Theorem ablfacrplem 17691

Description: Lemma for ablfacrp2 17693. (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 14643	. . . . . . 7 |- ( p e. Prime -> -. p || 1 )
2	1	adantl 468	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> -. p || 1 )
3		ablfacrp.1	. . . . . . . 8 |- ( ph -> ( M gcd N ) = 1 )
4	3	adantr 467	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> ( M gcd N ) = 1 )
5	4	breq2d 4413	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> ( p || ( M gcd N ) <-> p || 1 ) )
6	2, 5	mtbird 303	. . . . 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 11036	. . . . . . . . . . . . . . 15 |- ( ph -> M e. ZZ )
11		ablfacrp.o	. . . . . . . . . . . . . . . 16 |- O = ( od ` G )
12		ablfacrp.b	. . . . . . . . . . . . . . . 16 |- B = ( Base ` G )
13	11, 12	oddvdssubg 17486	. . . . . . . . . . . . . . 15 |- ( ( G e. Abel /\ M e. ZZ ) -> { x e. B | ( O ` x ) || M } e. (SubGrp ` G ) )
14	8, 10, 13	syl2anc 666	. . . . . . . . . . . . . 14 |- ( ph -> { x e. B | ( O ` x ) || M } e. (SubGrp ` G ) )
15	7, 14	syl5eqel 2532	. . . . . . . . . . . . 13 |- ( ph -> K e. (SubGrp ` G ) )
16	15	ad2antrr 731	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> K e. (SubGrp ` G ) )
17		eqid 2450	. . . . . . . . . . . . 13 |- ( G ↾s K ) = ( G ↾s K )
18	17	subggrp 16813	. . . . . . . . . . . 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 16814	. . . . . . . . . . . . 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 10922	. . . . . . . . . . . . . . . . 17 |- ( ph -> M e. NN0 )
24		ablfacrp.n	. . . . . . . . . . . . . . . . . 18 |- ( ph -> N e. NN )
25	24	nnnn0d 10922	. . . . . . . . . . . . . . . . 17 |- ( ph -> N e. NN0 )
26	23, 25	nn0mulcld 10927	. . . . . . . . . . . . . . . 16 |- ( ph -> ( M x. N ) e. NN0 )
27	22, 26	eqeltrd 2528	. . . . . . . . . . . . . . 15 |- ( ph -> ( # ` B ) e. NN0 )
28		fvex 5873	. . . . . . . . . . . . . . . . 17 |- ( Base ` G ) e. _V
29	12, 28	eqeltri 2524	. . . . . . . . . . . . . . . 16 |- B e. _V
30		hashclb 12537	. . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . 14 |- ( ph -> B e. Fin )
33		ssrab2 3513	. . . . . . . . . . . . . . 15 |- { x e. B | ( O ` x ) || M } C_ B
34	7, 33	eqsstri 3461	. . . . . . . . . . . . . 14 |- K C_ B
35		ssfi 7789	. . . . . . . . . . . . . 14 |- ( ( B e. Fin /\ K C_ B ) -> K e. Fin )
36	32, 34, 35	sylancl 667	. . . . . . . . . . . . 13 |- ( ph -> K e. Fin )
37	36	ad2antrr 731	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> K e. Fin )
38	21, 37	eqeltrrd 2529	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> ( Base ` ( G ↾s K ) ) e. Fin )
39		simplr 761	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> p e. Prime )
40		simpr 463	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> p || ( # ` K ) )
41	21	fveq2d 5867	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> ( # ` K ) = ( # ` ( Base ` ( G ↾s K ) ) ) )
42	40, 41	breqtrd 4426	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> p || ( # ` ( Base ` ( G ↾s K ) ) ) )
43		eqid 2450	. . . . . . . . . . . 12 |- ( Base ` ( G ↾s K ) ) = ( Base ` ( G ↾s K ) )
44		eqid 2450	. . . . . . . . . . . 12 |- ( od ` ( G ↾s K ) ) = ( od ` ( G ↾s K ) )
45	43, 44	odcau 17249	. . . . . . . . . . 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 1270	. . . . . . . . . 10 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> E. g e. ( Base ` ( G ↾s K ) ) ( ( od ` ( G ↾s K ) ) ` g ) = p )
47	21	rexeqdv 2993	. . . . . . . . . 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 236	. . . . . . . . 9 |- ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) -> E. g e. K ( ( od ` ( G ↾s K ) ) ` g ) = p )
49	17, 11, 44	subgod 17212	. . . . . . . . . . . . 13 |- ( ( K e. (SubGrp ` G ) /\ g e. K ) -> ( O ` g ) = ( ( od ` ( G ↾s K ) ) ` g ) )
50	16, 49	sylan 474	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) /\ g e. K ) -> ( O ` g ) = ( ( od ` ( G ↾s K ) ) ` g ) )
51		fveq2 5863	. . . . . . . . . . . . . . . 16 |- ( x = g -> ( O ` x ) = ( O ` g ) )
52	51	breq1d 4411	. . . . . . . . . . . . . . 15 |- ( x = g -> ( ( O ` x ) || M <-> ( O ` g ) || M ) )
53	52, 7	elrab2 3197	. . . . . . . . . . . . . 14 |- ( g e. K <-> ( g e. B /\ ( O ` g ) || M ) )
54	53	simprbi 466	. . . . . . . . . . . . 13 |- ( g e. K -> ( O ` g ) || M )
55	54	adantl 468	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) /\ g e. K ) -> ( O ` g ) || M )
56	50, 55	eqbrtrrd 4424	. . . . . . . . . . 11 |- ( ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) /\ g e. K ) -> ( ( od ` ( G ↾s K ) ) ` g ) || M )
57		breq1 4404	. . . . . . . . . . 11 |- ( ( ( od ` ( G ↾s K ) ) ` g ) = p -> ( ( ( od ` ( G ↾s K ) ) ` g ) || M <-> p || M ) )
58	56, 57	syl5ibcom 224	. . . . . . . . . 10 |- ( ( ( ( ph /\ p e. Prime ) /\ p || ( # ` K ) ) /\ g e. K ) -> ( ( ( od ` ( G ↾s K ) ) ` g ) = p -> p || M ) )
59	58	rexlimdva 2878	. . . . . . . . 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 436	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> ( p || ( # ` K ) -> p || M ) )
62	61	anim1d 567	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> ( ( p || ( # ` K ) /\ p || N ) -> ( p || M /\ p || N ) ) )
63		prmz 14619	. . . . . . . 8 |- ( p e. Prime -> p e. ZZ )
64	63	adantl 468	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> p e. ZZ )
65		hashcl 12535	. . . . . . . . . 10 |- ( K e. Fin -> ( # ` K ) e. NN0 )
66	36, 65	syl 17	. . . . . . . . 9 |- ( ph -> ( # ` K ) e. NN0 )
67	66	nn0zd 11035	. . . . . . . 8 |- ( ph -> ( # ` K ) e. ZZ )
68	67	adantr 467	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> ( # ` K ) e. ZZ )
69	24	nnzd 11036	. . . . . . . 8 |- ( ph -> N e. ZZ )
70	69	adantr 467	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> N e. ZZ )
71		dvdsgcdb 14505	. . . . . . 7 |- ( ( p e. ZZ /\ ( # ` K ) e. ZZ /\ N e. ZZ ) -> ( ( p || ( # ` K ) /\ p || N ) <-> p || ( ( # ` K ) gcd N ) ) )
72	64, 68, 70, 71	syl3anc 1267	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> ( ( p || ( # ` K ) /\ p || N ) <-> p || ( ( # ` K ) gcd N ) ) )
73	10	adantr 467	. . . . . . 7 |- ( ( ph /\ p e. Prime ) -> M e. ZZ )
74		dvdsgcdb 14505	. . . . . . 7 |- ( ( p e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( p || M /\ p || N ) <-> p || ( M gcd N ) ) )
75	64, 73, 70, 74	syl3anc 1267	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> ( ( p || M /\ p || N ) <-> p || ( M gcd N ) ) )
76	62, 72, 75	3imtr3d 271	. . . . 5 |- ( ( ph /\ p e. Prime ) -> ( p || ( ( # ` K ) gcd N ) -> p || ( M gcd N ) ) )
77	6, 76	mtod 181	. . . 4 |- ( ( ph /\ p e. Prime ) -> -. p || ( ( # ` K ) gcd N ) )
78	77	nrexdv 2842	. . 3 |- ( ph -> -. E. p e. Prime p || ( ( # ` K ) gcd N ) )
79		exprmfct 14641	. . 3 |- ( ( ( # ` K ) gcd N ) e. ( ZZ>= ` 2 ) -> E. p e. Prime p || ( ( # ` K ) gcd N ) )
80	78, 79	nsyl 125	. 2 |- ( ph -> -. ( ( # ` K ) gcd N ) e. ( ZZ>= ` 2 ) )
81	24	nnne0d 10651	. . . . . 6 |- ( ph -> N =/= 0 )
82		simpr 463	. . . . . . 7 |- ( ( ( # ` K ) = 0 /\ N = 0 ) -> N = 0 )
83	82	necon3ai 2648	. . . . . 6 |- ( N =/= 0 -> -. ( ( # ` K ) = 0 /\ N = 0 ) )
84	81, 83	syl 17	. . . . 5 |- ( ph -> -. ( ( # ` K ) = 0 /\ N = 0 ) )
85		gcdn0cl 14469	. . . . 5 |- ( ( ( ( # ` K ) e. ZZ /\ N e. ZZ ) /\ -. ( ( # ` K ) = 0 /\ N = 0 ) ) -> ( ( # ` K ) gcd N ) e. NN )
86	67, 69, 84, 85	syl21anc 1266	. . . 4 |- ( ph -> ( ( # ` K ) gcd N ) e. NN )
87		elnn1uz2 11232	. . . 4 |- ( ( ( # ` K ) gcd N ) e. NN <-> ( ( ( # ` K ) gcd N ) = 1 \/ ( ( # ` K ) gcd N ) e. ( ZZ>= ` 2 ) ) )
88	86, 87	sylib 200	. . 3 |- ( ph -> ( ( ( # ` K ) gcd N ) = 1 \/ ( ( # ` K ) gcd N ) e. ( ZZ>= ` 2 ) ) )
89	88	ord 379	. 2 |- ( ph -> ( -. ( ( # ` K ) gcd N ) = 1 -> ( ( # ` K ) gcd N ) e. ( ZZ>= ` 2 ) ) )
90	80, 89	mt3d 129	1 |- ( ph -> ( ( # ` K ) gcd N ) = 1 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   = wceq 1443   e. wcel 1886   =/= wne 2621   E.wrex 2737   {crab 2740   _Vcvv 3044   C_ wss 3403   class class class wbr 4401   ` cfv 5581  (class class class)co 6288   Fincfn 7566   0cc0 9536   1c1 9537   x. cmul 9541   NNcn 10606   2c2 10656   NN0cn0 10866   ZZcz 10934   ZZ>=cuz 11156   #chash 12512   || cdvds 14298   gcd cgcd 14461   Primecprime 14615   Basecbs 15114   ↾_s cress 15115   Grpcgrp 16662  SubGrpcsubg 16804   odcod 17158   Abelcabl 17424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-disj 4373  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-omul 7184  df-er 7360  df-ec 7362  df-qs 7366  df-map 7471  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-acn 8373  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-q 11262  df-rp 11300  df-fz 11782  df-fzo 11913  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-sum 13746  df-dvds 14299  df-gcd 14462  df-prm 14616  df-pc 14780  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-0g 15333  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-grp 16666  df-minusg 16667  df-sbg 16668  df-mulg 16669  df-subg 16807  df-eqg 16809  df-ga 16937  df-od 17165  df-cmn 17425  df-abl 17426

This theorem is referenced by:  ablfacrp2  17693