 6 |- ( G dom DProd S -> ( G DProd S ) e. (SubGrp ` G ) )
17	15, 16	syl 17	. . . . 5 |- ( ph -> ( G DProd S ) e. (SubGrp ` G ) )
18	2	lagsubg 17656	. . . . 5 |- ( ( ( G DProd S ) e. (SubGrp ` G ) /\ B e. Fin ) -> ( # ` ( G DProd S ) ) || ( # ` B ) )
19	17, 1, 18	syl2anc 693	. . . 4 |- ( ph -> ( # ` ( G DProd S ) ) || ( # ` B ) )
20		breq1 4656	. . . . . . . . . . 11 |- ( w = q -> ( w || ( # ` B ) <-> q || ( # ` B ) ) )
21		ablfac1c.d	. . . . . . . . . . 11 |- D = { w e. Prime | w || ( # ` B ) }
22	20, 21	elrab2 3366	. . . . . . . . . 10 |- ( q e. D <-> ( q e. Prime /\ q || ( # ` B ) ) )
23		ablfac1.2	. . . . . . . . . . 11 |- ( ph -> D C_ A )
24	23	sseld 3602	. . . . . . . . . 10 |- ( ph -> ( q e. D -> q e. A ) )
25	22, 24	syl5bir 233	. . . . . . . . 9 |- ( ph -> ( ( q e. Prime /\ q || ( # ` B ) ) -> q e. A ) )
26	25	impl 650	. . . . . . . 8 |- ( ( ( ph /\ q e. Prime ) /\ q || ( # ` B ) ) -> q e. A )
27	2, 11, 12, 13, 1, 14	ablfac1a 18468	. . . . . . . . . . 11 |- ( ( ph /\ q e. A ) -> ( # ` ( S ` q ) ) = ( q ^ ( q pCnt ( # ` B ) ) ) )
28		fvex 6201	. . . . . . . . . . . . . . . . . . . 20 |- ( Base ` G ) e. _V
29	2, 28	eqeltri 2697	. . . . . . . . . . . . . . . . . . 19 |- B e. _V
30	29	rabex 4813	. . . . . . . . . . . . . . . . . 18 |- { x e. B | ( O ` x ) || ( p ^ ( p pCnt ( # ` B ) ) ) } e. _V
31	30, 12	dmmpti 6023	. . . . . . . . . . . . . . . . 17 |- dom S = A
32	31	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ph -> dom S = A )
33	15, 32	dprdf2 18406	. . . . . . . . . . . . . . 15 |- ( ph -> S : A --> (SubGrp ` G ) )
34	33	ffvelrnda 6359	. . . . . . . . . . . . . 14 |- ( ( ph /\ q e. A ) -> ( S ` q ) e. (SubGrp ` G ) )
35	15	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ q e. A ) -> G dom DProd S )
36	31	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ph /\ q e. A ) -> dom S = A )
37		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ph /\ q e. A ) -> q e. A )
38	35, 36, 37	dprdub 18424	. . . . . . . . . . . . . 14 |- ( ( ph /\ q e. A ) -> ( S ` q ) C_ ( G DProd S ) )
39	17	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ q e. A ) -> ( G DProd S ) e. (SubGrp ` G ) )
40		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( G ↾s ( G DProd S ) ) = ( G ↾s ( G DProd S ) )
41	40	subsubg 17617	. . . . . . . . . . . . . . 15 |- ( ( G DProd S ) e. (SubGrp ` G ) -> ( ( S ` q ) e. (SubGrp ` ( G ↾s ( G DProd S ) ) ) <-> ( ( S ` q ) e. (SubGrp ` G ) /\ ( S ` q ) C_ ( G DProd S ) ) ) )
42	39, 41	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ q e. A ) -> ( ( S ` q ) e. (SubGrp ` ( G ↾s ( G DProd S ) ) ) <-> ( ( S ` q ) e. (SubGrp ` G ) /\ ( S ` q ) C_ ( G DProd S ) ) ) )
43	34, 38, 42	mpbir2and 957	. . . . . . . . . . . . 13 |- ( ( ph /\ q e. A ) -> ( S ` q ) e. (SubGrp ` ( G ↾s ( G DProd S ) ) ) )
44	40	subgbas 17598	. . . . . . . . . . . . . . 15 |- ( ( G DProd S ) e. (SubGrp ` G ) -> ( G DProd S ) = ( Base ` ( G ↾s ( G DProd S ) ) ) )
45	39, 44	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ q e. A ) -> ( G DProd S ) = ( Base ` ( G ↾s ( G DProd S ) ) ) )
46	6	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ q e. A ) -> ( G DProd S ) e. Fin )
47	45, 46	eqeltrrd 2702	. . . . . . . . . . . . 13 |- ( ( ph /\ q e. A ) -> ( Base ` ( G ↾s ( G DProd S ) ) ) e. Fin )
48		eqid 2622	. . . . . . . . . . . . . 14 |- ( Base ` ( G ↾s ( G DProd S ) ) ) = ( Base ` ( G ↾s ( G DProd S ) ) )
49	48	lagsubg 17656	. . . . . . . . . . . . 13 |- ( ( ( S ` q ) e. (SubGrp ` ( G ↾s ( G DProd S ) ) ) /\ ( Base ` ( G ↾s ( G DProd S ) ) ) e. Fin ) -> ( # ` ( S ` q ) ) || ( # ` ( Base ` ( G ↾s ( G DProd S ) ) ) ) )
50	43, 47, 49	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ q e. A ) -> ( # ` ( S ` q ) ) || ( # ` ( Base ` ( G ↾s ( G DProd S ) ) ) ) )
51	45	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ph /\ q e. A ) -> ( # ` ( G DProd S ) ) = ( # ` ( Base ` ( G ↾s ( G DProd S ) ) ) ) )
52	50, 51	breqtrrd 4681	. . . . . . . . . . 11 |- ( ( ph /\ q e. A ) -> ( # ` ( S ` q ) ) || ( # ` ( G DProd S ) ) )
53	27, 52	eqbrtrrd 4677	. . . . . . . . . 10 |- ( ( ph /\ q e. A ) -> ( q ^ ( q pCnt ( # ` B ) ) ) || ( # ` ( G DProd S ) ) )
54	14	sselda 3603	. . . . . . . . . . 11 |- ( ( ph /\ q e. A ) -> q e. Prime )
55	8	nn0zd 11480	. . . . . . . . . . . 12 |- ( ph -> ( # ` ( G DProd S ) ) e. ZZ )
56	55	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ q e. A ) -> ( # ` ( G DProd S ) ) e. ZZ )
57		simpr 477	. . . . . . . . . . . . 13 |- ( ( ph /\ q e. Prime ) -> q e. Prime )
58		ablgrp 18198	. . . . . . . . . . . . . . . 16 |- ( G e. Abel -> G e. Grp )
59	2	grpbn0 17451	. . . . . . . . . . . . . . . 16 |- ( G e. Grp -> B =/= (/) )
60	13, 58, 59	3syl 18	. . . . . . . . . . . . . . 15 |- ( ph -> B =/= (/) )
61		hashnncl 13157	. . . . . . . . . . . . . . . 16 |- ( B e. Fin -> ( ( # ` B ) e. NN <-> B =/= (/) ) )
62	1, 61	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( # ` B ) e. NN <-> B =/= (/) ) )
63	60, 62	mpbird 247	. . . . . . . . . . . . . 14 |- ( ph -> ( # ` B ) e. NN )
64	63	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ q e. Prime ) -> ( # ` B ) e. NN )
65	57, 64	pccld 15555	. . . . . . . . . . . 12 |- ( ( ph /\ q e. Prime ) -> ( q pCnt ( # ` B ) ) e. NN0 )
66	54, 65	syldan 487	. . . . . . . . . . 11 |- ( ( ph /\ q e. A ) -> ( q pCnt ( # ` B ) ) e. NN0 )
67		pcdvdsb 15573	. . . . . . . . . . 11 |- ( ( q e. Prime /\ ( # ` ( G DProd S ) ) e. ZZ /\ ( q pCnt ( # ` B ) ) e. NN0 ) -> ( ( q pCnt ( # ` B ) ) <_ ( q pCnt ( # ` ( G DProd S ) ) ) <-> ( q ^ ( q pCnt ( # ` B ) ) ) || ( # ` ( G DProd S ) ) ) )
68	54, 56, 66, 67	syl3anc 1326	. . . . . . . . . 10 |- ( ( ph /\ q e. A ) -> ( ( q pCnt ( # ` B ) ) <_ ( q pCnt ( # ` ( G DProd S ) ) ) <-> ( q ^ ( q pCnt ( # ` B ) ) ) || ( # ` ( G DProd S ) ) ) )
69	53, 68	mpbird 247	. . . . . . . . 9 |- ( ( ph /\ q e. A ) -> ( q pCnt ( # ` B ) ) <_ ( q pCnt ( # ` ( G DProd S ) ) ) )
70	69	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ q e. Prime ) /\ q e. A ) -> ( q pCnt ( # ` B ) ) <_ ( q pCnt ( # ` ( G DProd S ) ) ) )
71	26, 70	syldan 487	. . . . . . 7 |- ( ( ( ph /\ q e. Prime ) /\ q || ( # ` B ) ) -> ( q pCnt ( # ` B ) ) <_ ( q pCnt ( # ` ( G DProd S ) ) ) )
72		pceq0 15575	. . . . . . . . . 10 |- ( ( q e. Prime /\ ( # ` B ) e. NN ) -> ( ( q pCnt ( # ` B ) ) = 0 <-> -. q || ( # ` B ) ) )
73	57, 64, 72	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ q e. Prime ) -> ( ( q pCnt ( # ` B ) ) = 0 <-> -. q || ( # ` B ) ) )
74	73	biimpar 502	. . . . . . . 8 |- ( ( ( ph /\ q e. Prime ) /\ -. q || ( # ` B ) ) -> ( q pCnt ( # ` B ) ) = 0 )
75		eqid 2622	. . . . . . . . . . . . . . 15 |- ( 0g ` G ) = ( 0g ` G )
76	75	subg0cl 17602	. . . . . . . . . . . . . 14 |- ( ( G DProd S ) e. (SubGrp ` G ) -> ( 0g ` G ) e. ( G DProd S ) )
77		ne0i 3921	. . . . . . . . . . . . . 14 |- ( ( 0g ` G ) e. ( G DProd S ) -> ( G DProd S ) =/= (/) )
78	17, 76, 77	3syl 18	. . . . . . . . . . . . 13 |- ( ph -> ( G DProd S ) =/= (/) )
79		hashnncl 13157	. . . . . . . . . . . . . 14 |- ( ( G DProd S ) e. Fin -> ( ( # ` ( G DProd S ) ) e. NN <-> ( G DProd S ) =/= (/) ) )
80	6, 79	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( ( # ` ( G DProd S ) ) e. NN <-> ( G DProd S ) =/= (/) ) )
81	78, 80	mpbird 247	. . . . . . . . . . . 12 |- ( ph -> ( # ` ( G DProd S ) ) e. NN )
82	81	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ q e. Prime ) -> ( # ` ( G DProd S ) ) e. NN )
83	57, 82	pccld 15555	. . . . . . . . . 10 |- ( ( ph /\ q e. Prime ) -> ( q pCnt ( # ` ( G DProd S ) ) ) e. NN0 )
84	83	nn0ge0d 11354	. . . . . . . . 9 |- ( ( ph /\ q e. Prime ) -> 0 <_ ( q pCnt ( # ` ( G DProd S ) ) ) )
85	84	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ q e. Prime ) /\ -. q || ( # ` B ) ) -> 0 <_ ( q pCnt ( # ` ( G DProd S ) ) ) )
86	74, 85	eqbrtrd 4675	. . . . . . 7 |- ( ( ( ph /\ q e. Prime ) /\ -. q || ( # ` B ) ) -> ( q pCnt ( # ` B ) ) <_ ( q pCnt ( # ` ( G DProd S ) ) ) )
87	71, 86	pm2.61dan 832	. . . . . 6 |- ( ( ph /\ q e. Prime ) -> ( q pCnt ( # ` B ) ) <_ ( q pCnt ( # ` ( G DProd S ) ) ) )
88	87	ralrimiva 2966	. . . . 5 |- ( ph -> A. q e. Prime ( q pCnt ( # ` B ) ) <_ ( q pCnt ( # ` ( G DProd S ) ) ) )
89	10	nn0zd 11480	. . . . . 6 |- ( ph -> ( # ` B ) e. ZZ )
90		pc2dvds 15583	. . . . . 6 |- ( ( ( # ` B ) e. ZZ /\ ( # ` ( G DProd S ) ) e. ZZ ) -> ( ( # ` B ) || ( # ` ( G DProd S ) ) <-> A. q e. Prime ( q pCnt ( # ` B ) ) <_ ( q pCnt ( # ` ( G DProd S ) ) ) ) )
91	89, 55, 90	syl2anc 693	. . . . 5 |- ( ph -> ( ( # ` B ) || ( # ` ( G DProd S ) ) <-> A. q e. Prime ( q pCnt ( # ` B ) ) <_ ( q pCnt ( # ` ( G DProd S ) ) ) ) )
92	88, 91	mpbird 247	. . . 4 |- ( ph -> ( # ` B ) || ( # ` ( G DProd S ) ) )
93		dvdseq 15036	. . . 4 |- ( ( ( ( # ` ( G DProd S ) ) e. NN0 /\ ( # ` B ) e. NN0 ) /\ ( ( # ` ( G DProd S ) ) || ( # ` B ) /\ ( # ` B ) || ( # ` ( G DProd S ) ) ) ) -> ( # ` ( G DProd S ) ) = ( # ` B ) )
94	8, 10, 19, 92, 93	syl22anc 1327	. . 3 |- ( ph -> ( # ` ( G DProd S ) ) = ( # ` B ) )
95		hashen 13135	. . . 4 |- ( ( ( G DProd S ) e. Fin /\ B e. Fin ) -> ( ( # ` ( G DProd S ) ) = ( # ` B ) <-> ( G DProd S ) ~~ B ) )
96	6, 1, 95	syl2anc 693	. . 3 |- ( ph -> ( ( # ` ( G DProd S ) ) = ( # ` B ) <-> ( G DProd S ) ~~ B ) )
97	94, 96	mpbid 222	. 2 |- ( ph -> ( G DProd S ) ~~ B )
98		fisseneq 8171	. 2 |- ( ( B e. Fin /\ ( G DProd S ) C_ B /\ ( G DProd S ) ~~ B ) -> ( G DProd S ) = B )
99	1, 4, 97, 98	syl3anc 1326	1 |- ( ph -> ( G DProd S ) = B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ` cfv 5888  (class class class)co 6650   ~~ cen 7952   Fincfn 7955   0cc0 9936   <_ cle 10075   NNcn 11020   NN0cn0 11292   ZZcz 11377   ^cexp 12860   #chash 13117   || cdvds 14983   Primecprime 15385   pCnt cpc 15541   Basecbs 15857   ↾_s cress 15858   0gc0g 16100   Grpcgrp 17422  SubGrpcsubg 17588   odcod 17944   Abelcabl 18194   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  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-disj 4621  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-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  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-acn 8768  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  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-eqg 17593  df-ghm 17658  df-gim 17701  df-ga 17723  df-cntz 17750  df-oppg 17776  df-od 17948  df-lsm 18051  df-pj1 18052  df-cmn 18195  df-abl 18196  df-dprd 18394

This theorem is referenced by:  ablfaclem2  18485