Theorem odmod 17965

Description: Reduce the argument of a group multiple by modding out the order of the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 6-Sep-2015.)

Hypotheses

Ref	Expression
odcl.1	|- X = ( Base ` G )
odcl.2	|- O = ( od ` G )
odid.3	|- .x. = (.g ` G )
odid.4	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
odmod	|- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( N mod ( O ` A ) ) .x. A ) = ( N .x. A ) )

Proof of Theorem odmod

Step	Hyp	Ref	Expression
1		simpl3 1066	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> N e. ZZ )
2	1	zred 11482	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> N e. RR )
3		simpr 477	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( O ` A ) e. NN )
4	3	nnrpd 11870	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( O ` A ) e. RR+ )
5		modval 12670	. . . 4 |- ( ( N e. RR /\ ( O ` A ) e. RR+ ) -> ( N mod ( O ` A ) ) = ( N - ( ( O ` A ) x. ( |_ ` ( N / ( O ` A ) ) ) ) ) )
6	2, 4, 5	syl2anc 693	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( N mod ( O ` A ) ) = ( N - ( ( O ` A ) x. ( |_ ` ( N / ( O ` A ) ) ) ) ) )
7	6	oveq1d 6665	. 2 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( N mod ( O ` A ) ) .x. A ) = ( ( N - ( ( O ` A ) x. ( |_ ` ( N / ( O ` A ) ) ) ) ) .x. A ) )
8		simpl1 1064	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> G e. Grp )
9	3	nnzd 11481	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( O ` A ) e. ZZ )
10	2, 3	nndivred 11069	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( N / ( O ` A ) ) e. RR )
11	10	flcld 12599	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( |_ ` ( N / ( O ` A ) ) ) e. ZZ )
12	9, 11	zmulcld 11488	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( O ` A ) x. ( |_ ` ( N / ( O ` A ) ) ) ) e. ZZ )
13		simpl2 1065	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> A e. X )
14		odcl.1	. . . 4 |- X = ( Base ` G )
15		odid.3	. . . 4 |- .x. = (.g ` G )
16		eqid 2622	. . . 4 |- ( -g ` G ) = ( -g ` G )
17	14, 15, 16	mulgsubdir 17582	. . 3 |- ( ( G e. Grp /\ ( N e. ZZ /\ ( ( O ` A ) x. ( |_ ` ( N / ( O ` A ) ) ) ) e. ZZ /\ A e. X ) ) -> ( ( N - ( ( O ` A ) x. ( |_ ` ( N / ( O ` A ) ) ) ) ) .x. A ) = ( ( N .x. A ) ( -g ` G ) ( ( ( O ` A ) x. ( |_ ` ( N / ( O ` A ) ) ) ) .x. A ) ) )
18	8, 1, 12, 13, 17	syl13anc 1328	. 2 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( N - ( ( O ` A ) x. ( |_ ` ( N / ( O ` A ) ) ) ) ) .x. A ) = ( ( N .x. A ) ( -g ` G ) ( ( ( O ` A ) x. ( |_ ` ( N / ( O ` A ) ) ) ) .x. A ) ) )
19		nncn 11028	. . . . . . . 8 |- ( ( O ` A ) e. NN -> ( O ` A ) e. CC )
20		zcn 11382	. . . . . . . 8 |- ( ( |_ ` ( N / ( O ` A ) ) ) e. ZZ -> ( |_ ` ( N / ( O ` A ) ) ) e. CC )
21		mulcom 10022	. . . . . . . 8 |- ( ( ( O ` A ) e. CC /\ ( |_ ` ( N / ( O ` A ) ) ) e. CC ) -> ( ( O ` A ) x. ( |_ ` ( N / ( O ` A ) ) ) ) = ( ( |_ ` ( N / ( O ` A ) ) ) x. ( O ` A ) ) )
22	19, 20, 21	syl2an 494	. . . . . . 7 |- ( ( ( O ` A ) e. NN /\ ( |_ ` ( N / ( O ` A ) ) ) e. ZZ ) -> ( ( O ` A ) x. ( |_ ` ( N / ( O ` A ) ) ) ) = ( ( |_ ` ( N / ( O ` A ) ) ) x. ( O ` A ) ) )
23	3, 11, 22	syl2anc 693	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( O ` A ) x. ( |_ ` ( N / ( O ` A ) ) ) ) = ( ( |_ ` ( N / ( O ` A ) ) ) x. ( O ` A ) ) )
24	23	oveq1d 6665	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( ( O ` A ) x. ( |_ ` ( N / ( O ` A ) ) ) ) .x. A ) = ( ( ( |_ ` ( N / ( O ` A ) ) ) x. ( O ` A ) ) .x. A ) )
25	14, 15	mulgass 17579	. . . . . 6 |- ( ( G e. Grp /\ ( ( |_ ` ( N / ( O ` A ) ) ) e. ZZ /\ ( O ` A ) e. ZZ /\ A e. X ) ) -> ( ( ( |_ ` ( N / ( O ` A ) ) ) x. ( O ` A ) ) .x. A ) = ( ( |_ ` ( N / ( O ` A ) ) ) .x. ( ( O ` A ) .x. A ) ) )
26	8, 11, 9, 13, 25	syl13anc 1328	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( ( |_ ` ( N / ( O ` A ) ) ) x. ( O ` A ) ) .x. A ) = ( ( |_ ` ( N / ( O ` A ) ) ) .x. ( ( O ` A ) .x. A ) ) )
27		odcl.2	. . . . . . . . 9 |- O = ( od ` G )
28		odid.4	. . . . . . . . 9 |- .0. = ( 0g ` G )
29	14, 27, 15, 28	odid 17957	. . . . . . . 8 |- ( A e. X -> ( ( O ` A ) .x. A ) = .0. )
30	13, 29	syl 17	. . . . . . 7 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( O ` A ) .x. A ) = .0. )
31	30	oveq2d 6666	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( |_ ` ( N / ( O ` A ) ) ) .x. ( ( O ` A ) .x. A ) ) = ( ( |_ ` ( N / ( O ` A ) ) ) .x. .0. ) )
32	14, 15, 28	mulgz 17568	. . . . . . 7 |- ( ( G e. Grp /\ ( |_ ` ( N / ( O ` A ) ) ) e. ZZ ) -> ( ( |_ ` ( N / ( O ` A ) ) ) .x. .0. ) = .0. )
33	8, 11, 32	syl2anc 693	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( |_ ` ( N / ( O ` A ) ) ) .x. .0. ) = .0. )
34	31, 33	eqtrd 2656	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( |_ ` ( N / ( O ` A ) ) ) .x. ( ( O ` A ) .x. A ) ) = .0. )
35	24, 26, 34	3eqtrd 2660	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( ( O ` A ) x. ( |_ ` ( N / ( O ` A ) ) ) ) .x. A ) = .0. )
36	35	oveq2d 6666	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( N .x. A ) ( -g ` G ) ( ( ( O ` A ) x. ( |_ ` ( N / ( O ` A ) ) ) ) .x. A ) ) = ( ( N .x. A ) ( -g ` G ) .0. ) )
37	14, 15	mulgcl 17559	. . . . 5 |- ( ( G e. Grp /\ N e. ZZ /\ A e. X ) -> ( N .x. A ) e. X )
38	8, 1, 13, 37	syl3anc 1326	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( N .x. A ) e. X )
39	14, 28, 16	grpsubid1 17500	. . . 4 |- ( ( G e. Grp /\ ( N .x. A ) e. X ) -> ( ( N .x. A ) ( -g ` G ) .0. ) = ( N .x. A ) )
40	8, 38, 39	syl2anc 693	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( N .x. A ) ( -g ` G ) .0. ) = ( N .x. A ) )
41	36, 40	eqtrd 2656	. 2 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( N .x. A ) ( -g ` G ) ( ( ( O ` A ) x. ( |_ ` ( N / ( O ` A ) ) ) ) .x. A ) ) = ( N .x. A ) )
42	7, 18, 41	3eqtrd 2660	1 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( N mod ( O ` A ) ) .x. A ) = ( N .x. A ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   x. cmul 9941   - cmin 10266   / cdiv 10684   NNcn 11020   ZZcz 11377   RR+crp 11832   |_cfl 12591   mod cmo 12668   Basecbs 15857   0gc0g 16100   Grpcgrp 17422   -gcsg 17424  ._gcmg 17540   odcod 17944

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-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-iun 4522  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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-od 17948

This theorem is referenced by:  oddvds  17966  odf1o2  17988