Theorem ablfacrplem 17436

Description: Lemma for ablfacrp2 17438. (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 14461	. . . . . . 7 |- ( p e. Prime -> -. p || 1 )
2	1	adantl 464	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> -. p || 1 )
3		ablfacrp.1	. . . . . . . 8 |- ( ph -> ( M gcd N ) = 1 )
4	3	adantr 463	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> ( M gcd N ) = 1 )
5	4	breq2d 4407	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> ( p || ( M gcd N ) <-> p || 1 ) )
6	2, 5	mtbird 299	. . . . 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 11007	. . . . . . . . . . . . . . 15 |- ( ph -> M e. ZZ )
11		ablfacrp.o	. . . . . . . . . . . . . . . 16 |- O = ( od ` G )
12		ablfacrp.b	. . . . . . . . . . . . . . . 16 |- B = ( Base ` G )
13	11, 12	oddvdssubg 17185	. . . . . . . . . . . . . . 15 |- ( ( G e. Abel /\ M e. ZZ ) -> { x e. B | ( O ` x ) || M } e. (SubGrp ` G ) )
14	8, 10, 13	syl2anc 659	. . . . . . . . . . . . . 14 |- ( ph -> { x e. B | ( O ` x ) || M } e. (SubGrp ` G ) )
15	7, 14	syl5eqel 2494	. . . . . . . . . . . . 13 |- ( ph -> K e. (SubGrp ` G ) )
16	15	ad2antrr 724	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> K e. (SubGrp ` G ) )
17		eqid 2402	. . . . . . . . . . . . 13 |- ( G ↾s K ) = ( G ↾s K )
18	17	subggrp 16528	. . . . . . . . . . . 12 |- ( K e. (SubGrp ` G ) -> ( G ↾s K ) e. Grp )
19	16, 18	syl 17	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> ( G ↾s K ) e. Grp )
20	17	subgbas 16529	. . . . . . . . . . . . 13 |- ( K e. (SubGrp ` G ) -> K = ( Base ` ( G ↾s K ) ) )
21	16, 20	syl 17	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> K = ( Base ` ( G ↾s K ) ) )
22		ablfacrp.2	. . . . . . . . . . . . . . . 16 |- ( ph -> ( # ` B ) = ( M x. N ) )
23	9	nnnn0d 10893	. . . . . . . . . . . . . . . . 17 |- ( ph -> M e. NN0 )
24		ablfacrp.n	. . . . . . . . . . . . . . . . . 18 |- ( ph -> N e. NN )
25	24	nnnn0d 10893	. . . . . . . . . . . . . . . . 17 |- ( ph -> N e. NN0 )
26	23, 25	nn0mulcld 10898	. . . . . . . . . . . . . . . 16 |- ( ph -> ( M x. N ) e. NN0 )
27	22, 26	eqeltrd 2490	. . . . . . . . . . . . . . 15 |- ( ph -> ( # ` B ) e. NN0 )
28		fvex 5859	. . . . . . . . . . . . . . . . 17 |- ( Base ` G ) e. _V
29	12, 28	eqeltri 2486	. . . . . . . . . . . . . . . 16 |- B e. _V
30		hashclb 12477	. . . . . . . . . . . . . . . 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 3524	. . . . . . . . . . . . . . 15 |- { x e. B | ( O ` x ) || M } C_ B
34	7, 33	eqsstri 3472	. . . . . . . . . . . . . 14 |- K C_ B
35		ssfi 7775	. . . . . . . . . . . . . 14 |- ( ( B e. Fin /\ K C_ B ) -> K e. Fin )
36	32, 34, 35	sylancl 660	. . . . . . . . . . . . 13 |- ( ph -> K e. Fin )
37	36	ad2antrr 724	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> K e. Fin )
38	21, 37	eqeltrrd 2491	. . . . . . . . . . 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 459	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> p || ( # ` K ) )
41	21	fveq2d 5853	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> ( # ` K ) = ( # ` ( Base ` ( G ↾s K ) ) ) )
42	40, 41	breqtrd 4419	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> p || ( # ` ( Base ` ( G ↾s K ) ) ) )
43		eqid 2402	. . . . . . . . . . . 12 |- ( Base ` ( G ↾s K ) ) = ( Base ` ( G ↾s K ) )
44		eqid 2402	. . . . . . . . . . . 12 |- ( od ` ( G ↾s K ) ) = ( od ` ( G ↾s K ) )
45	43, 44	odcau 16948	. . . . . . . . . . 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 1233	. . . . . . . . . 10 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> E. g e. ( Base ` ( G ↾s K ) ) ( ( od ` ( G ↾s K ) ) ` g ) = p )
47	21	rexeqdv 3011	. . . . . . . . . 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 16914	. . . . . . . . . . . . 13 |- ( ( K e. (SubGrp ` G ) /\ g e. K ) -> ( O ` g ) = ( ( od ` ( G ↾s K ) ) ` g ) )
50	16, 49	sylan 469	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) /\ g e. K ) -> ( O ` g ) = ( ( od ` ( G ↾s K ) ) ` g ) )
51		fveq2 5849	. . . . . . . . . . . . . . . 16 |- ( x = g -> ( O ` x ) = ( O ` g ) )
52	51	breq1d 4405	. . . . . . . . . . . . . . 15 |- ( x = g -> ( ( O ` x ) || M <-> ( O ` g ) || M ) )
53	52, 7	elrab2 3209	. . . . . . . . . . . . . 14 |- ( g e. K <-> ( g e. B /\ ( O ` g ) || M ) )
54	53	simprbi 462	. . . . . . . . . . . . 13 |- ( g e. K -> ( O ` g ) || M )
55	54	adantl 464	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) /\ g e. K ) -> ( O ` g ) || M )
56	50, 55	eqbrtrrd 4417	. . . . . . . . . . 11 |- ( ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) /\ g e. K ) -> ( ( od ` ( G ↾s K ) ) ` g ) || M )
57		breq1 4398	. . . . . . . . . . 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 2896	. . . . . . . . 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 432	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> ( p || ( # ` K ) -> p || M ) )
62	61	anim1d 562	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> ( ( p || ( # ` K ) /\ p || N ) -> ( p || M /\ p || N ) ) )
63		prmz 14430	. . . . . . . 8 |- ( p e. Prime -> p e. ZZ )
64	63	adantl 464	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> p e. ZZ )
65		hashcl 12475	. . . . . . . . . 10 |- ( K e. Fin -> ( # ` K ) e. NN0 )
66	36, 65	syl 17	. . . . . . . . 9 |- ( ph -> ( # ` K ) e. NN0 )
67	66	nn0zd 11006	. . . . . . . 8 |- ( ph -> ( # ` K ) e. ZZ )
68	67	adantr 463	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> ( # ` K ) e. ZZ )
69	24	nnzd 11007	. . . . . . . 8 |- ( ph -> N e. ZZ )
70	69	adantr 463	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> N e. ZZ )
71		dvdsgcdb 14391	. . . . . . 7 |- ( ( p e. ZZ /\ ( # ` K ) e. ZZ /\ N e. ZZ ) -> ( ( p || ( # ` K ) /\ p || N ) <-> p || ( ( # ` K ) gcd N ) ) )
72	64, 68, 70, 71	syl3anc 1230	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> ( ( p || ( # ` K ) /\ p || N ) <-> p || ( ( # ` K ) gcd N ) ) )
73	10	adantr 463	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> M e. ZZ )
74		dvdsgcdb 14391	. . . . . . 7 |- ( ( p e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( p || M /\ p || N ) <-> p || ( M gcd N ) ) )
75	64, 73, 70, 74	syl3anc 1230	. . . . . 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 2860	. . 3 |- ( ph -> -. E. p e. Prime p || ( ( # ` K ) gcd N ) )
79		exprmfct 14460	. . 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 10621	. . . . . 6 |- ( ph -> N =/= 0 )
82		simpr 459	. . . . . . 7 |- ( ( ( # ` K ) = 0 /\ N = 0 ) -> N = 0 )
83	82	necon3ai 2631	. . . . . 6 |- ( N =/= 0 -> -. ( ( # ` K ) = 0 /\ N = 0 ) )
84	81, 83	syl 17	. . . . 5 |- ( ph -> -. ( ( # ` K ) = 0 /\ N = 0 ) )
85		gcdn0cl 14361	. . . . 5 |- ( ( ( ( # ` K ) e. ZZ /\ N e. ZZ ) /\ -. ( ( # ` K ) = 0 /\ N = 0 ) ) -> ( ( # ` K ) gcd N ) e. NN )
86	67, 69, 84, 85	syl21anc 1229	. . . 4 |- ( ph -> ( ( # ` K ) gcd N ) e. NN )
87		elnn1uz2 11203	. . . 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 375	. 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 366   /\ wa 367   = wceq 1405   e. wcel 1842   =/= wne 2598   E.wrex 2755   {crab 2758   _Vcvv 3059   C_ wss 3414   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   Fincfn 7554   0cc0 9522   1c1 9523   x. cmul 9527   NNcn 10576   2c2 10626   NN0cn0 10836   ZZcz 10905   ZZ>=cuz 11127   #chash 12452   || cdvds 14195   gcd cgcd 14353   Primecprime 14426   Basecbs 14841   ↾_s cress 14842   Grpcgrp 16377  SubGrpcsubg 16519   odcod 16873   Abelcabl 17123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-disj 4367  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-omul 7172  df-er 7348  df-ec 7350  df-qs 7354  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-acn 8355  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-q 11228  df-rp 11266  df-fz 11727  df-fzo 11855  df-fl 11966  df-mod 12035  df-seq 12152  df-exp 12211  df-fac 12398  df-bc 12425  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-sum 13658  df-dvds 14196  df-gcd 14354  df-prm 14427  df-pc 14570  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-0g 15056  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-grp 16381  df-minusg 16382  df-sbg 16383  df-mulg 16384  df-subg 16522  df-eqg 16524  df-ga 16652  df-od 16877  df-cmn 17124  df-abl 17125

This theorem is referenced by:  ablfacrp2  17438