Theorem ablfac1lem 17701

Description: Lemma for ablfac1b 17703. 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 3432	. . . . . 6 |- ( ( ph /\ P e. A ) -> P e. Prime )
4		prmnn 14625	. . . . . 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 17435	. . . . . . . . 9 |- ( G e. Abel -> G e. Grp )
8		ablfac1.b	. . . . . . . . . 10 |- B = ( Base ` G )
9	8	grpbn0 16695	. . . . . . . . 9 |- ( G e. Grp -> B =/= (/) )
10	6, 7, 9	3syl 18	. . . . . . . 8 |- ( ph -> B =/= (/) )
11		ablfac1.f	. . . . . . . . 9 |- ( ph -> B e. Fin )
12		hashnncl 12547	. . . . . . . . 9 |- ( B e. Fin -> ( ( # ` B ) e. NN <-> B =/= (/) ) )
13	11, 12	syl 17	. . . . . . . 8 |- ( ph -> ( ( # ` B ) e. NN <-> B =/= (/) ) )
14	10, 13	mpbird 236	. . . . . . 7 |- ( ph -> ( # ` B ) e. NN )
15	14	adantr 467	. . . . . 6 |- ( ( ph /\ P e. A ) -> ( # ` B ) e. NN )
16	3, 15	pccld 14800	. . . . 5 |- ( ( ph /\ P e. A ) -> ( P pCnt ( # ` B ) ) e. NN0 )
17	5, 16	nnexpcld 12437	. . . 4 |- ( ( ph /\ P e. A ) -> ( P ^ ( P pCnt ( # ` B ) ) ) e. NN )
18	1, 17	syl5eqel 2533	. . 3 |- ( ( ph /\ P e. A ) -> M e. NN )
19		ablfac1.n	. . . 4 |- N = ( ( # ` B ) / M )
20		pcdvds 14813	. . . . . . 7 |- ( ( P e. Prime /\ ( # ` B ) e. NN ) -> ( P ^ ( P pCnt ( # ` B ) ) ) || ( # ` B ) )
21	3, 15, 20	syl2anc 667	. . . . . 6 |- ( ( ph /\ P e. A ) -> ( P ^ ( P pCnt ( # ` B ) ) ) || ( # ` B ) )
22	1, 21	syl5eqbr 4436	. . . . 5 |- ( ( ph /\ P e. A ) -> M || ( # ` B ) )
23		nndivdvds 14311	. . . . . 6 |- ( ( ( # ` B ) e. NN /\ M e. NN ) -> ( M || ( # ` B ) <-> ( ( # ` B ) / M ) e. NN ) )
24	15, 18, 23	syl2anc 667	. . . . 5 |- ( ( ph /\ P e. A ) -> ( M || ( # ` B ) <-> ( ( # ` B ) / M ) e. NN ) )
25	22, 24	mpbid 214	. . . 4 |- ( ( ph /\ P e. A ) -> ( ( # ` B ) / M ) e. NN )
26	19, 25	syl5eqel 2533	. . 3 |- ( ( ph /\ P e. A ) -> N e. NN )
27	18, 26	jca 535	. 2 |- ( ( ph /\ P e. A ) -> ( M e. NN /\ N e. NN ) )
28	1	oveq1i 6300	. . 3 |- ( M gcd N ) = ( ( P ^ ( P pCnt ( # ` B ) ) ) gcd N )
29		pcndvds2 14817	. . . . . . 7 |- ( ( P e. Prime /\ ( # ` B ) e. NN ) -> -. P || ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) )
30	3, 15, 29	syl2anc 667	. . . . . 6 |- ( ( ph /\ P e. A ) -> -. P || ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) )
31	1	oveq2i 6301	. . . . . . . 8 |- ( ( # ` B ) / M ) = ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) )
32	19, 31	eqtri 2473	. . . . . . 7 |- N = ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) )
33	32	breq2i 4410	. . . . . 6 |- ( P || N <-> P || ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) )
34	30, 33	sylnibr 307	. . . . 5 |- ( ( ph /\ P e. A ) -> -. P || N )
35	26	nnzd 11039	. . . . . 6 |- ( ( ph /\ P e. A ) -> N e. ZZ )
36		coprm 14657	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> ( -. P || N <-> ( P gcd N ) = 1 ) )
37	3, 35, 36	syl2anc 667	. . . . 5 |- ( ( ph /\ P e. A ) -> ( -. P || N <-> ( P gcd N ) = 1 ) )
38	34, 37	mpbid 214	. . . 4 |- ( ( ph /\ P e. A ) -> ( P gcd N ) = 1 )
39		prmz 14626	. . . . . 6 |- ( P e. Prime -> P e. ZZ )
40	3, 39	syl 17	. . . . 5 |- ( ( ph /\ P e. A ) -> P e. ZZ )
41		rpexp1i 14673	. . . . 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 1268	. . . 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 2497	. 2 |- ( ( ph /\ P e. A ) -> ( M gcd N ) = 1 )
45	19	oveq2i 6301	. . 3 |- ( M x. N ) = ( M x. ( ( # ` B ) / M ) )
46	15	nncnd 10625	. . . 4 |- ( ( ph /\ P e. A ) -> ( # ` B ) e. CC )
47	18	nncnd 10625	. . . 4 |- ( ( ph /\ P e. A ) -> M e. CC )
48	18	nnne0d 10654	. . . 4 |- ( ( ph /\ P e. A ) -> M =/= 0 )
49	46, 47, 48	divcan2d 10385	. . 3 |- ( ( ph /\ P e. A ) -> ( M x. ( ( # ` B ) / M ) ) = ( # ` B ) )
50	45, 49	syl5req 2498	. 2 |- ( ( ph /\ P e. A ) -> ( # ` B ) = ( M x. N ) )
51	27, 44, 50	3jca 1188	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 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   {crab 2741   C_ wss 3404   (/)c0 3731   class class class wbr 4402   |-> cmpt 4461   ` cfv 5582  (class class class)co 6290   Fincfn 7569   1c1 9540   x. cmul 9544   / cdiv 10269   NNcn 10609   NN0cn0 10869   ZZcz 10937   ^cexp 12272   #chash 12515   || cdvds 14305   gcd cgcd 14468   Primecprime 14622   pCnt cpc 14786   Basecbs 15121   Grpcgrp 16669   odcod 17165   Abelcabl 17431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-fz 11785  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-dvds 14306  df-gcd 14469  df-prm 14623  df-pc 14787  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-grp 16673  df-abl 17433

This theorem is referenced by:  ablfac1a  17702  ablfac1b  17703