Theorem mulgnn0ass 13744

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 13503	. . . . . . . 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 531	. . . . . 6 |- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ ( M e. NN /\ N e. NN ) ) -> M e. NN )
5		simprr 533	. . . . . 6 |- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ ( M e. NN /\ N e. NN ) ) -> N e. NN )
6		simpr3 1031	. . . . . . 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 13743	. . . . . 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 1275	. . . . 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 2231	. . . . . . . . 9 |- ( 0g ` G ) = ( 0g ` G )
14	8, 13, 9	mulg0 13711	. . . . . . . 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 1029	. . . . . . . . . 10 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> M e. NN0 )
17	16	nn0cnd 9456	. . . . . . . . 9 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> M e. CC )
18	17	mul01d 8571	. . . . . . . 8 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( M x. 0 ) = 0 )
19	18	oveq1d 6032	. . . . . . 7 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( ( M x. 0 ) .x. X ) = ( 0 .x. X ) )
20	15	oveq2d 6033	. . . . . . . 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 13735	. . . . . . . . 9 |- ( ( G e. Mnd /\ M e. NN0 ) -> ( M .x. ( 0g ` G ) ) = ( 0g ` G ) )
22	21	3ad2antr1 1188	. . . . . . . 8 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( M .x. ( 0g ` G ) ) = ( 0g ` G ) )
23	20, 22	eqtrd 2264	. . . . . . 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 2274	. . . . . 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 6025	. . . . . . 7 |- ( N = 0 -> ( M x. N ) = ( M x. 0 ) )
27	26	oveq1d 6032	. . . . . 6 |- ( N = 0 -> ( ( M x. N ) .x. X ) = ( ( M x. 0 ) .x. X ) )
28		oveq1 6024	. . . . . . 7 |- ( N = 0 -> ( N .x. X ) = ( 0 .x. X ) )
29	28	oveq2d 6033	. . . . . 6 |- ( N = 0 -> ( M .x. ( N .x. X ) ) = ( M .x. ( 0 .x. X ) ) )
30	27, 29	eqeq12d 2246	. . . . 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 157	. . . 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 1030	. . . . . 6 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> N e. NN0 )
33		elnn0 9403	. . . . . 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 725	. . 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 9456	. . . . . 6 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> N e. CC )
39	38	mul02d 8570	. . . . 5 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( 0 x. N ) = 0 )
40	39	oveq1d 6032	. . . 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 13724	. . . . . 6 |- ( ( G e. Mnd /\ N e. NN0 /\ X e. B ) -> ( N .x. X ) e. B )
42	41	3adant3r1 1238	. . . . 5 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( N .x. X ) e. B )
43	8, 13, 9	mulg0 13711	. . . . 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 2274	. . 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 6024	. . . . 5 |- ( M = 0 -> ( M x. N ) = ( 0 x. N ) )
47	46	oveq1d 6032	. . . 4 |- ( M = 0 -> ( ( M x. N ) .x. X ) = ( ( 0 x. N ) .x. X ) )
48		oveq1 6024	. . . 4 |- ( M = 0 -> ( M .x. ( N .x. X ) ) = ( 0 .x. ( N .x. X ) ) )
49	47, 48	eqeq12d 2246	. . 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 157	. 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 9403	. . 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 725	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 715   /\ w3a 1004   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   0cc0 8031   x. cmul 8036   NNcn 9142   NN0cn0 9401   Basecbs 13081   0gc0g 13338  Smgrpcsgrp 13483   Mndcmnd 13498  ._gcmg 13705

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-2 9201  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243  df-fzo 10377  df-seqfrec 10709  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-minusg 13586  df-mulg 13706

This theorem is referenced by:  mulgass  13745