Theorem mulgnn0ass 12874

Description: Product of group multiples, generalized to NN0. (Contributed by Mario Carneiro, 13-Dec-2014.)

Hypotheses

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

Assertion

Ref	Expression
mulgnn0ass	|- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( ( M x. N ) .x. X ) = ( M .x. ( N .x. X ) ) )

Proof of Theorem mulgnn0ass

Step	Hyp	Ref	Expression
1		mndsgrp 12684	. . . . . . . 8 |- ( G e. Mnd -> G e. Smgrp )
2	1	adantr 276	. . . . . . 7 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> G e. Smgrp )
3	2	adantr 276	. . . . . 6 |- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ ( M e. NN /\ N e. NN ) ) -> G e. Smgrp )
4		simprl 529	. . . . . 6 |- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ ( M e. NN /\ N e. NN ) ) -> M e. NN )
5		simprr 530	. . . . . 6 |- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ ( M e. NN /\ N e. NN ) ) -> N e. NN )
6		simpr3 1003	. . . . . . 7 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> X e. B )
7	6	adantr 276	. . . . . 6 |- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ ( M e. NN /\ N e. NN ) ) -> X e. B )
8		mulgass.b	. . . . . . 7 |- B = ( Base ` G )
9		mulgass.t	. . . . . . 7 |- .x. = (.g ` G )
10	8, 9	mulgnnass 12873	. . . . . 6 |- ( ( G e. Smgrp /\ ( M e. NN /\ N e. NN /\ X e. B ) ) -> ( ( M x. N ) .x. X ) = ( M .x. ( N .x. X ) ) )
11	3, 4, 5, 7, 10	syl13anc 1238	. . . . 5 |- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ ( M e. NN /\ N e. NN ) ) -> ( ( M x. N ) .x. X ) = ( M .x. ( N .x. X ) ) )
12	11	expr 375	. . . 4 |- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M e. NN ) -> ( N e. NN -> ( ( M x. N ) .x. X ) = ( M .x. ( N .x. X ) ) ) )
13		eqid 2173	. . . . . . . . 9 |- ( 0g ` G ) = ( 0g ` G )
14	8, 13, 9	mulg0 12844	. . . . . . . 8 |- ( X e. B -> ( 0 .x. X ) = ( 0g ` G ) )
15	6, 14	syl 14	. . . . . . 7 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( 0 .x. X ) = ( 0g ` G ) )
16		simpr1 1001	. . . . . . . . . 10 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> M e. NN0 )
17	16	nn0cnd 9199	. . . . . . . . 9 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> M e. CC )
18	17	mul01d 8321	. . . . . . . 8 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( M x. 0 ) = 0 )
19	18	oveq1d 5877	. . . . . . 7 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( ( M x. 0 ) .x. X ) = ( 0 .x. X ) )
20	15	oveq2d 5878	. . . . . . . 8 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( M .x. ( 0 .x. X ) ) = ( M .x. ( 0g ` G ) ) )
21	8, 9, 13	mulgnn0z 12865	. . . . . . . . 9 |- ( ( G e. Mnd /\ M e. NN0 ) -> ( M .x. ( 0g ` G ) ) = ( 0g ` G ) )
22	21	3ad2antr1 1160	. . . . . . . 8 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( M .x. ( 0g ` G ) ) = ( 0g ` G ) )
23	20, 22	eqtrd 2206	. . . . . . 7 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( M .x. ( 0 .x. X ) ) = ( 0g ` G ) )
24	15, 19, 23	3eqtr4d 2216	. . . . . 6 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( ( M x. 0 ) .x. X ) = ( M .x. ( 0 .x. X ) ) )
25	24	adantr 276	. . . . 5 |- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M e. NN ) -> ( ( M x. 0 ) .x. X ) = ( M .x. ( 0 .x. X ) ) )
26		oveq2 5870	. . . . . . 7 |- ( N = 0 -> ( M x. N ) = ( M x. 0 ) )
27	26	oveq1d 5877	. . . . . 6 |- ( N = 0 -> ( ( M x. N ) .x. X ) = ( ( M x. 0 ) .x. X ) )
28		oveq1 5869	. . . . . . 7 |- ( N = 0 -> ( N .x. X ) = ( 0 .x. X ) )
29	28	oveq2d 5878	. . . . . 6 |- ( N = 0 -> ( M .x. ( N .x. X ) ) = ( M .x. ( 0 .x. X ) ) )
30	27, 29	eqeq12d 2188	. . . . 5 |- ( N = 0 -> ( ( ( M x. N ) .x. X ) = ( M .x. ( N .x. X ) ) <-> ( ( M x. 0 ) .x. X ) = ( M .x. ( 0 .x. X ) ) ) )
31	25, 30	syl5ibrcom 158	. . . 4 |- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M e. NN ) -> ( N = 0 -> ( ( M x. N ) .x. X ) = ( M .x. ( N .x. X ) ) ) )
32		simpr2 1002	. . . . . 6 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> N e. NN0 )
33		elnn0 9146	. . . . . 6 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
34	32, 33	sylib 122	. . . . 5 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( N e. NN \/ N = 0 ) )
35	34	adantr 276	. . . 4 |- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M e. NN ) -> ( N e. NN \/ N = 0 ) )
36	12, 31, 35	mpjaod 716	. . 3 |- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ M e. NN ) -> ( ( M x. N ) .x. X ) = ( M .x. ( N .x. X ) ) )
37	36	ex 115	. 2 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( M e. NN -> ( ( M x. N ) .x. X ) = ( M .x. ( N .x. X ) ) ) )
38	32	nn0cnd 9199	. . . . . 6 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> N e. CC )
39	38	mul02d 8320	. . . . 5 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( 0 x. N ) = 0 )
40	39	oveq1d 5877	. . . 4 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( ( 0 x. N ) .x. X ) = ( 0 .x. X ) )
41	8, 9	mulgnn0cl 12855	. . . . . 6 |- ( ( G e. Mnd /\ N e. NN0 /\ X e. B ) -> ( N .x. X ) e. B )
42	41	3adant3r1 1210	. . . . 5 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( N .x. X ) e. B )
43	8, 13, 9	mulg0 12844	. . . . 5 |- ( ( N .x. X ) e. B -> ( 0 .x. ( N .x. X ) ) = ( 0g ` G ) )
44	42, 43	syl 14	. . . 4 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( 0 .x. ( N .x. X ) ) = ( 0g ` G ) )
45	15, 40, 44	3eqtr4d 2216	. . 3 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( ( 0 x. N ) .x. X ) = ( 0 .x. ( N .x. X ) ) )
46		oveq1 5869	. . . . 5 |- ( M = 0 -> ( M x. N ) = ( 0 x. N ) )
47	46	oveq1d 5877	. . . 4 |- ( M = 0 -> ( ( M x. N ) .x. X ) = ( ( 0 x. N ) .x. X ) )
48		oveq1 5869	. . . 4 |- ( M = 0 -> ( M .x. ( N .x. X ) ) = ( 0 .x. ( N .x. X ) ) )
49	47, 48	eqeq12d 2188	. . 3 |- ( M = 0 -> ( ( ( M x. N ) .x. X ) = ( M .x. ( N .x. X ) ) <-> ( ( 0 x. N ) .x. X ) = ( 0 .x. ( N .x. X ) ) ) )
50	45, 49	syl5ibrcom 158	. 2 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( M = 0 -> ( ( M x. N ) .x. X ) = ( M .x. ( N .x. X ) ) ) )
51		elnn0 9146	. . 3 |- ( M e. NN0 <-> ( M e. NN \/ M = 0 ) )
52	16, 51	sylib 122	. 2 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( M e. NN \/ M = 0 ) )
53	37, 50, 52	mpjaod 716	1 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( ( M x. N ) .x. X ) = ( M .x. ( N .x. X ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 706   /\ w3a 976   = wceq 1351   e. wcel 2144   ` cfv 5205  (class class class)co 5862   0cc0 7783   x. cmul 7788   NNcn 8887   NN0cn0 9144   Basecbs 12425   0gc0g 12623  Smgrpcsgrp 12669   Mndcmnd 12679  ._gcmg 12839

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 612  ax-in2 613  ax-io 707  ax-5 1443  ax-7 1444  ax-gen 1445  ax-ie1 1489  ax-ie2 1490  ax-8 1500  ax-10 1501  ax-11 1502  ax-i12 1503  ax-bndl 1505  ax-4 1506  ax-17 1522  ax-i9 1526  ax-ial 1530  ax-i5r 1531  ax-13 2146  ax-14 2147  ax-ext 2155  ax-coll 4110  ax-sep 4113  ax-nul 4121  ax-pow 4166  ax-pr 4200  ax-un 4424  ax-setind 4527  ax-iinf 4578  ax-cnex 7874  ax-resscn 7875  ax-1cn 7876  ax-1re 7877  ax-icn 7878  ax-addcl 7879  ax-addrcl 7880  ax-mulcl 7881  ax-addcom 7883  ax-mulcom 7884  ax-addass 7885  ax-mulass 7886  ax-distr 7887  ax-i2m1 7888  ax-0lt1 7889  ax-1rid 7890  ax-0id 7891  ax-rnegex 7892  ax-cnre 7894  ax-pre-ltirr 7895  ax-pre-ltwlin 7896  ax-pre-lttrn 7897  ax-pre-ltadd 7899

This theorem depends on definitions:  df-bi 117  df-dc 833  df-3or 977  df-3an 978  df-tru 1354  df-fal 1357  df-nf 1457  df-sb 1759  df-eu 2025  df-mo 2026  df-clab 2160  df-cleq 2166  df-clel 2169  df-nfc 2304  df-ne 2344  df-nel 2439  df-ral 2456  df-rex 2457  df-reu 2458  df-rmo 2459  df-rab 2460  df-v 2735  df-sbc 2959  df-csb 3053  df-dif 3126  df-un 3128  df-in 3130  df-ss 3137  df-nul 3418  df-if 3530  df-pw 3571  df-sn 3592  df-pr 3593  df-op 3595  df-uni 3803  df-int 3838  df-iun 3881  df-br 3996  df-opab 4057  df-mpt 4058  df-tr 4094  df-id 4284  df-iord 4357  df-on 4359  df-ilim 4360  df-suc 4362  df-iom 4581  df-xp 4623  df-rel 4624  df-cnv 4625  df-co 4626  df-dm 4627  df-rn 4628  df-res 4629  df-ima 4630  df-iota 5167  df-fun 5207  df-fn 5208  df-f 5209  df-f1 5210  df-fo 5211  df-f1o 5212  df-fv 5213  df-riota 5818  df-ov 5865  df-oprab 5866  df-mpo 5867  df-1st 6128  df-2nd 6129  df-recs 6293  df-frec 6379  df-pnf 7965  df-mnf 7966  df-xr 7967  df-ltxr 7968  df-le 7969  df-sub 8101  df-neg 8102  df-inn 8888  df-2 8946  df-n0 9145  df-z 9222  df-uz 9497  df-fz 9975  df-fzo 10108  df-seqfrec 10411  df-ndx 12428  df-slot 12429  df-base 12431  df-plusg 12502  df-0g 12625  df-mgm 12637  df-sgrp 12670  df-mnd 12680  df-minusg 12739  df-mulg 12840

This theorem is referenced by:  mulgass  12875