Theorem ablfac1lem 16574

Description: Lemma for ablfac1b 16576. 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 3361	. . . . . 6 |- ( ( ph /\ P e. A ) -> P e. Prime )
4		prmnn 13771	. . . . . 6 |- ( P e. Prime -> P e. NN )
5	3, 4	syl 16	. . . . 5 |- ( ( ph /\ P e. A ) -> P e. NN )
6		ablfac1.g	. . . . . . . . 9 |- ( ph -> G e. Abel )
7		ablgrp 16287	. . . . . . . . 9 |- ( G e. Abel -> G e. Grp )
8		ablfac1.b	. . . . . . . . . 10 |- B = ( Base ` G )
9	8	grpbn0 15572	. . . . . . . . 9 |- ( G e. Grp -> B =/= (/) )
10	6, 7, 9	3syl 20	. . . . . . . 8 |- ( ph -> B =/= (/) )
11		ablfac1.f	. . . . . . . . 9 |- ( ph -> B e. Fin )
12		hashnncl 12139	. . . . . . . . 9 |- ( B e. Fin -> ( ( # ` B ) e. NN <-> B =/= (/) ) )
13	11, 12	syl 16	. . . . . . . 8 |- ( ph -> ( ( # ` B ) e. NN <-> B =/= (/) ) )
14	10, 13	mpbird 232	. . . . . . 7 |- ( ph -> ( # ` B ) e. NN )
15	14	adantr 465	. . . . . 6 |- ( ( ph /\ P e. A ) -> ( # ` B ) e. NN )
16	3, 15	pccld 13922	. . . . 5 |- ( ( ph /\ P e. A ) -> ( P pCnt ( # ` B ) ) e. NN0 )
17	5, 16	nnexpcld 12034	. . . 4 |- ( ( ph /\ P e. A ) -> ( P ^ ( P pCnt ( # ` B ) ) ) e. NN )
18	1, 17	syl5eqel 2527	. . 3 |- ( ( ph /\ P e. A ) -> M e. NN )
19		ablfac1.n	. . . 4 |- N = ( ( # ` B ) / M )
20		pcdvds 13935	. . . . . . 7 |- ( ( P e. Prime /\ ( # ` B ) e. NN ) -> ( P ^ ( P pCnt ( # ` B ) ) ) || ( # ` B ) )
21	3, 15, 20	syl2anc 661	. . . . . 6 |- ( ( ph /\ P e. A ) -> ( P ^ ( P pCnt ( # ` B ) ) ) || ( # ` B ) )
22	1, 21	syl5eqbr 4330	. . . . 5 |- ( ( ph /\ P e. A ) -> M || ( # ` B ) )
23		nndivdvds 13546	. . . . . 6 |- ( ( ( # ` B ) e. NN /\ M e. NN ) -> ( M || ( # ` B ) <-> ( ( # ` B ) / M ) e. NN ) )
24	15, 18, 23	syl2anc 661	. . . . 5 |- ( ( ph /\ P e. A ) -> ( M || ( # ` B ) <-> ( ( # ` B ) / M ) e. NN ) )
25	22, 24	mpbid 210	. . . 4 |- ( ( ph /\ P e. A ) -> ( ( # ` B ) / M ) e. NN )
26	19, 25	syl5eqel 2527	. . 3 |- ( ( ph /\ P e. A ) -> N e. NN )
27	18, 26	jca 532	. 2 |- ( ( ph /\ P e. A ) -> ( M e. NN /\ N e. NN ) )
28	1	oveq1i 6106	. . 3 |- ( M gcd N ) = ( ( P ^ ( P pCnt ( # ` B ) ) ) gcd N )
29		pcndvds2 13939	. . . . . . 7 |- ( ( P e. Prime /\ ( # ` B ) e. NN ) -> -. P || ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) )
30	3, 15, 29	syl2anc 661	. . . . . 6 |- ( ( ph /\ P e. A ) -> -. P || ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) )
31	1	oveq2i 6107	. . . . . . . 8 |- ( ( # ` B ) / M ) = ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) )
32	19, 31	eqtri 2463	. . . . . . 7 |- N = ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) )
33	32	breq2i 4305	. . . . . 6 |- ( P || N <-> P || ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) )
34	30, 33	sylnibr 305	. . . . 5 |- ( ( ph /\ P e. A ) -> -. P || N )
35	26	nnzd 10751	. . . . . 6 |- ( ( ph /\ P e. A ) -> N e. ZZ )
36		coprm 13791	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> ( -. P || N <-> ( P gcd N ) = 1 ) )
37	3, 35, 36	syl2anc 661	. . . . 5 |- ( ( ph /\ P e. A ) -> ( -. P || N <-> ( P gcd N ) = 1 ) )
38	34, 37	mpbid 210	. . . 4 |- ( ( ph /\ P e. A ) -> ( P gcd N ) = 1 )
39		prmz 13772	. . . . . 6 |- ( P e. Prime -> P e. ZZ )
40	3, 39	syl 16	. . . . 5 |- ( ( ph /\ P e. A ) -> P e. ZZ )
41		rpexp1i 13812	. . . . 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 1218	. . . 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 2487	. 2 |- ( ( ph /\ P e. A ) -> ( M gcd N ) = 1 )
45	19	oveq2i 6107	. . 3 |- ( M x. N ) = ( M x. ( ( # ` B ) / M ) )
46	15	nncnd 10343	. . . 4 |- ( ( ph /\ P e. A ) -> ( # ` B ) e. CC )
47	18	nncnd 10343	. . . 4 |- ( ( ph /\ P e. A ) -> M e. CC )
48	18	nnne0d 10371	. . . 4 |- ( ( ph /\ P e. A ) -> M =/= 0 )
49	46, 47, 48	divcan2d 10114	. . 3 |- ( ( ph /\ P e. A ) -> ( M x. ( ( # ` B ) / M ) ) = ( # ` B ) )
50	45, 49	syl5req 2488	. 2 |- ( ( ph /\ P e. A ) -> ( # ` B ) = ( M x. N ) )
51	27, 44, 50	3jca 1168	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 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2611   {crab 2724   C_ wss 3333   (/)c0 3642   class class class wbr 4297   e. cmpt 4355   ` cfv 5423  (class class class)co 6096   Fincfn 7315   1c1 9288   x. cmul 9292   / cdiv 9998   NNcn 10327   NN0cn0 10584   ZZcz 10651   ^cexp 11870   #chash 12108   || cdivides 13540   gcd cgcd 13695   Primecprime 13768   pCnt cpc 13908   Basecbs 14179   Grpcgrp 15415   odcod 16033   Abelcabel 16283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-fz 11443  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-dvds 13541  df-gcd 13696  df-prm 13769  df-pc 13909  df-0g 14385  df-mnd 15420  df-grp 15550  df-abl 16285

This theorem is referenced by:  ablfac1a  16575  ablfac1b  16576