Theorem ablfac1lem 17779

Description: Lemma for ablfac1b 17781. Satisfy the assumptions of ablfacrp. (Contributed by Mario Carneiro, 26-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 )
ablfac1.m	|- M = ( P ^ ( P pCnt ( # ` B ) ) )
ablfac1.n	|- N = ( ( # ` B ) / M )

Assertion

Ref	Expression
ablfac1lem	|- ( ( ph /\ P e. A ) -> ( ( M e. NN /\ N e. NN ) /\ ( M gcd N ) = 1 /\ ( # ` B ) = ( M x. N ) ) )

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

Allowed substitution hints:   S( x, p)   M( x, p)   N( x, p)

Proof of Theorem ablfac1lem

Step	Hyp	Ref	Expression
1		ablfac1.m	. . . 4 |- M = ( P ^ ( P pCnt ( # ` B ) ) )
2		ablfac1.1	. . . . . . 7 |- ( ph -> A C_ Prime )
3	2	sselda 3418	. . . . . 6 |- ( ( ph /\ P e. A ) -> P e. Prime )
4		prmnn 14704	. . . . . 6 |- ( P e. Prime -> P e. NN )
5	3, 4	syl 17	. . . . 5 |- ( ( ph /\ P e. A ) -> P e. NN )
6		ablfac1.g	. . . . . . . . 9 |- ( ph -> G e. Abel )
7		ablgrp 17513	. . . . . . . . 9 |- ( G e. Abel -> G e. Grp )
8		ablfac1.b	. . . . . . . . . 10 |- B = ( Base ` G )
9	8	grpbn0 16773	. . . . . . . . 9 |- ( G e. Grp -> B =/= (/) )
10	6, 7, 9	3syl 18	. . . . . . . 8 |- ( ph -> B =/= (/) )
11		ablfac1.f	. . . . . . . . 9 |- ( ph -> B e. Fin )
12		hashnncl 12585	. . . . . . . . 9 |- ( B e. Fin -> ( ( # ` B ) e. NN <-> B =/= (/) ) )
13	11, 12	syl 17	. . . . . . . 8 |- ( ph -> ( ( # ` B ) e. NN <-> B =/= (/) ) )
14	10, 13	mpbird 240	. . . . . . 7 |- ( ph -> ( # ` B ) e. NN )
15	14	adantr 472	. . . . . 6 |- ( ( ph /\ P e. A ) -> ( # ` B ) e. NN )
16	3, 15	pccld 14879	. . . . 5 |- ( ( ph /\ P e. A ) -> ( P pCnt ( # ` B ) ) e. NN0 )
17	5, 16	nnexpcld 12475	. . . 4 |- ( ( ph /\ P e. A ) -> ( P ^ ( P pCnt ( # ` B ) ) ) e. NN )
18	1, 17	syl5eqel 2553	. . 3 |- ( ( ph /\ P e. A ) -> M e. NN )
19		ablfac1.n	. . . 4 |- N = ( ( # ` B ) / M )
20		pcdvds 14892	. . . . . . 7 |- ( ( P e. Prime /\ ( # ` B ) e. NN ) -> ( P ^ ( P pCnt ( # ` B ) ) ) || ( # ` B ) )
21	3, 15, 20	syl2anc 673	. . . . . 6 |- ( ( ph /\ P e. A ) -> ( P ^ ( P pCnt ( # ` B ) ) ) || ( # ` B ) )
22	1, 21	syl5eqbr 4429	. . . . 5 |- ( ( ph /\ P e. A ) -> M || ( # ` B ) )
23		nndivdvds 14388	. . . . . 6 |- ( ( ( # ` B ) e. NN /\ M e. NN ) -> ( M || ( # ` B ) <-> ( ( # ` B ) / M ) e. NN ) )
24	15, 18, 23	syl2anc 673	. . . . 5 |- ( ( ph /\ P e. A ) -> ( M || ( # ` B ) <-> ( ( # ` B ) / M ) e. NN ) )
25	22, 24	mpbid 215	. . . 4 |- ( ( ph /\ P e. A ) -> ( ( # ` B ) / M ) e. NN )
26	19, 25	syl5eqel 2553	. . 3 |- ( ( ph /\ P e. A ) -> N e. NN )
27	18, 26	jca 541	. 2 |- ( ( ph /\ P e. A ) -> ( M e. NN /\ N e. NN ) )
28	1	oveq1i 6318	. . 3 |- ( M gcd N ) = ( ( P ^ ( P pCnt ( # ` B ) ) ) gcd N )
29		pcndvds2 14896	. . . . . . 7 |- ( ( P e. Prime /\ ( # ` B ) e. NN ) -> -. P || ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) )
30	3, 15, 29	syl2anc 673	. . . . . 6 |- ( ( ph /\ P e. A ) -> -. P || ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) )
31	1	oveq2i 6319	. . . . . . . 8 |- ( ( # ` B ) / M ) = ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) )
32	19, 31	eqtri 2493	. . . . . . 7 |- N = ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) )
33	32	breq2i 4403	. . . . . 6 |- ( P || N <-> P || ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) )
34	30, 33	sylnibr 312	. . . . 5 |- ( ( ph /\ P e. A ) -> -. P || N )
35	26	nnzd 11062	. . . . . 6 |- ( ( ph /\ P e. A ) -> N e. ZZ )
36		coprm 14736	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> ( -. P || N <-> ( P gcd N ) = 1 ) )
37	3, 35, 36	syl2anc 673	. . . . 5 |- ( ( ph /\ P e. A ) -> ( -. P || N <-> ( P gcd N ) = 1 ) )
38	34, 37	mpbid 215	. . . 4 |- ( ( ph /\ P e. A ) -> ( P gcd N ) = 1 )
39		prmz 14705	. . . . . 6 |- ( P e. Prime -> P e. ZZ )
40	3, 39	syl 17	. . . . 5 |- ( ( ph /\ P e. A ) -> P e. ZZ )
41		rpexp1i 14752	. . . . 5 |- ( ( P e. ZZ /\ N e. ZZ /\ ( P pCnt ( # ` B ) ) e. NN0 ) -> ( ( P gcd N ) = 1 -> ( ( P ^ ( P pCnt ( # ` B ) ) ) gcd N ) = 1 ) )
42	40, 35, 16, 41	syl3anc 1292	. . . 4 |- ( ( ph /\ P e. A ) -> ( ( P gcd N ) = 1 -> ( ( P ^ ( P pCnt ( # ` B ) ) ) gcd N ) = 1 ) )
43	38, 42	mpd 15	. . 3 |- ( ( ph /\ P e. A ) -> ( ( P ^ ( P pCnt ( # ` B ) ) ) gcd N ) = 1 )
44	28, 43	syl5eq 2517	. 2 |- ( ( ph /\ P e. A ) -> ( M gcd N ) = 1 )
45	19	oveq2i 6319	. . 3 |- ( M x. N ) = ( M x. ( ( # ` B ) / M ) )
46	15	nncnd 10647	. . . 4 |- ( ( ph /\ P e. A ) -> ( # ` B ) e. CC )
47	18	nncnd 10647	. . . 4 |- ( ( ph /\ P e. A ) -> M e. CC )
48	18	nnne0d 10676	. . . 4 |- ( ( ph /\ P e. A ) -> M =/= 0 )
49	46, 47, 48	divcan2d 10407	. . 3 |- ( ( ph /\ P e. A ) -> ( M x. ( ( # ` B ) / M ) ) = ( # ` B ) )
50	45, 49	syl5req 2518	. 2 |- ( ( ph /\ P e. A ) -> ( # ` B ) = ( M x. N ) )
51	27, 44, 50	3jca 1210	1 |- ( ( ph /\ P e. A ) -> ( ( M e. NN /\ N e. NN ) /\ ( M gcd N ) = 1 /\ ( # ` B ) = ( M x. N ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   {crab 2760   C_ wss 3390   (/)c0 3722   class class class wbr 4395   |-> cmpt 4454   ` cfv 5589  (class class class)co 6308   Fincfn 7587   1c1 9558   x. cmul 9562   / cdiv 10291   NNcn 10631   NN0cn0 10893   ZZcz 10961   ^cexp 12310   #chash 12553   || cdvds 14382   gcd cgcd 14547   Primecprime 14701   pCnt cpc 14865   Basecbs 15199   Grpcgrp 16747   odcod 17243   Abelcabl 17509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-fz 11811  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-prm 14702  df-pc 14866  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751  df-abl 17511

This theorem is referenced by:  ablfac1a  17780  ablfac1b  17781