Theorem mulgnn0ass 13911

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 13682	. . . . . . . 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 1032	. . . . . . 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 13910	. . . . . 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 1276	. . . . 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 2234	. . . . . . . . 9 |- ( 0g ` G ) = ( 0g ` G )
14	8, 13, 9	mulg0 13878	. . . . . . . 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 1030	. . . . . . . . . 10 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> M e. NN0 )
17	16	nn0cnd 9572	. . . . . . . . 9 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> M e. CC )
18	17	mul01d 8683	. . . . . . . 8 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( M x. 0 ) = 0 )
19	18	oveq1d 6073	. . . . . . 7 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( ( M x. 0 ) .x. X ) = ( 0 .x. X ) )
20	15	oveq2d 6074	. . . . . . . 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 13902	. . . . . . . . 9 |- ( ( G e. Mnd /\ M e. NN0 ) -> ( M .x. ( 0g ` G ) ) = ( 0g ` G ) )
22	21	3ad2antr1 1189	. . . . . . . 8 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( M .x. ( 0g ` G ) ) = ( 0g ` G ) )
23	20, 22	eqtrd 2267	. . . . . . 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 2277	. . . . . 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 6066	. . . . . . 7 |- ( N = 0 -> ( M x. N ) = ( M x. 0 ) )
27	26	oveq1d 6073	. . . . . 6 |- ( N = 0 -> ( ( M x. N ) .x. X ) = ( ( M x. 0 ) .x. X ) )
28		oveq1 6065	. . . . . . 7 |- ( N = 0 -> ( N .x. X ) = ( 0 .x. X ) )
29	28	oveq2d 6074	. . . . . 6 |- ( N = 0 -> ( M .x. ( N .x. X ) ) = ( M .x. ( 0 .x. X ) ) )
30	27, 29	eqeq12d 2249	. . . . 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 1031	. . . . . 6 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> N e. NN0 )
33		elnn0 9515	. . . . . 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 726	. . 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 9572	. . . . . 6 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> N e. CC )
39	38	mul02d 8682	. . . . 5 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( 0 x. N ) = 0 )
40	39	oveq1d 6073	. . . 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 13891	. . . . . 6 |- ( ( G e. Mnd /\ N e. NN0 /\ X e. B ) -> ( N .x. X ) e. B )
42	41	3adant3r1 1239	. . . . 5 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( N .x. X ) e. B )
43	8, 13, 9	mulg0 13878	. . . . 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 2277	. . 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 6065	. . . . 5 |- ( M = 0 -> ( M x. N ) = ( 0 x. N ) )
47	46	oveq1d 6073	. . . 4 |- ( M = 0 -> ( ( M x. N ) .x. X ) = ( ( 0 x. N ) .x. X ) )
48		oveq1 6065	. . . 4 |- ( M = 0 -> ( M .x. ( N .x. X ) ) = ( 0 .x. ( N .x. X ) ) )
49	47, 48	eqeq12d 2249	. . 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 9515	. . 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 726	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 716   /\ w3a 1005   = wceq 1398   e. wcel 2205   ` cfv 5357  (class class class)co 6058   0cc0 8143   x. cmul 8148   NNcn 9254   NN0cn0 9513   Basecbs 13296   0gc0g 13553  Smgrpcsgrp 13664   Mndcmnd 13677  ._gcmg 13872

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-minusg 13759  df-mulg 13873

This theorem is referenced by:  mulgass  13912