Theorem mulgz 14012

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 13871	. . . 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 14011	. . 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 531	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> G e. Grp )
9		nn0z 9675	. . . . 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 13893	. . . . 5 |- ( G e. Grp -> .0. e. B )
12	11	ad2antrr 492	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> .0. e. B )
13		eqid 2238	. . . . 5 |- ( invg ` G ) = ( invg ` G )
14	3, 4, 13	mulgneg 14002	. . . 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 1278	. . 3 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( -u -u N .x. .0. ) = ( ( invg ` G ) ` ( -u N .x. .0. ) ) )
16		zcn 9660	. . . . . 6 |- ( N e. ZZ -> N e. CC )
17	16	ad2antlr 493	. . . . 5 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> N e. CC )
18	17	negnegd 8635	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> -u -u N = N )
19	18	oveq1d 6103	. . 3 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( -u -u N .x. .0. ) = ( N .x. .0. ) )
20	3, 4, 5	mulgnn0z 14011	. . . . . 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 5702	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( ( invg ` G ) ` ( -u N .x. .0. ) ) = ( ( invg ` G ) ` .0. ) )
23	5, 13	grpinvid 13924	. . . . 5 |- ( G e. Grp -> ( ( invg ` G ) ` .0. ) = .0. )
24	23	ad2antrr 492	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( ( invg ` G ) ` .0. ) = .0. )
25	22, 24	eqtrd 2271	. . 3 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( ( invg ` G ) ` ( -u N .x. .0. ) ) = .0. )
26	15, 19, 25	3eqtr3d 2279	. 2 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( N .x. .0. ) = .0. )
27		elznn0 9670	. . . 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 810	1 |- ( ( G e. Grp /\ N e. ZZ ) -> ( N .x. .0. ) = .0. )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   ` cfv 5380  (class class class)co 6088   CCcc 8183   RRcr 8184   -ucneg 8505   NN0cn0 9574   ZZcz 9655   Basecbs 13410   0gc0g 13669   Mndcmnd 13788   Grpcgrp 13864   invgcminusg 13865  ._gcmg 13981

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-ltadd 8301

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-0g 13671  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-grp 13867  df-minusg 13868  df-mulg 13982

This theorem is used by:  mulgmodid  14023