Theorem ablfacrplem 16569

Description: Lemma for ablfacrp2 16571. (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 13800	. . . . . . 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 4307	. . . . . 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 10749	. . . . . . . . . . . . . . 15 |- ( ph -> M e. ZZ )
11		ablfacrp.o	. . . . . . . . . . . . . . . 16 |- O = ( od ` G )
12		ablfacrp.b	. . . . . . . . . . . . . . . 16 |- B = ( Base ` G )
13	11, 12	oddvdssubg 16340	. . . . . . . . . . . . . . 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 2527	. . . . . . . . . . . . 13 |- ( ph -> K e. (SubGrp ` G ) )
16	15	ad2antrr 725	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> K e. (SubGrp ` G ) )
17		eqid 2443	. . . . . . . . . . . . 13 |- ( G ↾s K ) = ( G ↾s K )
18	17	subggrp 15687	. . . . . . . . . . . 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 15688	. . . . . . . . . . . . 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 10639	. . . . . . . . . . . . . . . . 17 |- ( ph -> M e. NN0 )
24		ablfacrp.n	. . . . . . . . . . . . . . . . . 18 |- ( ph -> N e. NN )
25	24	nnnn0d 10639	. . . . . . . . . . . . . . . . 17 |- ( ph -> N e. NN0 )
26	23, 25	nn0mulcld 10644	. . . . . . . . . . . . . . . 16 |- ( ph -> ( M x. N ) e. NN0 )
27	22, 26	eqeltrd 2517	. . . . . . . . . . . . . . 15 |- ( ph -> ( # ` B ) e. NN0 )
28		fvex 5704	. . . . . . . . . . . . . . . . 17 |- ( Base ` G ) e. _V
29	12, 28	eqeltri 2513	. . . . . . . . . . . . . . . 16 |- B e. _V
30		hashclb 12131	. . . . . . . . . . . . . . . 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 3440	. . . . . . . . . . . . . . 15 |- { x e. B | ( O ` x ) || M } C_ B
34	7, 33	eqsstri 3389	. . . . . . . . . . . . . 14 |- K C_ B
35		ssfi 7536	. . . . . . . . . . . . . 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 2518	. . . . . . . . . . 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 5698	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> ( # ` K ) = ( # ` ( Base ` ( G ↾s K ) ) ) )
42	40, 41	breqtrd 4319	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> p || ( # ` ( Base ` ( G ↾s K ) ) ) )
43		eqid 2443	. . . . . . . . . . . 12 |- ( Base ` ( G ↾s K ) ) = ( Base ` ( G ↾s K ) )
44		eqid 2443	. . . . . . . . . . . 12 |- ( od ` ( G ↾s K ) ) = ( od ` ( G ↾s K ) )
45	43, 44	odcau 16106	. . . . . . . . . . 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 1221	. . . . . . . . . 10 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> E. g e. ( Base ` ( G ↾s K ) ) ( ( od ` ( G ↾s K ) ) ` g ) = p )
47	21	rexeqdv 2927	. . . . . . . . . 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 16072	. . . . . . . . . . . . 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 5694	. . . . . . . . . . . . . . . 16 |- ( x = g -> ( O ` x ) = ( O ` g ) )
52	51	breq1d 4305	. . . . . . . . . . . . . . 15 |- ( x = g -> ( ( O ` x ) || M <-> ( O ` g ) || M ) )
53	52, 7	elrab2 3122	. . . . . . . . . . . . . 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 4317	. . . . . . . . . . 11 |- ( ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) /\ g e. K ) -> ( ( od ` ( G ↾s K ) ) ` g ) || M )
57		breq1 4298	. . . . . . . . . . 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 2844	. . . . . . . . 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 13770	. . . . . . . 8 |- ( p e. Prime -> p e. ZZ )
64	63	adantl 466	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> p e. ZZ )
65		hashcl 12129	. . . . . . . . . 10 |- ( K e. Fin -> ( # ` K ) e. NN0 )
66	36, 65	syl 16	. . . . . . . . 9 |- ( ph -> ( # ` K ) e. NN0 )
67	66	nn0zd 10748	. . . . . . . 8 |- ( ph -> ( # ` K ) e. ZZ )
68	67	adantr 465	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> ( # ` K ) e. ZZ )
69	24	nnzd 10749	. . . . . . . 8 |- ( ph -> N e. ZZ )
70	69	adantr 465	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> N e. ZZ )
71		dvdsgcdb 13731	. . . . . . 7 |- ( ( p e. ZZ /\ ( # ` K ) e. ZZ /\ N e. ZZ ) -> ( ( p || ( # ` K ) /\ p || N ) <-> p || ( ( # ` K ) gcd N ) ) )
72	64, 68, 70, 71	syl3anc 1218	. . . . . 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 13731	. . . . . . 7 |- ( ( p e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( p || M /\ p || N ) <-> p || ( M gcd N ) ) )
75	64, 73, 70, 74	syl3anc 1218	. . . . . 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 2822	. . 3 |- ( ph -> -. E. p e. Prime p || ( ( # ` K ) gcd N ) )
79		exprmfct 13799	. . 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 10369	. . . . . 6 |- ( ph -> N =/= 0 )
82		simpr 461	. . . . . . 7 |- ( ( ( # ` K ) = 0 /\ N = 0 ) -> N = 0 )
83	82	necon3ai 2654	. . . . . 6 |- ( N =/= 0 -> -. ( ( # ` K ) = 0 /\ N = 0 ) )
84	81, 83	syl 16	. . . . 5 |- ( ph -> -. ( ( # ` K ) = 0 /\ N = 0 ) )
85		gcdn0cl 13701	. . . . 5 |- ( ( ( ( # ` K ) e. ZZ /\ N e. ZZ ) /\ -. ( ( # ` K ) = 0 /\ N = 0 ) ) -> ( ( # ` K ) gcd N ) e. NN )
86	67, 69, 84, 85	syl21anc 1217	. . . 4 |- ( ph -> ( ( # ` K ) gcd N ) e. NN )
87		elnn1uz2 10934	. . . 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 1369   e. wcel 1756   =/= wne 2609   E.wrex 2719   {crab 2722   _Vcvv 2975   C_ wss 3331   class class class wbr 4295   ` cfv 5421  (class class class)co 6094   Fincfn 7313   0cc0 9285   1c1 9286   x. cmul 9290   NNcn 10325   2c2 10374   NN0cn0 10582   ZZcz 10649   ZZ>=cuz 10864   #chash 12106   || cdivides 13538   gcd cgcd 13693   Primecprime 13766   Basecbs 14177   ↾_s cress 14178   Grpcgrp 15413  SubGrpcsubg 15678   odcod 16031   Abelcabel 16281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-disj 4266  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-2o 6924  df-oadd 6927  df-omul 6928  df-er 7104  df-ec 7106  df-qs 7110  df-map 7219  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-sup 7694  df-oi 7727  df-card 8112  df-acn 8115  df-cda 8340  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-q 10957  df-rp 10995  df-fz 11441  df-fzo 11552  df-fl 11645  df-mod 11712  df-seq 11810  df-exp 11869  df-fac 12055  df-bc 12082  df-hash 12107  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-clim 12969  df-sum 13167  df-dvds 13539  df-gcd 13694  df-prm 13767  df-pc 13907  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-ress 14184  df-plusg 14254  df-0g 14383  df-mnd 15418  df-submnd 15468  df-grp 15548  df-minusg 15549  df-sbg 15550  df-mulg 15551  df-subg 15681  df-eqg 15683  df-ga 15811  df-od 16035  df-cmn 16282  df-abl 16283

This theorem is referenced by:  ablfacrp2  16571