Theorem mulgz 13486

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 13339	. . . 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 13485	. . 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 9392	. . . . 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 13361	. . . . 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 2205	. . . . 5 |- ( invg ` G ) = ( invg ` G )
14	3, 4, 13	mulgneg 13476	. . . 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 1250	. . 3 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( -u -u N .x. .0. ) = ( ( invg ` G ) ` ( -u N .x. .0. ) ) )
16		zcn 9377	. . . . . 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 8374	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> -u -u N = N )
19	18	oveq1d 5959	. . 3 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( -u -u N .x. .0. ) = ( N .x. .0. ) )
20	3, 4, 5	mulgnn0z 13485	. . . . . 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 5580	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( ( invg ` G ) ` ( -u N .x. .0. ) ) = ( ( invg ` G ) ` .0. ) )
23	5, 13	grpinvid 13392	. . . . 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 2238	. . 3 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( ( invg ` G ) ` ( -u N .x. .0. ) ) = .0. )
26	15, 19, 25	3eqtr3d 2246	. 2 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( N .x. .0. ) = .0. )
27		elznn0 9387	. . . 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 800	1 |- ( ( G e. Grp /\ N e. ZZ ) -> ( N .x. .0. ) = .0. )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 710   = wceq 1373   e. wcel 2176   ` cfv 5271  (class class class)co 5944   CCcc 7923   RRcr 7924   -ucneg 8244   NN0cn0 9295   ZZcz 9372   Basecbs 12832   0gc0g 13088   Mndcmnd 13248   Grpcgrp 13332   invgcminusg 13333  ._gcmg 13455

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-ltadd 8041

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-2 9095  df-n0 9296  df-z 9373  df-uz 9649  df-fz 10131  df-fzo 10265  df-seqfrec 10593  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-minusg 13336  df-mulg 13456

This theorem is referenced by:  mulgmodid  13497