Theorem mulgnn0ass 13961

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 13734	. . . . . . . 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 535	. . . . . 6 |- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ ( M e. NN /\ N e. NN ) ) -> M e. NN )
5		simprr 537	. . . . . 6 |- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ ( M e. NN /\ N e. NN ) ) -> N e. NN )
6		simpr3 1036	. . . . . . 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 13960	. . . . . 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 1280	. . . . 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 2238	. . . . . . . . 9 |- ( 0g ` G ) = ( 0g ` G )
14	8, 13, 9	mulg0 13928	. . . . . . . 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 1034	. . . . . . . . . 10 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> M e. NN0 )
17	16	nn0cnd 9622	. . . . . . . . 9 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> M e. CC )
18	17	mul01d 8720	. . . . . . . 8 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( M x. 0 ) = 0 )
19	18	oveq1d 6100	. . . . . . 7 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( ( M x. 0 ) .x. X ) = ( 0 .x. X ) )
20	15	oveq2d 6101	. . . . . . . 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 13952	. . . . . . . . 9 |- ( ( G e. Mnd /\ M e. NN0 ) -> ( M .x. ( 0g ` G ) ) = ( 0g ` G ) )
22	21	3ad2antr1 1193	. . . . . . . 8 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( M .x. ( 0g ` G ) ) = ( 0g ` G ) )
23	20, 22	eqtrd 2271	. . . . . . 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 2281	. . . . . 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 6093	. . . . . . 7 |- ( N = 0 -> ( M x. N ) = ( M x. 0 ) )
27	26	oveq1d 6100	. . . . . 6 |- ( N = 0 -> ( ( M x. N ) .x. X ) = ( ( M x. 0 ) .x. X ) )
28		oveq1 6092	. . . . . . 7 |- ( N = 0 -> ( N .x. X ) = ( 0 .x. X ) )
29	28	oveq2d 6101	. . . . . 6 |- ( N = 0 -> ( M .x. ( N .x. X ) ) = ( M .x. ( 0 .x. X ) ) )
30	27, 29	eqeq12d 2253	. . . . 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 1035	. . . . . 6 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> N e. NN0 )
33		elnn0 9565	. . . . . 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 730	. . 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 9622	. . . . . 6 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> N e. CC )
39	38	mul02d 8719	. . . . 5 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( 0 x. N ) = 0 )
40	39	oveq1d 6100	. . . 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 13941	. . . . . 6 |- ( ( G e. Mnd /\ N e. NN0 /\ X e. B ) -> ( N .x. X ) e. B )
42	41	3adant3r1 1243	. . . . 5 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( N .x. X ) e. B )
43	8, 13, 9	mulg0 13928	. . . . 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 2281	. . 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 6092	. . . . 5 |- ( M = 0 -> ( M x. N ) = ( 0 x. N ) )
47	46	oveq1d 6100	. . . 4 |- ( M = 0 -> ( ( M x. N ) .x. X ) = ( ( 0 x. N ) .x. X ) )
48		oveq1 6092	. . . 4 |- ( M = 0 -> ( M .x. ( N .x. X ) ) = ( 0 .x. ( N .x. X ) ) )
49	47, 48	eqeq12d 2253	. . 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 9565	. . 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 730	1 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( ( M x. N ) .x. X ) = ( M .x. ( N .x. X ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085   0cc0 8179   x. cmul 8184   NNcn 9304   NN0cn0 9563   Basecbs 13352   0gc0g 13610  Smgrpcsgrp 13716   Mndcmnd 13729  ._gcmg 13922

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-ndx 13355  df-slot 13356  df-base 13358  df-plusg 13444  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-minusg 13809  df-mulg 13923

This theorem is used by:  mulgass  13962