Theorem mulgmodid 13747

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		zq 9859	. . . . . . 7 |- ( N e. ZZ -> N e. QQ )
2	1	adantr 276	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> N e. QQ )
3		nnq 9866	. . . . . . 7 |- ( M e. NN -> M e. QQ )
4	3	adantl 277	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> M e. QQ )
5		nngt0 9167	. . . . . . 7 |- ( M e. NN -> 0 < M )
6	5	adantl 277	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> 0 < M )
7		modqval 10585	. . . . . 6 |- ( ( N e. QQ /\ M e. QQ /\ 0 < M ) -> ( N mod M ) = ( N - ( M x. ( |_ ` ( N / M ) ) ) ) )
8	2, 4, 6, 7	syl3anc 1273	. . . . 5 |- ( ( N e. ZZ /\ M e. NN ) -> ( N mod M ) = ( N - ( M x. ( |_ ` ( N / M ) ) ) ) )
9	8	3ad2ant2 1045	. . . 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 ) ) ) ) )
10	9	oveq1d 6032	. . 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 ) )
11		zcn 9483	. . . . . . 7 |- ( N e. ZZ -> N e. CC )
12	11	adantr 276	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> N e. CC )
13		nnz 9497	. . . . . . . . 9 |- ( M e. NN -> M e. ZZ )
14	13	adantl 277	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN ) -> M e. ZZ )
15		znq 9857	. . . . . . . . 9 |- ( ( N e. ZZ /\ M e. NN ) -> ( N / M ) e. QQ )
16	15	flqcld 10536	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN ) -> ( |_ ` ( N / M ) ) e. ZZ )
17	14, 16	zmulcld 9607	. . . . . . 7 |- ( ( N e. ZZ /\ M e. NN ) -> ( M x. ( |_ ` ( N / M ) ) ) e. ZZ )
18	17	zcnd 9602	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> ( M x. ( |_ ` ( N / M ) ) ) e. CC )
19	12, 18	negsubd 8495	. . . . 5 |- ( ( N e. ZZ /\ M e. NN ) -> ( N + -u ( M x. ( |_ ` ( N / M ) ) ) ) = ( N - ( M x. ( |_ ` ( N / M ) ) ) ) )
20	19	3ad2ant2 1045	. . . 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 6032	. . 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 1023	. . . 4 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> G e. Grp )
23		simpl 109	. . . . 5 |- ( ( N e. ZZ /\ M e. NN ) -> N e. ZZ )
24	23	3ad2ant2 1045	. . . 4 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> N e. ZZ )
25	14	3ad2ant2 1045	. . . . . 6 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> M e. ZZ )
26	16	3ad2ant2 1045	. . . . . 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 9607	. . . . 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 9603	. . . 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 109	. . . . 5 |- ( ( X e. B /\ ( M .x. X ) = .0. ) -> X e. B )
30	29	3ad2ant3 1046	. . . 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 2231	. . . . 5 |- ( +g ` G ) = ( +g ` G )
34	31, 32, 33	mulgdir 13740	. . . 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 1275	. . 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	10, 21, 35	3eqtr2d 2270	. 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 9150	. . . . . . . 8 |- ( M e. NN -> M e. CC )
38	37	adantl 277	. . . . . . 7 |- ( ( N e. ZZ /\ M e. NN ) -> M e. CC )
39	16	zcnd 9602	. . . . . . 7 |- ( ( N e. ZZ /\ M e. NN ) -> ( |_ ` ( N / M ) ) e. CC )
40	38, 39	mulneg2d 8590	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> ( M x. -u ( |_ ` ( N / M ) ) ) = -u ( M x. ( |_ ` ( N / M ) ) ) )
41	40	3ad2ant2 1045	. . . . 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 6032	. . . 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 1045	. . . . . . . 8 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( N / M ) e. QQ )
44	43	flqcld 10536	. . . . . . 7 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( |_ ` ( N / M ) ) e. ZZ )
45	44	znegcld 9603	. . . . . 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 13746	. . . . . 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 1275	. . . . 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 6025	. . . . . . 7 |- ( ( M .x. X ) = .0. -> ( -u ( |_ ` ( N / M ) ) .x. ( M .x. X ) ) = ( -u ( |_ ` ( N / M ) ) .x. .0. ) )
49	48	adantl 277	. . . . . 6 |- ( ( X e. B /\ ( M .x. X ) = .0. ) -> ( -u ( |_ ` ( N / M ) ) .x. ( M .x. X ) ) = ( -u ( |_ ` ( N / M ) ) .x. .0. ) )
50	49	3ad2ant3 1046	. . . . 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 13736	. . . . . 6 |- ( ( G e. Grp /\ -u ( |_ ` ( N / M ) ) e. ZZ ) -> ( -u ( |_ ` ( N / M ) ) .x. .0. ) = .0. )
53	22, 45, 52	syl2anc 411	. . . . 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 2268	. . . 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 2266	. . 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 6033	. 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 19	. . . 4 |- ( G e. Grp -> G e. Grp )
58	31, 32	mulgcl 13725	. . . 4 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( N .x. X ) e. B )
59	57, 23, 29, 58	syl3an 1315	. . 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 13614	. . 3 |- ( ( G e. Grp /\ ( N .x. X ) e. B ) -> ( ( N .x. X ) ( +g ` G ) .0. ) = ( N .x. X ) )
61	22, 59, 60	syl2anc 411	. 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 2268	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 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   CCcc 8029   0cc0 8031   + caddc 8034   x. cmul 8036   < clt 8213   - cmin 8349   -ucneg 8350   / cdiv 8851   NNcn 9142   ZZcz 9478   QQcq 9852   |_cfl 10527   mod cmo 10583   Basecbs 13081   +g cplusg 13159   0gc0g 13338   Grpcgrp 13582  ._gcmg 13705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-mulg 13706

This theorem is referenced by: (None)