Theorem mulgz 13601

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 13454	. . . 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 13600	. . 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 9427	. . . . 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 13476	. . . . 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 2207	. . . . 5 |- ( invg ` G ) = ( invg ` G )
14	3, 4, 13	mulgneg 13591	. . . 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 9412	. . . . . 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 8409	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> -u -u N = N )
19	18	oveq1d 5982	. . 3 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( -u -u N .x. .0. ) = ( N .x. .0. ) )
20	3, 4, 5	mulgnn0z 13600	. . . . . 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 5603	. . . 4 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( ( invg ` G ) ` ( -u N .x. .0. ) ) = ( ( invg ` G ) ` .0. ) )
23	5, 13	grpinvid 13507	. . . . 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 2240	. . 3 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( ( invg ` G ) ` ( -u N .x. .0. ) ) = .0. )
26	15, 19, 25	3eqtr3d 2248	. 2 |- ( ( ( G e. Grp /\ N e. ZZ ) /\ -u N e. NN0 ) -> ( N .x. .0. ) = .0. )
27		elznn0 9422	. . . 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 2178   ` cfv 5290  (class class class)co 5967   CCcc 7958   RRcr 7959   -ucneg 8279   NN0cn0 9330   ZZcz 9407   Basecbs 12947   0gc0g 13203   Mndcmnd 13363   Grpcgrp 13447   invgcminusg 13448  ._gcmg 13570

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 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076

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 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-2 9130  df-n0 9331  df-z 9408  df-uz 9684  df-fz 10166  df-fzo 10300  df-seqfrec 10630  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-mulg 13571

This theorem is referenced by:  mulgmodid  13612