Theorem ablfacrplem 16899

Description: Lemma for ablfacrp2 16901. (Contributed by Mario Carneiro, 19-Apr-2016.)

Hypotheses

Ref	Expression
ablfacrp.b	|- B = ( Base ` G )
ablfacrp.o	|- O = ( od ` G )
ablfacrp.k	|- K = { x e. B | ( O ` x ) || M }
ablfacrp.l	|- L = { x e. B | ( O ` x ) || N }
ablfacrp.g	|- ( ph -> G e. Abel )
ablfacrp.m	|- ( ph -> M e. NN )
ablfacrp.n	|- ( ph -> N e. NN )
ablfacrp.1	|- ( ph -> ( M gcd N ) = 1 )
ablfacrp.2	|- ( ph -> ( # ` B ) = ( M x. N ) )

Assertion

Ref	Expression
ablfacrplem	|- ( ph -> ( ( # ` K ) gcd N ) = 1 )

Distinct variable groups:   x, B   x, G   x, O   x, M   x, N   ph, x

Allowed substitution hints:   K( x)   L( x)

Proof of Theorem ablfacrplem

Dummy variables g p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nprmdvds1 14100	. . . . . . 7 |- ( p e. Prime -> -. p || 1 )
2	1	adantl 466	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> -. p || 1 )
3		ablfacrp.1	. . . . . . . 8 |- ( ph -> ( M gcd N ) = 1 )
4	3	adantr 465	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> ( M gcd N ) = 1 )
5	4	breq2d 4452	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> ( p || ( M gcd N ) <-> p || 1 ) )
6	2, 5	mtbird 301	. . . . 5 |- ( ( ph /\ p e. Prime ) -> -. p || ( M gcd N ) )
7		ablfacrp.k	. . . . . . . . . . . . . 14 |- K = { x e. B | ( O ` x ) || M }
8		ablfacrp.g	. . . . . . . . . . . . . . 15 |- ( ph -> G e. Abel )
9		ablfacrp.m	. . . . . . . . . . . . . . . 16 |- ( ph -> M e. NN )
10	9	nnzd 10954	. . . . . . . . . . . . . . 15 |- ( ph -> M e. ZZ )
11		ablfacrp.o	. . . . . . . . . . . . . . . 16 |- O = ( od ` G )
12		ablfacrp.b	. . . . . . . . . . . . . . . 16 |- B = ( Base ` G )
13	11, 12	oddvdssubg 16647	. . . . . . . . . . . . . . 15 |- ( ( G e. Abel /\ M e. ZZ ) -> { x e. B | ( O ` x ) || M } e. (SubGrp ` G ) )
14	8, 10, 13	syl2anc 661	. . . . . . . . . . . . . 14 |- ( ph -> { x e. B | ( O ` x ) || M } e. (SubGrp ` G ) )
15	7, 14	syl5eqel 2552	. . . . . . . . . . . . 13 |- ( ph -> K e. (SubGrp ` G ) )
16	15	ad2antrr 725	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> K e. (SubGrp ` G ) )
17		eqid 2460	. . . . . . . . . . . . 13 |- ( G ↾s K ) = ( G ↾s K )
18	17	subggrp 15992	. . . . . . . . . . . 12 |- ( K e. (SubGrp ` G ) -> ( G ↾s K ) e. Grp )
19	16, 18	syl 16	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> ( G ↾s K ) e. Grp )
20	17	subgbas 15993	. . . . . . . . . . . . 13 |- ( K e. (SubGrp ` G ) -> K = ( Base ` ( G ↾s K ) ) )
21	16, 20	syl 16	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> K = ( Base ` ( G ↾s K ) ) )
22		ablfacrp.2	. . . . . . . . . . . . . . . 16 |- ( ph -> ( # ` B ) = ( M x. N ) )
23	9	nnnn0d 10841	. . . . . . . . . . . . . . . . 17 |- ( ph -> M e. NN0 )
24		ablfacrp.n	. . . . . . . . . . . . . . . . . 18 |- ( ph -> N e. NN )
25	24	nnnn0d 10841	. . . . . . . . . . . . . . . . 17 |- ( ph -> N e. NN0 )
26	23, 25	nn0mulcld 10846	. . . . . . . . . . . . . . . 16 |- ( ph -> ( M x. N ) e. NN0 )
27	22, 26	eqeltrd 2548	. . . . . . . . . . . . . . 15 |- ( ph -> ( # ` B ) e. NN0 )
28		fvex 5867	. . . . . . . . . . . . . . . . 17 |- ( Base ` G ) e. _V
29	12, 28	eqeltri 2544	. . . . . . . . . . . . . . . 16 |- B e. _V
30		hashclb 12385	. . . . . . . . . . . . . . . 16 |- ( B e. _V -> ( B e. Fin <-> ( # ` B ) e. NN0 ) )
31	29, 30	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( B e. Fin <-> ( # ` B ) e. NN0 )
32	27, 31	sylibr 212	. . . . . . . . . . . . . 14 |- ( ph -> B e. Fin )
33		ssrab2 3578	. . . . . . . . . . . . . . 15 |- { x e. B | ( O ` x ) || M } C_ B
34	7, 33	eqsstri 3527	. . . . . . . . . . . . . 14 |- K C_ B
35		ssfi 7730	. . . . . . . . . . . . . 14 |- ( ( B e. Fin /\ K C_ B ) -> K e. Fin )
36	32, 34, 35	sylancl 662	. . . . . . . . . . . . 13 |- ( ph -> K e. Fin )
37	36	ad2antrr 725	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> K e. Fin )
38	21, 37	eqeltrrd 2549	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> ( Base ` ( G ↾s K ) ) e. Fin )
39		simplr 754	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> p e. Prime )
40		simpr 461	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> p || ( # ` K ) )
41	21	fveq2d 5861	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> ( # ` K ) = ( # ` ( Base ` ( G ↾s K ) ) ) )
42	40, 41	breqtrd 4464	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> p || ( # ` ( Base ` ( G ↾s K ) ) ) )
43		eqid 2460	. . . . . . . . . . . 12 |- ( Base ` ( G ↾s K ) ) = ( Base ` ( G ↾s K ) )
44		eqid 2460	. . . . . . . . . . . 12 |- ( od ` ( G ↾s K ) ) = ( od ` ( G ↾s K ) )
45	43, 44	odcau 16413	. . . . . . . . . . 11 |- ( ( ( ( G ↾s K ) e. Grp /\ ( Base ` ( G ↾s K ) ) e. Fin /\ p e. Prime ) /\ p || ( # ` ( Base ` ( G ↾s K ) ) ) ) -> E. g e. ( Base ` ( G ↾s K ) ) ( ( od ` ( G ↾s K ) ) ` g ) = p )
46	19, 38, 39, 42, 45	syl31anc 1226	. . . . . . . . . 10 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> E. g e. ( Base ` ( G ↾s K ) ) ( ( od ` ( G ↾s K ) ) ` g ) = p )
47	21	rexeqdv 3058	. . . . . . . . . 10 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> ( E. g e. K ( ( od ` ( G ↾s K ) ) ` g ) = p <-> E. g e. ( Base ` ( G ↾s K ) ) ( ( od ` ( G ↾s K ) ) ` g ) = p ) )
48	46, 47	mpbird 232	. . . . . . . . 9 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> E. g e. K ( ( od ` ( G ↾s K ) ) ` g ) = p )
49	17, 11, 44	subgod 16379	. . . . . . . . . . . . 13 |- ( ( K e. (SubGrp ` G ) /\ g e. K ) -> ( O ` g ) = ( ( od ` ( G ↾s K ) ) ` g ) )
50	16, 49	sylan 471	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) /\ g e. K ) -> ( O ` g ) = ( ( od ` ( G ↾s K ) ) ` g ) )
51		fveq2 5857	. . . . . . . . . . . . . . . 16 |- ( x = g -> ( O ` x ) = ( O ` g ) )
52	51	breq1d 4450	. . . . . . . . . . . . . . 15 |- ( x = g -> ( ( O ` x ) || M <-> ( O ` g ) || M ) )
53	52, 7	elrab2 3256	. . . . . . . . . . . . . 14 |- ( g e. K <-> ( g e. B /\ ( O ` g ) || M ) )
54	53	simprbi 464	. . . . . . . . . . . . 13 |- ( g e. K -> ( O ` g ) || M )
55	54	adantl 466	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) /\ g e. K ) -> ( O ` g ) || M )
56	50, 55	eqbrtrrd 4462	. . . . . . . . . . 11 |- ( ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) /\ g e. K ) -> ( ( od ` ( G ↾s K ) ) ` g ) || M )
57		breq1 4443	. . . . . . . . . . 11 |- ( ( ( od ` ( G ↾s K ) ) ` g ) = p -> ( ( ( od ` ( G ↾s K ) ) ` g ) || M <-> p || M ) )
58	56, 57	syl5ibcom 220	. . . . . . . . . 10 |- ( ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) /\ g e. K ) -> ( ( ( od ` ( G ↾s K ) ) ` g ) = p -> p || M ) )
59	58	rexlimdva 2948	. . . . . . . . 9 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> ( E. g e. K ( ( od ` ( G ↾s K ) ) ` g ) = p -> p || M ) )
60	48, 59	mpd 15	. . . . . . . 8 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> p || M )
61	60	ex 434	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> ( p || ( # ` K ) -> p || M ) )
62	61	anim1d 564	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> ( ( p || ( # ` K ) /\ p || N ) -> ( p || M /\ p || N ) ) )
63		prmz 14069	. . . . . . . 8 |- ( p e. Prime -> p e. ZZ )
64	63	adantl 466	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> p e. ZZ )
65		hashcl 12383	. . . . . . . . . 10 |- ( K e. Fin -> ( # ` K ) e. NN0 )
66	36, 65	syl 16	. . . . . . . . 9 |- ( ph -> ( # ` K ) e. NN0 )
67	66	nn0zd 10953	. . . . . . . 8 |- ( ph -> ( # ` K ) e. ZZ )
68	67	adantr 465	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> ( # ` K ) e. ZZ )
69	24	nnzd 10954	. . . . . . . 8 |- ( ph -> N e. ZZ )
70	69	adantr 465	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> N e. ZZ )
71		dvdsgcdb 14030	. . . . . . 7 |- ( ( p e. ZZ /\ ( # ` K ) e. ZZ /\ N e. ZZ ) -> ( ( p || ( # ` K ) /\ p || N ) <-> p || ( ( # ` K ) gcd N ) ) )
72	64, 68, 70, 71	syl3anc 1223	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> ( ( p || ( # ` K ) /\ p || N ) <-> p || ( ( # ` K ) gcd N ) ) )
73	10	adantr 465	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> M e. ZZ )
74		dvdsgcdb 14030	. . . . . . 7 |- ( ( p e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( p || M /\ p || N ) <-> p || ( M gcd N ) ) )
75	64, 73, 70, 74	syl3anc 1223	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> ( ( p || M /\ p || N ) <-> p || ( M gcd N ) ) )
76	62, 72, 75	3imtr3d 267	. . . . 5 |- ( ( ph /\ p e. Prime ) -> ( p || ( ( # ` K ) gcd N ) -> p || ( M gcd N ) ) )
77	6, 76	mtod 177	. . . 4 |- ( ( ph /\ p e. Prime ) -> -. p || ( ( # ` K ) gcd N ) )
78	77	nrexdv 2913	. . 3 |- ( ph -> -. E. p e. Prime p || ( ( # ` K ) gcd N ) )
79		exprmfct 14099	. . 3 |- ( ( ( # ` K ) gcd N ) e. ( ZZ>= ` 2 ) -> E. p e. Prime p || ( ( # ` K ) gcd N ) )
80	78, 79	nsyl 121	. 2 |- ( ph -> -. ( ( # ` K ) gcd N ) e. ( ZZ>= ` 2 ) )
81	24	nnne0d 10569	. . . . . 6 |- ( ph -> N =/= 0 )
82		simpr 461	. . . . . . 7 |- ( ( ( # ` K ) = 0 /\ N = 0 ) -> N = 0 )
83	82	necon3ai 2688	. . . . . 6 |- ( N =/= 0 -> -. ( ( # ` K ) = 0 /\ N = 0 ) )
84	81, 83	syl 16	. . . . 5 |- ( ph -> -. ( ( # ` K ) = 0 /\ N = 0 ) )
85		gcdn0cl 14000	. . . . 5 |- ( ( ( ( # ` K ) e. ZZ /\ N e. ZZ ) /\ -. ( ( # ` K ) = 0 /\ N = 0 ) ) -> ( ( # ` K ) gcd N ) e. NN )
86	67, 69, 84, 85	syl21anc 1222	. . . 4 |- ( ph -> ( ( # ` K ) gcd N ) e. NN )
87		elnn1uz2 11147	. . . 4 |- ( ( ( # ` K ) gcd N ) e. NN <-> ( ( ( # ` K ) gcd N ) = 1 \/ ( ( # ` K ) gcd N ) e. ( ZZ>= ` 2 ) ) )
88	86, 87	sylib 196	. . 3 |- ( ph -> ( ( ( # ` K ) gcd N ) = 1 \/ ( ( # ` K ) gcd N ) e. ( ZZ>= ` 2 ) ) )
89	88	ord 377	. 2 |- ( ph -> ( -. ( ( # ` K ) gcd N ) = 1 -> ( ( # ` K ) gcd N ) e. ( ZZ>= ` 2 ) ) )
90	80, 89	mt3d 125	1 |- ( ph -> ( ( # ` K ) gcd N ) = 1 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1374   e. wcel 1762   =/= wne 2655   E.wrex 2808   {crab 2811   _Vcvv 3106   C_ wss 3469   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Fincfn 7506   0cc0 9481   1c1 9482   x. cmul 9486   NNcn 10525   2c2 10574   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071   #chash 12360   || cdivides 13836   gcd cgcd 13992   Primecprime 14065   Basecbs 14479   ↾_s cress 14480   Grpcgrp 15716  SubGrpcsubg 15983   odcod 16338   Abelcabel 16588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-disj 4411  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-omul 7125  df-er 7301  df-ec 7303  df-qs 7307  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-acn 8312  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-q 11172  df-rp 11210  df-fz 11662  df-fzo 11782  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-fac 12309  df-bc 12336  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-dvds 13837  df-gcd 13993  df-prm 14066  df-pc 14209  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-0g 14686  df-mnd 15721  df-submnd 15771  df-grp 15851  df-minusg 15852  df-sbg 15853  df-mulg 15854  df-subg 15986  df-eqg 15988  df-ga 16116  df-od 16342  df-cmn 16589  df-abl 16590

This theorem is referenced by:  ablfacrp2  16901