Theorem ablfac1lem 16921

Description: Lemma for ablfac1b 16923. 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 3504	. . . . . 6 |- ( ( ph /\ P e. A ) -> P e. Prime )
4		prmnn 14079	. . . . . 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 16609	. . . . . . . . 9 |- ( G e. Abel -> G e. Grp )
8		ablfac1.b	. . . . . . . . . 10 |- B = ( Base ` G )
9	8	grpbn0 15889	. . . . . . . . 9 |- ( G e. Grp -> B =/= (/) )
10	6, 7, 9	3syl 20	. . . . . . . 8 |- ( ph -> B =/= (/) )
11		ablfac1.f	. . . . . . . . 9 |- ( ph -> B e. Fin )
12		hashnncl 12404	. . . . . . . . 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 14233	. . . . 5 |- ( ( ph /\ P e. A ) -> ( P pCnt ( # ` B ) ) e. NN0 )
17	5, 16	nnexpcld 12299	. . . 4 |- ( ( ph /\ P e. A ) -> ( P ^ ( P pCnt ( # ` B ) ) ) e. NN )
18	1, 17	syl5eqel 2559	. . 3 |- ( ( ph /\ P e. A ) -> M e. NN )
19		ablfac1.n	. . . 4 |- N = ( ( # ` B ) / M )
20		pcdvds 14246	. . . . . . 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 4480	. . . . 5 |- ( ( ph /\ P e. A ) -> M || ( # ` B ) )
23		nndivdvds 13853	. . . . . 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 2559	. . 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 6294	. . 3 |- ( M gcd N ) = ( ( P ^ ( P pCnt ( # ` B ) ) ) gcd N )
29		pcndvds2 14250	. . . . . . 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 6295	. . . . . . . 8 |- ( ( # ` B ) / M ) = ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) )
32	19, 31	eqtri 2496	. . . . . . 7 |- N = ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) )
33	32	breq2i 4455	. . . . . 6 |- ( P || N <-> P || ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) )
34	30, 33	sylnibr 305	. . . . 5 |- ( ( ph /\ P e. A ) -> -. P || N )
35	26	nnzd 10965	. . . . . 6 |- ( ( ph /\ P e. A ) -> N e. ZZ )
36		coprm 14100	. . . . . 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 14080	. . . . . 6 |- ( P e. Prime -> P e. ZZ )
40	3, 39	syl 16	. . . . 5 |- ( ( ph /\ P e. A ) -> P e. ZZ )
41		rpexp1i 14121	. . . . 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 1228	. . . 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 2520	. 2 |- ( ( ph /\ P e. A ) -> ( M gcd N ) = 1 )
45	19	oveq2i 6295	. . 3 |- ( M x. N ) = ( M x. ( ( # ` B ) / M ) )
46	15	nncnd 10552	. . . 4 |- ( ( ph /\ P e. A ) -> ( # ` B ) e. CC )
47	18	nncnd 10552	. . . 4 |- ( ( ph /\ P e. A ) -> M e. CC )
48	18	nnne0d 10580	. . . 4 |- ( ( ph /\ P e. A ) -> M =/= 0 )
49	46, 47, 48	divcan2d 10322	. . 3 |- ( ( ph /\ P e. A ) -> ( M x. ( ( # ` B ) / M ) ) = ( # ` B ) )
50	45, 49	syl5req 2521	. 2 |- ( ( ph /\ P e. A ) -> ( # ` B ) = ( M x. N ) )
51	27, 44, 50	3jca 1176	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 973   = wceq 1379   e. wcel 1767   =/= wne 2662   {crab 2818   C_ wss 3476   (/)c0 3785   class class class wbr 4447   |-> cmpt 4505   ` cfv 5588  (class class class)co 6284   Fincfn 7516   1c1 9493   x. cmul 9497   / cdiv 10206   NNcn 10536   NN0cn0 10795   ZZcz 10864   ^cexp 12134   #chash 12373   || cdivides 13847   gcd cgcd 14003   Primecprime 14076   pCnt cpc 14219   Basecbs 14490   Grpcgrp 15727   odcod 16355   Abelcabl 16605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-q 11183  df-rp 11221  df-fz 11673  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-dvds 13848  df-gcd 14004  df-prm 14077  df-pc 14220  df-0g 14697  df-mnd 15732  df-grp 15867  df-abl 16607

This theorem is referenced by:  ablfac1a  16922  ablfac1b  16923