Theorem ablfac1lem 17246

Description: Lemma for ablfac1b 17248. 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 3499	. . . . . 6 |- ( ( ph /\ P e. A ) -> P e. Prime )
4		prmnn 14232	. . . . . 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 16930	. . . . . . . . 9 |- ( G e. Abel -> G e. Grp )
8		ablfac1.b	. . . . . . . . . 10 |- B = ( Base ` G )
9	8	grpbn0 16206	. . . . . . . . 9 |- ( G e. Grp -> B =/= (/) )
10	6, 7, 9	3syl 20	. . . . . . . 8 |- ( ph -> B =/= (/) )
11		ablfac1.f	. . . . . . . . 9 |- ( ph -> B e. Fin )
12		hashnncl 12439	. . . . . . . . 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 14386	. . . . 5 |- ( ( ph /\ P e. A ) -> ( P pCnt ( # ` B ) ) e. NN0 )
17	5, 16	nnexpcld 12334	. . . 4 |- ( ( ph /\ P e. A ) -> ( P ^ ( P pCnt ( # ` B ) ) ) e. NN )
18	1, 17	syl5eqel 2549	. . 3 |- ( ( ph /\ P e. A ) -> M e. NN )
19		ablfac1.n	. . . 4 |- N = ( ( # ` B ) / M )
20		pcdvds 14399	. . . . . . 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 4489	. . . . 5 |- ( ( ph /\ P e. A ) -> M || ( # ` B ) )
23		nndivdvds 14004	. . . . . 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 2549	. . 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 6306	. . 3 |- ( M gcd N ) = ( ( P ^ ( P pCnt ( # ` B ) ) ) gcd N )
29		pcndvds2 14403	. . . . . . 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 6307	. . . . . . . 8 |- ( ( # ` B ) / M ) = ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) )
32	19, 31	eqtri 2486	. . . . . . 7 |- N = ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) )
33	32	breq2i 4464	. . . . . 6 |- ( P || N <-> P || ( ( # ` B ) / ( P ^ ( P pCnt ( # ` B ) ) ) ) )
34	30, 33	sylnibr 305	. . . . 5 |- ( ( ph /\ P e. A ) -> -. P || N )
35	26	nnzd 10989	. . . . . 6 |- ( ( ph /\ P e. A ) -> N e. ZZ )
36		coprm 14253	. . . . . 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 14233	. . . . . 6 |- ( P e. Prime -> P e. ZZ )
40	3, 39	syl 16	. . . . 5 |- ( ( ph /\ P e. A ) -> P e. ZZ )
41		rpexp1i 14274	. . . . 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 2510	. 2 |- ( ( ph /\ P e. A ) -> ( M gcd N ) = 1 )
45	19	oveq2i 6307	. . 3 |- ( M x. N ) = ( M x. ( ( # ` B ) / M ) )
46	15	nncnd 10572	. . . 4 |- ( ( ph /\ P e. A ) -> ( # ` B ) e. CC )
47	18	nncnd 10572	. . . 4 |- ( ( ph /\ P e. A ) -> M e. CC )
48	18	nnne0d 10601	. . . 4 |- ( ( ph /\ P e. A ) -> M =/= 0 )
49	46, 47, 48	divcan2d 10343	. . 3 |- ( ( ph /\ P e. A ) -> ( M x. ( ( # ` B ) / M ) ) = ( # ` B ) )
50	45, 49	syl5req 2511	. 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 1395   e. wcel 1819   =/= wne 2652   {crab 2811   C_ wss 3471   (/)c0 3793   class class class wbr 4456   |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   Fincfn 7535   1c1 9510   x. cmul 9514   / cdiv 10227   NNcn 10556   NN0cn0 10816   ZZcz 10885   ^cexp 12169   #chash 12408   || cdvds 13998   gcd cgcd 14156   Primecprime 14229   pCnt cpc 14372   Basecbs 14644   Grpcgrp 16180   odcod 16676   Abelcabl 16926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-fz 11698  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-dvds 13999  df-gcd 14157  df-prm 14230  df-pc 14373  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-abl 16928

This theorem is referenced by:  ablfac1a  17247  ablfac1b  17248