Theorem ablfacrp2 18466

Description: The factors K , L of ablfacrp 18465 have the expected orders (which allows for repeated application to decompose G into subgroups of prime-power order). Lemma 6.1C.2 of [Shapiro], p. 199. (Contributed by Mario Carneiro, 21-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
ablfacrp2	|- ( ph -> ( ( # ` K ) = M /\ ( # ` L ) = N ) )

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 ablfacrp2

Step	Hyp	Ref	Expression
1		ablfacrp.2	. . . . . . 7 |- ( ph -> ( # ` B ) = ( M x. N ) )
2		ablfacrp.m	. . . . . . . . 9 |- ( ph -> M e. NN )
3	2	nnnn0d 11351	. . . . . . . 8 |- ( ph -> M e. NN0 )
4		ablfacrp.n	. . . . . . . . 9 |- ( ph -> N e. NN )
5	4	nnnn0d 11351	. . . . . . . 8 |- ( ph -> N e. NN0 )
6	3, 5	nn0mulcld 11356	. . . . . . 7 |- ( ph -> ( M x. N ) e. NN0 )
7	1, 6	eqeltrd 2701	. . . . . 6 |- ( ph -> ( # ` B ) e. NN0 )
8		ablfacrp.b	. . . . . . . 8 |- B = ( Base ` G )
9		fvex 6201	. . . . . . . 8 |- ( Base ` G ) e. _V
10	8, 9	eqeltri 2697	. . . . . . 7 |- B e. _V
11		hashclb 13149	. . . . . . 7 |- ( B e. _V -> ( B e. Fin <-> ( # ` B ) e. NN0 ) )
12	10, 11	ax-mp 5	. . . . . 6 |- ( B e. Fin <-> ( # ` B ) e. NN0 )
13	7, 12	sylibr 224	. . . . 5 |- ( ph -> B e. Fin )
14		ablfacrp.k	. . . . . 6 |- K = { x e. B | ( O ` x ) || M }
15		ssrab2 3687	. . . . . 6 |- { x e. B | ( O ` x ) || M } C_ B
16	14, 15	eqsstri 3635	. . . . 5 |- K C_ B
17		ssfi 8180	. . . . 5 |- ( ( B e. Fin /\ K C_ B ) -> K e. Fin )
18	13, 16, 17	sylancl 694	. . . 4 |- ( ph -> K e. Fin )
19		hashcl 13147	. . . 4 |- ( K e. Fin -> ( # ` K ) e. NN0 )
20	18, 19	syl 17	. . 3 |- ( ph -> ( # ` K ) e. NN0 )
21		ablfacrp.g	. . . . . . . 8 |- ( ph -> G e. Abel )
22	2	nnzd 11481	. . . . . . . 8 |- ( ph -> M e. ZZ )
23		ablfacrp.o	. . . . . . . . 9 |- O = ( od ` G )
24	23, 8	oddvdssubg 18258	. . . . . . . 8 |- ( ( G e. Abel /\ M e. ZZ ) -> { x e. B | ( O ` x ) || M } e. (SubGrp ` G ) )
25	21, 22, 24	syl2anc 693	. . . . . . 7 |- ( ph -> { x e. B | ( O ` x ) || M } e. (SubGrp ` G ) )
26	14, 25	syl5eqel 2705	. . . . . 6 |- ( ph -> K e. (SubGrp ` G ) )
27	8	lagsubg 17656	. . . . . 6 |- ( ( K e. (SubGrp ` G ) /\ B e. Fin ) -> ( # ` K ) || ( # ` B ) )
28	26, 13, 27	syl2anc 693	. . . . 5 |- ( ph -> ( # ` K ) || ( # ` B ) )
29	2	nncnd 11036	. . . . . . 7 |- ( ph -> M e. CC )
30	4	nncnd 11036	. . . . . . 7 |- ( ph -> N e. CC )
31	29, 30	mulcomd 10061	. . . . . 6 |- ( ph -> ( M x. N ) = ( N x. M ) )
32	1, 31	eqtrd 2656	. . . . 5 |- ( ph -> ( # ` B ) = ( N x. M ) )
33	28, 32	breqtrd 4679	. . . 4 |- ( ph -> ( # ` K ) || ( N x. M ) )
34		ablfacrp.l	. . . . 5 |- L = { x e. B | ( O ` x ) || N }
35		ablfacrp.1	. . . . 5 |- ( ph -> ( M gcd N ) = 1 )
36	8, 23, 14, 34, 21, 2, 4, 35, 1	ablfacrplem 18464	. . . 4 |- ( ph -> ( ( # ` K ) gcd N ) = 1 )
37	20	nn0zd 11480	. . . . 5 |- ( ph -> ( # ` K ) e. ZZ )
38	4	nnzd 11481	. . . . 5 |- ( ph -> N e. ZZ )
39		coprmdvds 15366	. . . . 5 |- ( ( ( # ` K ) e. ZZ /\ N e. ZZ /\ M e. ZZ ) -> ( ( ( # ` K ) || ( N x. M ) /\ ( ( # ` K ) gcd N ) = 1 ) -> ( # ` K ) || M ) )
40	37, 38, 22, 39	syl3anc 1326	. . . 4 |- ( ph -> ( ( ( # ` K ) || ( N x. M ) /\ ( ( # ` K ) gcd N ) = 1 ) -> ( # ` K ) || M ) )
41	33, 36, 40	mp2and 715	. . 3 |- ( ph -> ( # ` K ) || M )
42	23, 8	oddvdssubg 18258	. . . . . . . . . . 11 |- ( ( G e. Abel /\ N e. ZZ ) -> { x e. B | ( O ` x ) || N } e. (SubGrp ` G ) )
43	21, 38, 42	syl2anc 693	. . . . . . . . . 10 |- ( ph -> { x e. B | ( O ` x ) || N } e. (SubGrp ` G ) )
44	34, 43	syl5eqel 2705	. . . . . . . . 9 |- ( ph -> L e. (SubGrp ` G ) )
45	8	lagsubg 17656	. . . . . . . . 9 |- ( ( L e. (SubGrp ` G ) /\ B e. Fin ) -> ( # ` L ) || ( # ` B ) )
46	44, 13, 45	syl2anc 693	. . . . . . . 8 |- ( ph -> ( # ` L ) || ( # ` B ) )
47	46, 1	breqtrd 4679	. . . . . . 7 |- ( ph -> ( # ` L ) || ( M x. N ) )
48		gcdcom 15235	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( N gcd M ) )
49	22, 38, 48	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( M gcd N ) = ( N gcd M ) )
50	49, 35	eqtr3d 2658	. . . . . . . 8 |- ( ph -> ( N gcd M ) = 1 )
51	8, 23, 34, 14, 21, 4, 2, 50, 32	ablfacrplem 18464	. . . . . . 7 |- ( ph -> ( ( # ` L ) gcd M ) = 1 )
52		ssrab2 3687	. . . . . . . . . . . 12 |- { x e. B | ( O ` x ) || N } C_ B
53	34, 52	eqsstri 3635	. . . . . . . . . . 11 |- L C_ B
54		ssfi 8180	. . . . . . . . . . 11 |- ( ( B e. Fin /\ L C_ B ) -> L e. Fin )
55	13, 53, 54	sylancl 694	. . . . . . . . . 10 |- ( ph -> L e. Fin )
56		hashcl 13147	. . . . . . . . . 10 |- ( L e. Fin -> ( # ` L ) e. NN0 )
57	55, 56	syl 17	. . . . . . . . 9 |- ( ph -> ( # ` L ) e. NN0 )
58	57	nn0zd 11480	. . . . . . . 8 |- ( ph -> ( # ` L ) e. ZZ )
59		coprmdvds 15366	. . . . . . . 8 |- ( ( ( # ` L ) e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( # ` L ) || ( M x. N ) /\ ( ( # ` L ) gcd M ) = 1 ) -> ( # ` L ) || N ) )
60	58, 22, 38, 59	syl3anc 1326	. . . . . . 7 |- ( ph -> ( ( ( # ` L ) || ( M x. N ) /\ ( ( # ` L ) gcd M ) = 1 ) -> ( # ` L ) || N ) )
61	47, 51, 60	mp2and 715	. . . . . 6 |- ( ph -> ( # ` L ) || N )
62		dvdscmul 15008	. . . . . . 7 |- ( ( ( # ` L ) e. ZZ /\ N e. ZZ /\ M e. ZZ ) -> ( ( # ` L ) || N -> ( M x. ( # ` L ) ) || ( M x. N ) ) )
63	58, 38, 22, 62	syl3anc 1326	. . . . . 6 |- ( ph -> ( ( # ` L ) || N -> ( M x. ( # ` L ) ) || ( M x. N ) ) )
64	61, 63	mpd 15	. . . . 5 |- ( ph -> ( M x. ( # ` L ) ) || ( M x. N ) )
65		eqid 2622	. . . . . . . . . 10 |- ( 0g ` G ) = ( 0g ` G )
66		eqid 2622	. . . . . . . . . 10 |- ( LSSum ` G ) = ( LSSum ` G )
67	8, 23, 14, 34, 21, 2, 4, 35, 1, 65, 66	ablfacrp 18465	. . . . . . . . 9 |- ( ph -> ( ( K i^i L ) = { ( 0g ` G ) } /\ ( K ( LSSum ` G ) L ) = B ) )
68	67	simprd 479	. . . . . . . 8 |- ( ph -> ( K ( LSSum ` G ) L ) = B )
69	68	fveq2d 6195	. . . . . . 7 |- ( ph -> ( # ` ( K ( LSSum ` G ) L ) ) = ( # ` B ) )
70		eqid 2622	. . . . . . . 8 |- (Cntz ` G ) = (Cntz ` G )
71	67	simpld 475	. . . . . . . 8 |- ( ph -> ( K i^i L ) = { ( 0g ` G ) } )
72	70, 21, 26, 44	ablcntzd 18260	. . . . . . . 8 |- ( ph -> K C_ ( (Cntz ` G ) ` L ) )
73	66, 65, 70, 26, 44, 71, 72, 18, 55	lsmhash 18118	. . . . . . 7 |- ( ph -> ( # ` ( K ( LSSum ` G ) L ) ) = ( ( # ` K ) x. ( # ` L ) ) )
74	69, 73	eqtr3d 2658	. . . . . 6 |- ( ph -> ( # ` B ) = ( ( # ` K ) x. ( # ` L ) ) )
75	74, 1	eqtr3d 2658	. . . . 5 |- ( ph -> ( ( # ` K ) x. ( # ` L ) ) = ( M x. N ) )
76	64, 75	breqtrrd 4681	. . . 4 |- ( ph -> ( M x. ( # ` L ) ) || ( ( # ` K ) x. ( # ` L ) ) )
77	65	subg0cl 17602	. . . . . . . 8 |- ( L e. (SubGrp ` G ) -> ( 0g ` G ) e. L )
78		ne0i 3921	. . . . . . . 8 |- ( ( 0g ` G ) e. L -> L =/= (/) )
79	44, 77, 78	3syl 18	. . . . . . 7 |- ( ph -> L =/= (/) )
80		hashnncl 13157	. . . . . . . 8 |- ( L e. Fin -> ( ( # ` L ) e. NN <-> L =/= (/) ) )
81	55, 80	syl 17	. . . . . . 7 |- ( ph -> ( ( # ` L ) e. NN <-> L =/= (/) ) )
82	79, 81	mpbird 247	. . . . . 6 |- ( ph -> ( # ` L ) e. NN )
83	82	nnne0d 11065	. . . . 5 |- ( ph -> ( # ` L ) =/= 0 )
84		dvdsmulcr 15011	. . . . 5 |- ( ( M e. ZZ /\ ( # ` K ) e. ZZ /\ ( ( # ` L ) e. ZZ /\ ( # ` L ) =/= 0 ) ) -> ( ( M x. ( # ` L ) ) || ( ( # ` K ) x. ( # ` L ) ) <-> M || ( # ` K ) ) )
85	22, 37, 58, 83, 84	syl112anc 1330	. . . 4 |- ( ph -> ( ( M x. ( # ` L ) ) || ( ( # ` K ) x. ( # ` L ) ) <-> M || ( # ` K ) ) )
86	76, 85	mpbid 222	. . 3 |- ( ph -> M || ( # ` K ) )
87		dvdseq 15036	. . 3 |- ( ( ( ( # ` K ) e. NN0 /\ M e. NN0 ) /\ ( ( # ` K ) || M /\ M || ( # ` K ) ) ) -> ( # ` K ) = M )
88	20, 3, 41, 86, 87	syl22anc 1327	. 2 |- ( ph -> ( # ` K ) = M )
89		dvdsmulc 15009	. . . . . . 7 |- ( ( ( # ` K ) e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( # ` K ) || M -> ( ( # ` K ) x. N ) || ( M x. N ) ) )
90	37, 22, 38, 89	syl3anc 1326	. . . . . 6 |- ( ph -> ( ( # ` K ) || M -> ( ( # ` K ) x. N ) || ( M x. N ) ) )
91	41, 90	mpd 15	. . . . 5 |- ( ph -> ( ( # ` K ) x. N ) || ( M x. N ) )
92	91, 75	breqtrrd 4681	. . . 4 |- ( ph -> ( ( # ` K ) x. N ) || ( ( # ` K ) x. ( # ` L ) ) )
93	88, 2	eqeltrd 2701	. . . . . 6 |- ( ph -> ( # ` K ) e. NN )
94	93	nnne0d 11065	. . . . 5 |- ( ph -> ( # ` K ) =/= 0 )
95		dvdscmulr 15010	. . . . 5 |- ( ( N e. ZZ /\ ( # ` L ) e. ZZ /\ ( ( # ` K ) e. ZZ /\ ( # ` K ) =/= 0 ) ) -> ( ( ( # ` K ) x. N ) || ( ( # ` K ) x. ( # ` L ) ) <-> N || ( # ` L ) ) )
96	38, 58, 37, 94, 95	syl112anc 1330	. . . 4 |- ( ph -> ( ( ( # ` K ) x. N ) || ( ( # ` K ) x. ( # ` L ) ) <-> N || ( # ` L ) ) )
97	92, 96	mpbid 222	. . 3 |- ( ph -> N || ( # ` L ) )
98		dvdseq 15036	. . 3 |- ( ( ( ( # ` L ) e. NN0 /\ N e. NN0 ) /\ ( ( # ` L ) || N /\ N || ( # ` L ) ) ) -> ( # ` L ) = N )
99	57, 5, 61, 97, 98	syl22anc 1327	. 2 |- ( ph -> ( # ` L ) = N )
100	88, 99	jca 554	1 |- ( ph -> ( ( # ` K ) = M /\ ( # ` L ) = N ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   1c1 9937   x. cmul 9941   NNcn 11020   NN0cn0 11292   ZZcz 11377   #chash 13117   || cdvds 14983   gcd cgcd 15216   Basecbs 15857   0gc0g 16100  SubGrpcsubg 17588  Cntzccntz 17748   odcod 17944   LSSumclsm 18049   Abelcabl 18194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-eqg 17593  df-ga 17723  df-cntz 17750  df-od 17948  df-lsm 18051  df-pj1 18052  df-cmn 18195  df-abl 18196

This theorem is referenced by:  ablfac1a  18468