Theorem mulgmodid 17581

Description: Casting out multiples of the identity element leaves the group multiple unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)

Hypotheses

Ref	Expression
mulgmodid.b	|- B = ( Base ` G )
mulgmodid.o	|- .0. = ( 0g ` G )
mulgmodid.t	|- .x. = (.g ` G )

Assertion

Ref	Expression
mulgmodid	|- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( ( N mod M ) .x. X ) = ( N .x. X ) )

Proof of Theorem mulgmodid

Step	Hyp	Ref	Expression
1		zre 11381	. . . . . 6 |- ( N e. ZZ -> N e. RR )
2		nnrp 11842	. . . . . 6 |- ( M e. NN -> M e. RR+ )
3		modval 12670	. . . . . 6 |- ( ( N e. RR /\ M e. RR+ ) -> ( N mod M ) = ( N - ( M x. ( |_ ` ( N / M ) ) ) ) )
4	1, 2, 3	syl2an 494	. . . . 5 |- ( ( N e. ZZ /\ M e. NN ) -> ( N mod M ) = ( N - ( M x. ( |_ ` ( N / M ) ) ) ) )
5	4	3ad2ant2 1083	. . . 4 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( N mod M ) = ( N - ( M x. ( |_ ` ( N / M ) ) ) ) )
6	5	oveq1d 6665	. . 3 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( ( N mod M ) .x. X ) = ( ( N - ( M x. ( |_ ` ( N / M ) ) ) ) .x. X ) )
7		zcn 11382	. . . . . . 7 |- ( N e. ZZ -> N e. CC )
8	7	adantr 481	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> N e. CC )
9		nnz 11399	. . . . . . . . 9 |- ( M e. NN -> M e. ZZ )
10	9	adantl 482	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN ) -> M e. ZZ )
11		nnre 11027	. . . . . . . . . . 11 |- ( M e. NN -> M e. RR )
12		nnne0 11053	. . . . . . . . . . 11 |- ( M e. NN -> M =/= 0 )
13		redivcl 10744	. . . . . . . . . . 11 |- ( ( N e. RR /\ M e. RR /\ M =/= 0 ) -> ( N / M ) e. RR )
14	1, 11, 12, 13	syl3an 1368	. . . . . . . . . 10 |- ( ( N e. ZZ /\ M e. NN /\ M e. NN ) -> ( N / M ) e. RR )
15	14	3anidm23 1385	. . . . . . . . 9 |- ( ( N e. ZZ /\ M e. NN ) -> ( N / M ) e. RR )
16	15	flcld 12599	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN ) -> ( |_ ` ( N / M ) ) e. ZZ )
17	10, 16	zmulcld 11488	. . . . . . 7 |- ( ( N e. ZZ /\ M e. NN ) -> ( M x. ( |_ ` ( N / M ) ) ) e. ZZ )
18	17	zcnd 11483	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> ( M x. ( |_ ` ( N / M ) ) ) e. CC )
19	8, 18	negsubd 10398	. . . . 5 |- ( ( N e. ZZ /\ M e. NN ) -> ( N + -u ( M x. ( |_ ` ( N / M ) ) ) ) = ( N - ( M x. ( |_ ` ( N / M ) ) ) ) )
20	19	3ad2ant2 1083	. . . 4 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( N + -u ( M x. ( |_ ` ( N / M ) ) ) ) = ( N - ( M x. ( |_ ` ( N / M ) ) ) ) )
21	20	oveq1d 6665	. . 3 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( ( N + -u ( M x. ( |_ ` ( N / M ) ) ) ) .x. X ) = ( ( N - ( M x. ( |_ ` ( N / M ) ) ) ) .x. X ) )
22		simp1 1061	. . . 4 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> G e. Grp )
23		simpl 473	. . . . 5 |- ( ( N e. ZZ /\ M e. NN ) -> N e. ZZ )
24	23	3ad2ant2 1083	. . . 4 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> N e. ZZ )
25	10	3ad2ant2 1083	. . . . . 6 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> M e. ZZ )
26	16	3ad2ant2 1083	. . . . . 6 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( |_ ` ( N / M ) ) e. ZZ )
27	25, 26	zmulcld 11488	. . . . 5 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( M x. ( |_ ` ( N / M ) ) ) e. ZZ )
28	27	znegcld 11484	. . . 4 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> -u ( M x. ( |_ ` ( N / M ) ) ) e. ZZ )
29		simpl 473	. . . . 5 |- ( ( X e. B /\ ( M .x. X ) = .0. ) -> X e. B )
30	29	3ad2ant3 1084	. . . 4 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> X e. B )
31		mulgmodid.b	. . . . 5 |- B = ( Base ` G )
32		mulgmodid.t	. . . . 5 |- .x. = (.g ` G )
33		eqid 2622	. . . . 5 |- ( +g ` G ) = ( +g ` G )
34	31, 32, 33	mulgdir 17573	. . . 4 |- ( ( G e. Grp /\ ( N e. ZZ /\ -u ( M x. ( |_ ` ( N / M ) ) ) e. ZZ /\ X e. B ) ) -> ( ( N + -u ( M x. ( |_ ` ( N / M ) ) ) ) .x. X ) = ( ( N .x. X ) ( +g ` G ) ( -u ( M x. ( |_ ` ( N / M ) ) ) .x. X ) ) )
35	22, 24, 28, 30, 34	syl13anc 1328	. . 3 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( ( N + -u ( M x. ( |_ ` ( N / M ) ) ) ) .x. X ) = ( ( N .x. X ) ( +g ` G ) ( -u ( M x. ( |_ ` ( N / M ) ) ) .x. X ) ) )
36	6, 21, 35	3eqtr2d 2662	. 2 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( ( N mod M ) .x. X ) = ( ( N .x. X ) ( +g ` G ) ( -u ( M x. ( |_ ` ( N / M ) ) ) .x. X ) ) )
37		nncn 11028	. . . . . . . 8 |- ( M e. NN -> M e. CC )
38	37	adantl 482	. . . . . . 7 |- ( ( N e. ZZ /\ M e. NN ) -> M e. CC )
39	16	zcnd 11483	. . . . . . 7 |- ( ( N e. ZZ /\ M e. NN ) -> ( |_ ` ( N / M ) ) e. CC )
40	38, 39	mulneg2d 10484	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> ( M x. -u ( |_ ` ( N / M ) ) ) = -u ( M x. ( |_ ` ( N / M ) ) ) )
41	40	3ad2ant2 1083	. . . . 5 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( M x. -u ( |_ ` ( N / M ) ) ) = -u ( M x. ( |_ ` ( N / M ) ) ) )
42	41	oveq1d 6665	. . . 4 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( ( M x. -u ( |_ ` ( N / M ) ) ) .x. X ) = ( -u ( M x. ( |_ ` ( N / M ) ) ) .x. X ) )
43	15	3ad2ant2 1083	. . . . . . . 8 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( N / M ) e. RR )
44	43	flcld 12599	. . . . . . 7 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( |_ ` ( N / M ) ) e. ZZ )
45	44	znegcld 11484	. . . . . 6 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> -u ( |_ ` ( N / M ) ) e. ZZ )
46	31, 32	mulgassr 17580	. . . . . 6 |- ( ( G e. Grp /\ ( -u ( |_ ` ( N / M ) ) e. ZZ /\ M e. ZZ /\ X e. B ) ) -> ( ( M x. -u ( |_ ` ( N / M ) ) ) .x. X ) = ( -u ( |_ ` ( N / M ) ) .x. ( M .x. X ) ) )
47	22, 45, 25, 30, 46	syl13anc 1328	. . . . 5 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( ( M x. -u ( |_ ` ( N / M ) ) ) .x. X ) = ( -u ( |_ ` ( N / M ) ) .x. ( M .x. X ) ) )
48		oveq2 6658	. . . . . . 7 |- ( ( M .x. X ) = .0. -> ( -u ( |_ ` ( N / M ) ) .x. ( M .x. X ) ) = ( -u ( |_ ` ( N / M ) ) .x. .0. ) )
49	48	adantl 482	. . . . . 6 |- ( ( X e. B /\ ( M .x. X ) = .0. ) -> ( -u ( |_ ` ( N / M ) ) .x. ( M .x. X ) ) = ( -u ( |_ ` ( N / M ) ) .x. .0. ) )
50	49	3ad2ant3 1084	. . . . 5 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( -u ( |_ ` ( N / M ) ) .x. ( M .x. X ) ) = ( -u ( |_ ` ( N / M ) ) .x. .0. ) )
51		mulgmodid.o	. . . . . . 7 |- .0. = ( 0g ` G )
52	31, 32, 51	mulgz 17568	. . . . . 6 |- ( ( G e. Grp /\ -u ( |_ ` ( N / M ) ) e. ZZ ) -> ( -u ( |_ ` ( N / M ) ) .x. .0. ) = .0. )
53	22, 45, 52	syl2anc 693	. . . . 5 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( -u ( |_ ` ( N / M ) ) .x. .0. ) = .0. )
54	47, 50, 53	3eqtrd 2660	. . . 4 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( ( M x. -u ( |_ ` ( N / M ) ) ) .x. X ) = .0. )
55	42, 54	eqtr3d 2658	. . 3 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( -u ( M x. ( |_ ` ( N / M ) ) ) .x. X ) = .0. )
56	55	oveq2d 6666	. 2 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( ( N .x. X ) ( +g ` G ) ( -u ( M x. ( |_ ` ( N / M ) ) ) .x. X ) ) = ( ( N .x. X ) ( +g ` G ) .0. ) )
57		id 22	. . . 4 |- ( G e. Grp -> G e. Grp )
58	31, 32	mulgcl 17559	. . . 4 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( N .x. X ) e. B )
59	57, 23, 29, 58	syl3an 1368	. . 3 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( N .x. X ) e. B )
60	31, 33, 51	grprid 17453	. . 3 |- ( ( G e. Grp /\ ( N .x. X ) e. B ) -> ( ( N .x. X ) ( +g ` G ) .0. ) = ( N .x. X ) )
61	22, 59, 60	syl2anc 693	. 2 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( ( N .x. X ) ( +g ` G ) .0. ) = ( N .x. X ) )
62	36, 56, 61	3eqtrd 2660	1 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( ( N mod M ) .x. X ) = ( N .x. X ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   ZZcz 11377   RR+crp 11832   |_cfl 12591   mod cmo 12668   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422  ._gcmg 17540

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-mulg 17541

This theorem is referenced by: (None)