Theorem mulgmodid 13572

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 9767	. . . . . . 7 |- ( N e. ZZ -> N e. QQ )
2	1	adantr 276	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> N e. QQ )
3		nnq 9774	. . . . . . 7 |- ( M e. NN -> M e. QQ )
4	3	adantl 277	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> M e. QQ )
5		nngt0 9081	. . . . . . 7 |- ( M e. NN -> 0 < M )
6	5	adantl 277	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> 0 < M )
7		modqval 10491	. . . . . 6 |- ( ( N e. QQ /\ M e. QQ /\ 0 < M ) -> ( N mod M ) = ( N - ( M x. ( |_ ` ( N / M ) ) ) ) )
8	2, 4, 6, 7	syl3anc 1250	. . . . 5 |- ( ( N e. ZZ /\ M e. NN ) -> ( N mod M ) = ( N - ( M x. ( |_ ` ( N / M ) ) ) ) )
9	8	3ad2ant2 1022	. . . 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 5972	. . 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 9397	. . . . . . 7 |- ( N e. ZZ -> N e. CC )
12	11	adantr 276	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> N e. CC )
13		nnz 9411	. . . . . . . . 9 |- ( M e. NN -> M e. ZZ )
14	13	adantl 277	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN ) -> M e. ZZ )
15		znq 9765	. . . . . . . . 9 |- ( ( N e. ZZ /\ M e. NN ) -> ( N / M ) e. QQ )
16	15	flqcld 10442	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN ) -> ( |_ ` ( N / M ) ) e. ZZ )
17	14, 16	zmulcld 9521	. . . . . . 7 |- ( ( N e. ZZ /\ M e. NN ) -> ( M x. ( |_ ` ( N / M ) ) ) e. ZZ )
18	17	zcnd 9516	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> ( M x. ( |_ ` ( N / M ) ) ) e. CC )
19	12, 18	negsubd 8409	. . . . 5 |- ( ( N e. ZZ /\ M e. NN ) -> ( N + -u ( M x. ( |_ ` ( N / M ) ) ) ) = ( N - ( M x. ( |_ ` ( N / M ) ) ) ) )
20	19	3ad2ant2 1022	. . . 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 5972	. . 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 1000	. . . 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 1022	. . . 4 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> N e. ZZ )
25	14	3ad2ant2 1022	. . . . . 6 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> M e. ZZ )
26	16	3ad2ant2 1022	. . . . . 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 9521	. . . . 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 9517	. . . 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 1023	. . . 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 2206	. . . . 5 |- ( +g ` G ) = ( +g ` G )
34	31, 32, 33	mulgdir 13565	. . . 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 1252	. . 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 2245	. 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 9064	. . . . . . . 8 |- ( M e. NN -> M e. CC )
38	37	adantl 277	. . . . . . 7 |- ( ( N e. ZZ /\ M e. NN ) -> M e. CC )
39	16	zcnd 9516	. . . . . . 7 |- ( ( N e. ZZ /\ M e. NN ) -> ( |_ ` ( N / M ) ) e. CC )
40	38, 39	mulneg2d 8504	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> ( M x. -u ( |_ ` ( N / M ) ) ) = -u ( M x. ( |_ ` ( N / M ) ) ) )
41	40	3ad2ant2 1022	. . . . 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 5972	. . . 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 1022	. . . . . . . 8 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( N / M ) e. QQ )
44	43	flqcld 10442	. . . . . . 7 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( |_ ` ( N / M ) ) e. ZZ )
45	44	znegcld 9517	. . . . . 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 13571	. . . . . 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 1252	. . . . 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 5965	. . . . . . 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 1023	. . . . 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 13561	. . . . . 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 2243	. . . 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 2241	. . 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 5973	. 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 13550	. . . 4 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( N .x. X ) e. B )
59	57, 23, 29, 58	syl3an 1292	. . 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 13439	. . 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 2243	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 981   = wceq 1373   e. wcel 2177   class class class wbr 4051   ` cfv 5280  (class class class)co 5957   CCcc 7943   0cc0 7945   + caddc 7948   x. cmul 7950   < clt 8127   - cmin 8263   -ucneg 8264   / cdiv 8765   NNcn 9056   ZZcz 9392   QQcq 9760   |_cfl 10433   mod cmo 10489   Basecbs 12907   +g cplusg 12984   0gc0g 13163   Grpcgrp 13407  ._gcmg 13530

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063  ax-arch 8064

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-frec 6490  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766  df-inn 9057  df-2 9115  df-n0 9316  df-z 9393  df-uz 9669  df-q 9761  df-rp 9796  df-fz 10151  df-fzo 10285  df-fl 10435  df-mod 10490  df-seqfrec 10615  df-ndx 12910  df-slot 12911  df-base 12913  df-plusg 12997  df-0g 13165  df-mgm 13263  df-sgrp 13309  df-mnd 13324  df-grp 13410  df-minusg 13411  df-mulg 13531

This theorem is referenced by: (None)