Theorem mulgnn0z 13686

Description: A group multiple of the identity, for nonnegative 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
mulgnn0z	|- ( ( G e. Mnd /\ N e. NN0 ) -> ( N .x. .0. ) = .0. )

Proof of Theorem mulgnn0z

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnn0 9371	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		id 19	. . . . 5 |- ( N e. NN -> N e. NN )
3		mulgnn0z.b	. . . . . 6 |- B = ( Base ` G )
4		mulgnn0z.o	. . . . . 6 |- .0. = ( 0g ` G )
5	3, 4	mndidcl 13463	. . . . 5 |- ( G e. Mnd -> .0. e. B )
6		eqid 2229	. . . . . 6 |- ( +g ` G ) = ( +g ` G )
7		mulgnn0z.t	. . . . . 6 |- .x. = (.g ` G )
8		eqid 2229	. . . . . 6 |- seq 1 ( ( +g ` G ) , ( NN X. { .0. } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { .0. } ) )
9	3, 6, 7, 8	mulgnn 13663	. . . . 5 |- ( ( N e. NN /\ .0. e. B ) -> ( N .x. .0. ) = ( seq 1 ( ( +g ` G ) , ( NN X. { .0. } ) ) ` N ) )
10	2, 5, 9	syl2anr 290	. . . 4 |- ( ( G e. Mnd /\ N e. NN ) -> ( N .x. .0. ) = ( seq 1 ( ( +g ` G ) , ( NN X. { .0. } ) ) ` N ) )
11	3, 6, 4	mndlid 13468	. . . . . . 7 |- ( ( G e. Mnd /\ .0. e. B ) -> ( .0. ( +g ` G ) .0. ) = .0. )
12	5, 11	mpdan 421	. . . . . 6 |- ( G e. Mnd -> ( .0. ( +g ` G ) .0. ) = .0. )
13	12	adantr 276	. . . . 5 |- ( ( G e. Mnd /\ N e. NN ) -> ( .0. ( +g ` G ) .0. ) = .0. )
14		simpr 110	. . . . . 6 |- ( ( G e. Mnd /\ N e. NN ) -> N e. NN )
15		nnuz 9758	. . . . . 6 |- NN = ( ZZ>= ` 1 )
16	14, 15	eleqtrdi 2322	. . . . 5 |- ( ( G e. Mnd /\ N e. NN ) -> N e. ( ZZ>= ` 1 ) )
17	5	adantr 276	. . . . . 6 |- ( ( G e. Mnd /\ N e. NN ) -> .0. e. B )
18		elfznn 10250	. . . . . 6 |- ( x e. ( 1 ... N ) -> x e. NN )
19		fvconst2g 5853	. . . . . 6 |- ( ( .0. e. B /\ x e. NN ) -> ( ( NN X. { .0. } ) ` x ) = .0. )
20	17, 18, 19	syl2an 289	. . . . 5 |- ( ( ( G e. Mnd /\ N e. NN ) /\ x e. ( 1 ... N ) ) -> ( ( NN X. { .0. } ) ` x ) = .0. )
21	15, 17	ialgrlemconst 12565	. . . . 5 |- ( ( ( G e. Mnd /\ N e. NN ) /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { .0. } ) ` x ) e. B )
22	3, 6	mndcl 13456	. . . . . . 7 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
23	22	3expb 1228	. . . . . 6 |- ( ( G e. Mnd /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) e. B )
24	23	adantlr 477	. . . . 5 |- ( ( ( G e. Mnd /\ N e. NN ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) e. B )
25	13, 16, 20, 17, 21, 24	seq3id3 10746	. . . 4 |- ( ( G e. Mnd /\ N e. NN ) -> ( seq 1 ( ( +g ` G ) , ( NN X. { .0. } ) ) ` N ) = .0. )
26	10, 25	eqtrd 2262	. . 3 |- ( ( G e. Mnd /\ N e. NN ) -> ( N .x. .0. ) = .0. )
27		oveq1 6008	. . . 4 |- ( N = 0 -> ( N .x. .0. ) = ( 0 .x. .0. ) )
28	3, 4, 7	mulg0 13662	. . . . 5 |- ( .0. e. B -> ( 0 .x. .0. ) = .0. )
29	5, 28	syl 14	. . . 4 |- ( G e. Mnd -> ( 0 .x. .0. ) = .0. )
30	27, 29	sylan9eqr 2284	. . 3 |- ( ( G e. Mnd /\ N = 0 ) -> ( N .x. .0. ) = .0. )
31	26, 30	jaodan 802	. 2 |- ( ( G e. Mnd /\ ( N e. NN \/ N = 0 ) ) -> ( N .x. .0. ) = .0. )
32	1, 31	sylan2b 287	1 |- ( ( G e. Mnd /\ N e. NN0 ) -> ( N .x. .0. ) = .0. )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   {csn 3666   X. cxp 4717   ` cfv 5318  (class class class)co 6001   0cc0 7999   1c1 8000   NNcn 9110   NN0cn0 9369   ZZ>=cuz 9722   ...cfz 10204   seqcseq 10669   Basecbs 13032   +g cplusg 13110   0gc0g 13289   Mndcmnd 13449  ._gcmg 13656

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-2 9169  df-n0 9370  df-z 9447  df-uz 9723  df-fz 10205  df-fzo 10339  df-seqfrec 10670  df-ndx 13035  df-slot 13036  df-base 13038  df-plusg 13123  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-minusg 13537  df-mulg 13657

This theorem is referenced by:  mulgz  13687  mulgnn0ass  13695  srg1expzeq1  13958