Theorem mulgmodid 13941

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 10005	. . . . . . 7 |- ( N e. ZZ -> N e. QQ )
2	1	adantr 276	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> N e. QQ )
3		nnq 10012	. . . . . . 7 |- ( M e. NN -> M e. QQ )
4	3	adantl 277	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> M e. QQ )
5		nngt0 9308	. . . . . . 7 |- ( M e. NN -> 0 < M )
6	5	adantl 277	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> 0 < M )
7		modqval 10739	. . . . . 6 |- ( ( N e. QQ /\ M e. QQ /\ 0 < M ) -> ( N mod M ) = ( N - ( M x. ( |_ ` ( N / M ) ) ) ) )
8	2, 4, 6, 7	syl3anc 1278	. . . . 5 |- ( ( N e. ZZ /\ M e. NN ) -> ( N mod M ) = ( N - ( M x. ( |_ ` ( N / M ) ) ) ) )
9	8	3ad2ant2 1050	. . . 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 6090	. . 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 9628	. . . . . . 7 |- ( N e. ZZ -> N e. CC )
12	11	adantr 276	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> N e. CC )
13		nnz 9642	. . . . . . . . 9 |- ( M e. NN -> M e. ZZ )
14	13	adantl 277	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN ) -> M e. ZZ )
15		znq 10003	. . . . . . . . 9 |- ( ( N e. ZZ /\ M e. NN ) -> ( N / M ) e. QQ )
16	15	flqcld 10690	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN ) -> ( |_ ` ( N / M ) ) e. ZZ )
17	14, 16	zmulcld 9753	. . . . . . 7 |- ( ( N e. ZZ /\ M e. NN ) -> ( M x. ( |_ ` ( N / M ) ) ) e. ZZ )
18	17	zcnd 9748	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> ( M x. ( |_ ` ( N / M ) ) ) e. CC )
19	12, 18	negsubd 8633	. . . . 5 |- ( ( N e. ZZ /\ M e. NN ) -> ( N + -u ( M x. ( |_ ` ( N / M ) ) ) ) = ( N - ( M x. ( |_ ` ( N / M ) ) ) ) )
20	19	3ad2ant2 1050	. . . 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 6090	. . 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 1028	. . . 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 1050	. . . 4 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> N e. ZZ )
25	14	3ad2ant2 1050	. . . . . 6 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> M e. ZZ )
26	16	3ad2ant2 1050	. . . . . 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 9753	. . . . 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 9749	. . . 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 1051	. . . 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 2238	. . . . 5 |- ( +g ` G ) = ( +g ` G )
34	31, 32, 33	mulgdir 13934	. . . 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 1280	. . 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 2277	. 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 9291	. . . . . . . 8 |- ( M e. NN -> M e. CC )
38	37	adantl 277	. . . . . . 7 |- ( ( N e. ZZ /\ M e. NN ) -> M e. CC )
39	16	zcnd 9748	. . . . . . 7 |- ( ( N e. ZZ /\ M e. NN ) -> ( |_ ` ( N / M ) ) e. CC )
40	38, 39	mulneg2d 8729	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN ) -> ( M x. -u ( |_ ` ( N / M ) ) ) = -u ( M x. ( |_ ` ( N / M ) ) ) )
41	40	3ad2ant2 1050	. . . . 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 6090	. . . 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 1050	. . . . . . . 8 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( N / M ) e. QQ )
44	43	flqcld 10690	. . . . . . 7 |- ( ( G e. Grp /\ ( N e. ZZ /\ M e. NN ) /\ ( X e. B /\ ( M .x. X ) = .0. ) ) -> ( |_ ` ( N / M ) ) e. ZZ )
45	44	znegcld 9749	. . . . . 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 13940	. . . . . 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 1280	. . . . 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 6083	. . . . . . 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 1051	. . . . 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 13930	. . . . . 6 |- ( ( G e. Grp /\ -u ( |_ ` ( N / M ) ) e. ZZ ) -> ( -u ( |_ ` ( N / M ) ) .x. .0. ) = .0. )
53	22, 45, 52	syl2anc 415	. . . . 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 2275	. . . 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 2273	. . 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 6091	. 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 13919	. . . 4 |- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( N .x. X ) e. B )
59	57, 23, 29, 58	syl3an 1320	. . 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 13814	. . 3 |- ( ( G e. Grp /\ ( N .x. X ) e. B ) -> ( ( N .x. X ) ( +g ` G ) .0. ) = ( N .x. X ) )
61	22, 59, 60	syl2anc 415	. 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 2275	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 1009   = wceq 1402   e. wcel 2209   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   CCcc 8167   0cc0 8169   + caddc 8172   x. cmul 8174   < clt 8350   - cmin 8487   -ucneg 8488   / cdiv 8992   NNcn 9283   ZZcz 9623   QQcq 9998   |_cfl 10681   mod cmo 10737   Basecbs 13330   +g cplusg 13408   0gc0g 13587   Grpcgrp 13782  ._gcmg 13899

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-fl 10683  df-mod 10738  df-seqfrec 10863  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-mulg 13900

This theorem is referenced by: (None)