Theorem mulgz 13280

Description: A group multiple of the identity, for integer multiple. (Contributed by Mario Carneiro, 13-Dec-2014.)

Hypotheses

Ref	Expression
mulgnn0z.b	|- B = ( Base ` G )
mulgnn0z.t	|- .x. = (.g ` G )
mulgnn0z.o	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
mulgz	|- ( ( G e. Grp /\ N e. ZZ ) -> ( N .x. .0. ) = .0. )

Proof of Theorem mulgz

Step	Hyp	Ref	Expression
1		grpmnd 13139	. . . 4 |- ( G e. Grp -> G e. Mnd )
2	1	adantr 276	. . 3 |- ( ( G e. Grp /\ N e. ZZ ) -> G e. Mnd )
3		mulgnn0z.b	. . . 4 |- B = ( Base ` G )
4		mulgnn0z.t	. . . 4 |- .x. = (.g ` G )
5		mulgnn0z.o	. . . 4 |- .0. = ( 0g ` G )
6	3, 4, 5	mulgnn0z 13279	. . 3 |- ( ( G e. Mnd /\ N e. NN0 ) -> ( N .x. .0. ) = .0. )
7	2, 6	sylan 283	. 2 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ N e. NN0 ) -> ( N .x. .0. ) = .0. )
8		simpll 527	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> G e. Grp )
9		nn0z 9346	. . . . 5 |- ( -u N e. NN0 -> -u N e. ZZ )
10	9	adantl 277	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> -u N e. ZZ )
11	3, 5	grpidcl 13161	. . . . 5 |- ( G e. Grp -> .0. e. B )
12	11	ad2antrr 488	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> .0. e. B )
13		eqid 2196	. . . . 5 |- ( invg ` G ) = ( invg ` G )
14	3, 4, 13	mulgneg 13270	. . . 4 |- ( ( G e. Grp /\ -u N e. ZZ /\ .0. e. B ) -> ( -u -u N .x. .0. ) = ( ( invg ` G ) ` ( -u N .x. .0. ) ) )
15	8, 10, 12, 14	syl3anc 1249	. . 3 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( -u -u N .x. .0. ) = ( ( invg ` G ) ` ( -u N .x. .0. ) ) )
16		zcn 9331	. . . . . 6 |- ( N e. ZZ -> N e. CC )
17	16	ad2antlr 489	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> N e. CC )
18	17	negnegd 8328	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> -u -u N = N )
19	18	oveq1d 5937	. . 3 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( -u -u N .x. .0. ) = ( N .x. .0. ) )
20	3, 4, 5	mulgnn0z 13279	. . . . . 6 |- ( ( G e. Mnd /\ -u N e. NN0 ) -> ( -u N .x. .0. ) = .0. )
21	2, 20	sylan 283	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( -u N .x. .0. ) = .0. )
22	21	fveq2d 5562	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( ( invg ` G ) ` ( -u N .x. .0. ) ) = ( ( invg ` G ) ` .0. ) )
23	5, 13	grpinvid 13192	. . . . 5 |- ( G e. Grp -> ( ( invg ` G ) ` .0. ) = .0. )
24	23	ad2antrr 488	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( ( invg ` G ) ` .0. ) = .0. )
25	22, 24	eqtrd 2229	. . 3 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( ( invg ` G ) ` ( -u N .x. .0. ) ) = .0. )
26	15, 19, 25	3eqtr3d 2237	. 2 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( N .x. .0. ) = .0. )
27		elznn0 9341	. . . 4 |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) )
28	27	simprbi 275	. . 3 |- ( N e. ZZ -> ( N e. NN0 \/ -u N e. NN0 ) )
29	28	adantl 277	. 2 |- ( ( G e. Grp /\ N e. ZZ ) -> ( N e. NN0 \/ -u N e. NN0 ) )
30	7, 26, 29	mpjaodan 799	1 |- ( ( G e. Grp /\ N e. ZZ ) -> ( N .x. .0. ) = .0. )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2167   ` cfv 5258  (class class class)co 5922   CCcc 7877   RRcr 7878   -ucneg 8198   NN0cn0 9249   ZZcz 9326   Basecbs 12678   0gc0g 12927   Mndcmnd 13057   Grpcgrp 13132   invgcminusg 13133  ._gcmg 13249

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-2 9049  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084  df-fzo 10218  df-seqfrec 10540  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-mulg 13250

This theorem is referenced by:  mulgmodid  13291