Theorem ablfacrplem 16540

Description: Lemma for ablfacrp2 16542. (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 13780	. . . . . . 7 |- ( p e. Prime -> -. p || 1 )
2	1	adantl 463	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> -. p || 1 )
3		ablfacrp.1	. . . . . . . 8 |- ( ph -> ( M gcd N ) = 1 )
4	3	adantr 462	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> ( M gcd N ) = 1 )
5	4	breq2d 4292	. . . . . 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 10734	. . . . . . . . . . . . . . 15 |- ( ph -> M e. ZZ )
11		ablfacrp.o	. . . . . . . . . . . . . . . 16 |- O = ( od ` G )
12		ablfacrp.b	. . . . . . . . . . . . . . . 16 |- B = ( Base ` G )
13	11, 12	oddvdssubg 16317	. . . . . . . . . . . . . . 15 |- ( ( G e. Abel /\ M e. ZZ ) -> { x e. B | ( O ` x ) || M } e. (SubGrp ` G ) )
14	8, 10, 13	syl2anc 654	. . . . . . . . . . . . . 14 |- ( ph -> { x e. B | ( O ` x ) || M } e. (SubGrp ` G ) )
15	7, 14	syl5eqel 2517	. . . . . . . . . . . . 13 |- ( ph -> K e. (SubGrp ` G ) )
16	15	ad2antrr 718	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> K e. (SubGrp ` G ) )
17		eqid 2433	. . . . . . . . . . . . 13 |- ( G ↾s K ) = ( G ↾s K )
18	17	subggrp 15664	. . . . . . . . . . . 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 15665	. . . . . . . . . . . . 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 10624	. . . . . . . . . . . . . . . . 17 |- ( ph -> M e. NN0 )
24		ablfacrp.n	. . . . . . . . . . . . . . . . . 18 |- ( ph -> N e. NN )
25	24	nnnn0d 10624	. . . . . . . . . . . . . . . . 17 |- ( ph -> N e. NN0 )
26	23, 25	nn0mulcld 10629	. . . . . . . . . . . . . . . 16 |- ( ph -> ( M x. N ) e. NN0 )
27	22, 26	eqeltrd 2507	. . . . . . . . . . . . . . 15 |- ( ph -> ( # ` B ) e. NN0 )
28		fvex 5689	. . . . . . . . . . . . . . . . 17 |- ( Base ` G ) e. _V
29	12, 28	eqeltri 2503	. . . . . . . . . . . . . . . 16 |- B e. _V
30		hashclb 12112	. . . . . . . . . . . . . . . 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 3425	. . . . . . . . . . . . . . 15 |- { x e. B | ( O ` x ) || M } C_ B
34	7, 33	eqsstri 3374	. . . . . . . . . . . . . 14 |- K C_ B
35		ssfi 7521	. . . . . . . . . . . . . 14 |- ( ( B e. Fin /\ K C_ B ) -> K e. Fin )
36	32, 34, 35	sylancl 655	. . . . . . . . . . . . 13 |- ( ph -> K e. Fin )
37	36	ad2antrr 718	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> K e. Fin )
38	21, 37	eqeltrrd 2508	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> ( Base ` ( G ↾s K ) ) e. Fin )
39		simplr 747	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> p e. Prime )
40		simpr 458	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> p || ( # ` K ) )
41	21	fveq2d 5683	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> ( # ` K ) = ( # ` ( Base ` ( G ↾s K ) ) ) )
42	40, 41	breqtrd 4304	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> p || ( # ` ( Base ` ( G ↾s K ) ) ) )
43		eqid 2433	. . . . . . . . . . . 12 |- ( Base ` ( G ↾s K ) ) = ( Base ` ( G ↾s K ) )
44		eqid 2433	. . . . . . . . . . . 12 |- ( od ` ( G ↾s K ) ) = ( od ` ( G ↾s K ) )
45	43, 44	odcau 16083	. . . . . . . . . . 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 1214	. . . . . . . . . 10 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> E. g e. ( Base ` ( G ↾s K ) ) ( ( od ` ( G ↾s K ) ) ` g ) = p )
47	21	rexeqdv 2914	. . . . . . . . . 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 16049	. . . . . . . . . . . . 13 |- ( ( K e. (SubGrp ` G ) /\ g e. K ) -> ( O ` g ) = ( ( od ` ( G ↾s K ) ) ` g ) )
50	16, 49	sylan 468	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) /\ g e. K ) -> ( O ` g ) = ( ( od ` ( G ↾s K ) ) ` g ) )
51		fveq2 5679	. . . . . . . . . . . . . . . 16 |- ( x = g -> ( O ` x ) = ( O ` g ) )
52	51	breq1d 4290	. . . . . . . . . . . . . . 15 |- ( x = g -> ( ( O ` x ) || M <-> ( O ` g ) || M ) )
53	52, 7	elrab2 3108	. . . . . . . . . . . . . 14 |- ( g e. K <-> ( g e. B /\ ( O ` g ) || M ) )
54	53	simprbi 461	. . . . . . . . . . . . 13 |- ( g e. K -> ( O ` g ) || M )
55	54	adantl 463	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) /\ g e. K ) -> ( O ` g ) || M )
56	50, 55	eqbrtrrd 4302	. . . . . . . . . . 11 |- ( ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) /\ g e. K ) -> ( ( od ` ( G ↾s K ) ) ` g ) || M )
57		breq1 4283	. . . . . . . . . . 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 2831	. . . . . . . . 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 559	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> ( ( p || ( # ` K ) /\ p || N ) -> ( p || M /\ p || N ) ) )
63		prmz 13750	. . . . . . . 8 |- ( p e. Prime -> p e. ZZ )
64	63	adantl 463	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> p e. ZZ )
65		hashcl 12110	. . . . . . . . . 10 |- ( K e. Fin -> ( # ` K ) e. NN0 )
66	36, 65	syl 16	. . . . . . . . 9 |- ( ph -> ( # ` K ) e. NN0 )
67	66	nn0zd 10733	. . . . . . . 8 |- ( ph -> ( # ` K ) e. ZZ )
68	67	adantr 462	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> ( # ` K ) e. ZZ )
69	24	nnzd 10734	. . . . . . . 8 |- ( ph -> N e. ZZ )
70	69	adantr 462	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> N e. ZZ )
71		dvdsgcdb 13711	. . . . . . 7 |- ( ( p e. ZZ /\ ( # ` K ) e. ZZ /\ N e. ZZ ) -> ( ( p || ( # ` K ) /\ p || N ) <-> p || ( ( # ` K ) gcd N ) ) )
72	64, 68, 70, 71	syl3anc 1211	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> ( ( p || ( # ` K ) /\ p || N ) <-> p || ( ( # ` K ) gcd N ) ) )
73	10	adantr 462	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> M e. ZZ )
74		dvdsgcdb 13711	. . . . . . 7 |- ( ( p e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( p || M /\ p || N ) <-> p || ( M gcd N ) ) )
75	64, 73, 70, 74	syl3anc 1211	. . . . . 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 2809	. . 3 |- ( ph -> -. E. p e. Prime p || ( ( # ` K ) gcd N ) )
79		exprmfct 13779	. . 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 10354	. . . . . 6 |- ( ph -> N =/= 0 )
82		simpr 458	. . . . . . 7 |- ( ( ( # ` K ) = 0 /\ N = 0 ) -> N = 0 )
83	82	necon3ai 2641	. . . . . 6 |- ( N =/= 0 -> -. ( ( # ` K ) = 0 /\ N = 0 ) )
84	81, 83	syl 16	. . . . 5 |- ( ph -> -. ( ( # ` K ) = 0 /\ N = 0 ) )
85		gcdn0cl 13681	. . . . 5 |- ( ( ( ( # ` K ) e. ZZ /\ N e. ZZ ) /\ -. ( ( # ` K ) = 0 /\ N = 0 ) ) -> ( ( # ` K ) gcd N ) e. NN )
86	67, 69, 84, 85	syl21anc 1210	. . . 4 |- ( ph -> ( ( # ` K ) gcd N ) e. NN )
87		elnn1uz2 10919	. . . 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 1362   e. wcel 1755   =/= wne 2596   E.wrex 2706   {crab 2709   _Vcvv 2962   C_ wss 3316   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   Fincfn 7298   0cc0 9270   1c1 9271   x. cmul 9275   NNcn 10310   2c2 10359   NN0cn0 10567   ZZcz 10634   ZZ>=cuz 10849   #chash 12087   || cdivides 13518   gcd cgcd 13673   Primecprime 13746   Basecbs 14157   ↾_s cress 14158   Grpcgrp 15393  SubGrpcsubg 15655   odcod 16008   Abelcabel 16258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-disj 4251  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-omul 6913  df-er 7089  df-ec 7091  df-qs 7095  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-acn 8100  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-n0 10568  df-z 10635  df-uz 10850  df-q 10942  df-rp 10980  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-fac 12036  df-bc 12063  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-sum 13148  df-dvds 13519  df-gcd 13674  df-prm 13747  df-pc 13887  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-0g 14363  df-mnd 15398  df-submnd 15448  df-grp 15525  df-minusg 15526  df-sbg 15527  df-mulg 15528  df-subg 15658  df-eqg 15660  df-ga 15788  df-od 16012  df-cmn 16259  df-abl 16260

This theorem is referenced by:  ablfacrp2  16542