Theorem mulgnn0ass 13231

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 13005	. . . . . . . 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 531	. . . . . 6 |- ( ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) /\ ( M e. NN /\ N e. NN ) ) -> N e. NN )
6		simpr3 1007	. . . . . . 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 13230	. . . . . 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 1251	. . . . 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 2193	. . . . . . . . 9 |- ( 0g ` G ) = ( 0g ` G )
14	8, 13, 9	mulg0 13198	. . . . . . . 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 1005	. . . . . . . . . 10 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> M e. NN0 )
17	16	nn0cnd 9298	. . . . . . . . 9 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> M e. CC )
18	17	mul01d 8414	. . . . . . . 8 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( M x. 0 ) = 0 )
19	18	oveq1d 5934	. . . . . . 7 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( ( M x. 0 ) .x. X ) = ( 0 .x. X ) )
20	15	oveq2d 5935	. . . . . . . 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 13222	. . . . . . . . 9 |- ( ( G e. Mnd /\ M e. NN0 ) -> ( M .x. ( 0g ` G ) ) = ( 0g ` G ) )
22	21	3ad2antr1 1164	. . . . . . . 8 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( M .x. ( 0g ` G ) ) = ( 0g ` G ) )
23	20, 22	eqtrd 2226	. . . . . . 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 2236	. . . . . 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 5927	. . . . . . 7 |- ( N = 0 -> ( M x. N ) = ( M x. 0 ) )
27	26	oveq1d 5934	. . . . . 6 |- ( N = 0 -> ( ( M x. N ) .x. X ) = ( ( M x. 0 ) .x. X ) )
28		oveq1 5926	. . . . . . 7 |- ( N = 0 -> ( N .x. X ) = ( 0 .x. X ) )
29	28	oveq2d 5935	. . . . . 6 |- ( N = 0 -> ( M .x. ( N .x. X ) ) = ( M .x. ( 0 .x. X ) ) )
30	27, 29	eqeq12d 2208	. . . . 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 1006	. . . . . 6 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> N e. NN0 )
33		elnn0 9245	. . . . . 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 719	. . 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 9298	. . . . . 6 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> N e. CC )
39	38	mul02d 8413	. . . . 5 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( 0 x. N ) = 0 )
40	39	oveq1d 5934	. . . 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 13211	. . . . . 6 |- ( ( G e. Mnd /\ N e. NN0 /\ X e. B ) -> ( N .x. X ) e. B )
42	41	3adant3r1 1214	. . . . 5 |- ( ( G e. Mnd /\ ( M e. NN0 /\ N e. NN0 /\ X e. B ) ) -> ( N .x. X ) e. B )
43	8, 13, 9	mulg0 13198	. . . . 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 2236	. . 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 5926	. . . . 5 |- ( M = 0 -> ( M x. N ) = ( 0 x. N ) )
47	46	oveq1d 5934	. . . 4 |- ( M = 0 -> ( ( M x. N ) .x. X ) = ( ( 0 x. N ) .x. X ) )
48		oveq1 5926	. . . 4 |- ( M = 0 -> ( M .x. ( N .x. X ) ) = ( 0 .x. ( N .x. X ) ) )
49	47, 48	eqeq12d 2208	. . 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 9245	. . 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 719	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 709   /\ w3a 980   = wceq 1364   e. wcel 2164   ` cfv 5255  (class class class)co 5919   0cc0 7874   x. cmul 7879   NNcn 8984   NN0cn0 9243   Basecbs 12621   0gc0g 12870  Smgrpcsgrp 12987   Mndcmnd 13000  ._gcmg 13192

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-2 9043  df-n0 9244  df-z 9321  df-uz 9596  df-fz 10078  df-fzo 10212  df-seqfrec 10522  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-minusg 13079  df-mulg 13193

This theorem is referenced by:  mulgass  13232