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Theorem lmod1lem3 32388
Description: Lemma 3 for lmod1 32391. (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
StepHypRef Expression
1 eqidd 2468 . . 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 756 . . 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 simpr 461 . . . . . . . 8  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  R  e.  Ring )
43adantr 465 . . . . . . 7  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  R  e.  Ring )
5 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 ) >. } )
65lmodsca 14625 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  =  (Scalar `  M )
)
76fveq2d 5870 . . . . . . 7  |-  ( R  e.  Ring  ->  ( +g  `  R )  =  ( +g  `  (Scalar `  M ) ) )
84, 7syl 16 . . . . . 6  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  ( +g  `  R )  =  ( +g  `  (Scalar `  M ) ) )
98eqcomd 2475 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  ( +g  `  (Scalar `  M
) )  =  ( +g  `  R ) )
109oveqd 6302 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
q ( +g  `  (Scalar `  M ) ) r )  =  ( q ( +g  `  R
) r ) )
11 simprl 755 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  q  e.  ( Base `  R
) )
12 simprr 756 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  r  e.  ( Base `  R
) )
13 eqid 2467 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
14 eqid 2467 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
1513, 14rngacl 17039 . . . . 5  |-  ( ( R  e.  Ring  /\  q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
)  ->  ( q
( +g  `  R ) r )  e.  (
Base `  R )
)
164, 11, 12, 15syl3anc 1228 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
q ( +g  `  R
) r )  e.  ( Base `  R
) )
1710, 16eqeltrd 2555 . . 3  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
q ( +g  `  (Scalar `  M ) ) r )  e.  ( Base `  R ) )
18 snidg 4053 . . . . 5  |-  ( I  e.  V  ->  I  e.  { I } )
1918adantr 465 . . . 4  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  I  e.  { I } )
2019adantr 465 . . 3  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  I  e.  { I } )
21 simpl 457 . . . 4  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  I  e.  V )
2221adantr 465 . . 3  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  I  e.  V )
231, 2, 17, 20, 22ovmpt2d 6415 . 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 )
24 fvex 5876 . . . . . . 7  |-  ( Base `  R )  e.  _V
25 snex 4688 . . . . . . 7  |-  { I }  e.  _V
2624, 25pm3.2i 455 . . . . . 6  |-  ( (
Base `  R )  e.  _V  /\  { I }  e.  _V )
27 eqid 2467 . . . . . . 7  |-  ( x  e.  ( Base `  R
) ,  y  e. 
{ I }  |->  y )  =  ( x  e.  ( Base `  R
) ,  y  e. 
{ I }  |->  y )
2827mpt2exg 6859 . . . . . 6  |-  ( ( ( Base `  R
)  e.  _V  /\  { I }  e.  _V )  ->  ( x  e.  ( Base `  R
) ,  y  e. 
{ I }  |->  y )  e.  _V )
2926, 28mp1i 12 . . . . 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 )
305lmodvsca 14626 . . . . 5  |-  ( ( x  e.  ( Base `  R ) ,  y  e.  { I }  |->  y )  e.  _V  ->  ( x  e.  (
Base `  R ) ,  y  e.  { I }  |->  y )  =  ( .s `  M
) )
3129, 30syl 16 . . . 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 ) )
3231eqcomd 2475 . . 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 ) )
3332oveqd 6302 . 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 ) )
3432oveqd 6302 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
q ( .s `  M ) I )  =  ( q ( x  e.  ( Base `  R ) ,  y  e.  { I }  |->  y ) I ) )
35 simprr 756 . . . . . 6  |-  ( ( ( ( I  e.  V  /\  R  e. 
Ring )  /\  (
q  e.  ( Base `  R )  /\  r  e.  ( Base `  R
) ) )  /\  ( x  =  q  /\  y  =  I
) )  ->  y  =  I )
361, 35, 11, 20, 22ovmpt2d 6415 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
q ( x  e.  ( Base `  R
) ,  y  e. 
{ I }  |->  y ) I )  =  I )
3734, 36eqtrd 2508 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
q ( .s `  M ) I )  =  I )
3832oveqd 6302 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
r ( .s `  M ) I )  =  ( r ( x  e.  ( Base `  R ) ,  y  e.  { I }  |->  y ) I ) )
39 simprr 756 . . . . . 6  |-  ( ( ( ( I  e.  V  /\  R  e. 
Ring )  /\  (
q  e.  ( Base `  R )  /\  r  e.  ( Base `  R
) ) )  /\  ( x  =  r  /\  y  =  I
) )  ->  y  =  I )
401, 39, 12, 20, 22ovmpt2d 6415 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
r ( x  e.  ( Base `  R
) ,  y  e. 
{ I }  |->  y ) I )  =  I )
4138, 40eqtrd 2508 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
r ( .s `  M ) I )  =  I )
4237, 41oveq12d 6303 . . 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 ) )
43 snex 4688 . . . . . . 7  |-  { <. <.
I ,  I >. ,  I >. }  e.  _V
445lmodplusg 14624 . . . . . . 7  |-  ( {
<. <. I ,  I >. ,  I >. }  e.  _V  ->  { <. <. I ,  I >. ,  I >. }  =  ( +g  `  M
) )
4543, 44mp1i 12 . . . . . 6  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  { <. <.
I ,  I >. ,  I >. }  =  ( +g  `  M ) )
4645eqcomd 2475 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  ( +g  `  M )  =  { <. <. I ,  I >. ,  I >. } )
4746oveqd 6302 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
I ( +g  `  M
) I )  =  ( I { <. <.
I ,  I >. ,  I >. } I ) )
48 df-ov 6288 . . . . 5  |-  ( I { <. <. I ,  I >. ,  I >. } I
)  =  ( {
<. <. I ,  I >. ,  I >. } `  <. I ,  I >. )
49 opex 4711 . . . . . . . 8  |-  <. I ,  I >.  e.  _V
5021, 49jctil 537 . . . . . . 7  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( <. I ,  I >.  e.  _V  /\  I  e.  V ) )
5150adantr 465 . . . . . 6  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  ( <. I ,  I >.  e. 
_V  /\  I  e.  V ) )
52 fvsng 6096 . . . . . 6  |-  ( (
<. I ,  I >.  e. 
_V  /\  I  e.  V )  ->  ( { <. <. I ,  I >. ,  I >. } `  <. I ,  I >. )  =  I )
5351, 52syl 16 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  ( { <. <. I ,  I >. ,  I >. } `  <. I ,  I >. )  =  I )
5448, 53syl5eq 2520 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
I { <. <. I ,  I >. ,  I >. } I )  =  I )
5547, 54eqtrd 2508 . . 3  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
I ( +g  `  M
) I )  =  I )
5642, 55eqtrd 2508 . 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 )
5723, 33, 563eqtr4d 2518 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 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    u. cun 3474   {csn 4027   {ctp 4031   <.cop 4033   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   ndxcnx 14490   Basecbs 14493   +g cplusg 14558  Scalarcsca 14561   .scvsca 14562   Ringcrg 17012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-plusg 14571  df-sca 14574  df-vsca 14575  df-mnd 15735  df-grp 15871  df-rng 17014
This theorem is referenced by:  lmod1  32391
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