Theorem lmod1lem3 42278

Description: Lemma 3 for lmod1 42281. (Contributed by AV, 29-Apr-2019.)

Hypothesis

Ref	Expression
lmod1.m	|- M = ( { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , ( x e. ( Base ` R ) , y e. { I } |-> y ) >. } )

Assertion

Ref	Expression
lmod1lem3	|- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) )

Distinct variable groups:   I, r, x, y   R, r, x, y   V, r, x, y   I, q   R, q   V, q   x, M, y   x, q, y

Allowed substitution hints:   M( r, q)

Proof of Theorem lmod1lem3

Step	Hyp	Ref	Expression
1		eqidd 2623	. . 3 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( x e. ( Base ` R ) , y e. { I } |-> y ) = ( x e. ( Base ` R ) , y e. { I } |-> y ) )
2		simprr 796	. . 3 |- ( ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) /\ ( x = ( q ( +g ` (Scalar ` M ) ) r ) /\ y = I ) ) -> y = I )
3		simplr 792	. . . . . . 7 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> R e. Ring )
4		lmod1.m	. . . . . . . . 9 |- M = ( { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , ( x e. ( Base ` R ) , y e. { I } |-> y ) >. } )
5	4	lmodsca 16020	. . . . . . . 8 |- ( R e. Ring -> R = (Scalar ` M ) )
6	5	fveq2d 6195	. . . . . . 7 |- ( R e. Ring -> ( +g ` R ) = ( +g ` (Scalar ` M ) ) )
7	3, 6	syl 17	. . . . . 6 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( +g ` R ) = ( +g ` (Scalar ` M ) ) )
8	7	eqcomd 2628	. . . . 5 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( +g ` (Scalar ` M ) ) = ( +g ` R ) )
9	8	oveqd 6667	. . . 4 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( q ( +g ` (Scalar ` M ) ) r ) = ( q ( +g ` R ) r ) )
10		simprl 794	. . . . 5 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> q e. ( Base ` R ) )
11		simprr 796	. . . . 5 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> r e. ( Base ` R ) )
12		eqid 2622	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
13		eqid 2622	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
14	12, 13	ringacl 18578	. . . . 5 |- ( ( R e. Ring /\ q e. ( Base ` R ) /\ r e. ( Base ` R ) ) -> ( q ( +g ` R ) r ) e. ( Base ` R ) )
15	3, 10, 11, 14	syl3anc 1326	. . . 4 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( q ( +g ` R ) r ) e. ( Base ` R ) )
16	9, 15	eqeltrd 2701	. . 3 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( q ( +g ` (Scalar ` M ) ) r ) e. ( Base ` R ) )
17		snidg 4206	. . . . 5 |- ( I e. V -> I e. { I } )
18	17	adantr 481	. . . 4 |- ( ( I e. V /\ R e. Ring ) -> I e. { I } )
19	18	adantr 481	. . 3 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> I e. { I } )
20		simpl 473	. . . 4 |- ( ( I e. V /\ R e. Ring ) -> I e. V )
21	20	adantr 481	. . 3 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> I e. V )
22	1, 2, 16, 19, 21	ovmpt2d 6788	. 2 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( ( q ( +g ` (Scalar ` M ) ) r ) ( x e. ( Base ` R ) , y e. { I } |-> y ) I ) = I )
23		fvex 6201	. . . . . . 7 |- ( Base ` R ) e. _V
24		snex 4908	. . . . . . 7 |- { I } e. _V
25	23, 24	pm3.2i 471	. . . . . 6 |- ( ( Base ` R ) e. _V /\ { I } e. _V )
26		mpt2exga 7246	. . . . . 6 |- ( ( ( Base ` R ) e. _V /\ { I } e. _V ) -> ( x e. ( Base ` R ) , y e. { I } |-> y ) e. _V )
27	25, 26	mp1i 13	. . . . 5 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( x e. ( Base ` R ) , y e. { I } |-> y ) e. _V )
28	4	lmodvsca 16021	. . . . 5 |- ( ( x e. ( Base ` R ) , y e. { I } |-> y ) e. _V -> ( x e. ( Base ` R ) , y e. { I } |-> y ) = ( .s ` M ) )
29	27, 28	syl 17	. . . 4 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( x e. ( Base ` R ) , y e. { I } |-> y ) = ( .s ` M ) )
30	29	eqcomd 2628	. . 3 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( .s ` M ) = ( x e. ( Base ` R ) , y e. { I } |-> y ) )
31	30	oveqd 6667	. 2 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( ( q ( +g ` (Scalar ` M ) ) r ) ( x e. ( Base ` R ) , y e. { I } |-> y ) I ) )
32		simprr 796	. . . . 5 |- ( ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) /\ ( x = q /\ y = I ) ) -> y = I )
33	30, 32, 10, 19, 19	ovmpt2d 6788	. . . 4 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( q ( .s ` M ) I ) = I )
34		simprr 796	. . . . 5 |- ( ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) /\ ( x = r /\ y = I ) ) -> y = I )
35	30, 34, 11, 19, 19	ovmpt2d 6788	. . . 4 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( r ( .s ` M ) I ) = I )
36	33, 35	oveq12d 6668	. . 3 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) = ( I ( +g ` M ) I ) )
37		snex 4908	. . . . . 6 |- { <. <. I , I >. , I >. } e. _V
38	4	lmodplusg 16019	. . . . . 6 |- ( { <. <. I , I >. , I >. } e. _V -> { <. <. I , I >. , I >. } = ( +g ` M ) )
39	37, 38	mp1i 13	. . . . 5 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> { <. <. I , I >. , I >. } = ( +g ` M ) )
40	39	eqcomd 2628	. . . 4 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( +g ` M ) = { <. <. I , I >. , I >. } )
41	40	oveqd 6667	. . 3 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( I ( +g ` M ) I ) = ( I { <. <. I , I >. , I >. } I ) )
42		df-ov 6653	. . . 4 |- ( I { <. <. I , I >. , I >. } I ) = ( { <. <. I , I >. , I >. } ` <. I , I >. )
43		opex 4932	. . . . . . 7 |- <. I , I >. e. _V
44	20, 43	jctil 560	. . . . . 6 |- ( ( I e. V /\ R e. Ring ) -> ( <. I , I >. e. _V /\ I e. V ) )
45	44	adantr 481	. . . . 5 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( <. I , I >. e. _V /\ I e. V ) )
46		fvsng 6447	. . . . 5 |- ( ( <. I , I >. e. _V /\ I e. V ) -> ( { <. <. I , I >. , I >. } ` <. I , I >. ) = I )
47	45, 46	syl 17	. . . 4 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( { <. <. I , I >. , I >. } ` <. I , I >. ) = I )
48	42, 47	syl5eq 2668	. . 3 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( I { <. <. I , I >. , I >. } I ) = I )
49	36, 41, 48	3eqtrd 2660	. 2 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) = I )
50	22, 31, 49	3eqtr4d 2666	1 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   {csn 4177   {ctp 4181   <.cop 4183   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ndxcnx 15854   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   Ringcrg 18547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-sca 15957  df-vsca 15958  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ring 18549

This theorem is referenced by:  lmod1  42281