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Theorem lmod1lem3 33344
Description: Lemma 3 for lmod1 33347. (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 2455 . . 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 755 . . 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 753 . . . . . . 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 ) >. } )
54lmodsca 14855 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  =  (Scalar `  M )
)
65fveq2d 5852 . . . . . . 7  |-  ( R  e.  Ring  ->  ( +g  `  R )  =  ( +g  `  (Scalar `  M ) ) )
73, 6syl 16 . . . . . 6  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  ( +g  `  R )  =  ( +g  `  (Scalar `  M ) ) )
87eqcomd 2462 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  ( +g  `  (Scalar `  M
) )  =  ( +g  `  R ) )
98oveqd 6287 . . . 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 754 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  q  e.  ( Base `  R
) )
11 simprr 755 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  r  e.  ( Base `  R
) )
12 eqid 2454 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
13 eqid 2454 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
1412, 13ringacl 17421 . . . . 5  |-  ( ( R  e.  Ring  /\  q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
)  ->  ( q
( +g  `  R ) r )  e.  (
Base `  R )
)
153, 10, 11, 14syl3anc 1226 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
q ( +g  `  R
) r )  e.  ( Base `  R
) )
169, 15eqeltrd 2542 . . 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 4042 . . . . 5  |-  ( I  e.  V  ->  I  e.  { I } )
1817adantr 463 . . . 4  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  I  e.  { I } )
1918adantr 463 . . 3  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  I  e.  { I } )
20 simpl 455 . . . 4  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  I  e.  V )
2120adantr 463 . . 3  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  I  e.  V )
221, 2, 16, 19, 21ovmpt2d 6403 . 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 5858 . . . . . . 7  |-  ( Base `  R )  e.  _V
24 snex 4678 . . . . . . 7  |-  { I }  e.  _V
2523, 24pm3.2i 453 . . . . . 6  |-  ( (
Base `  R )  e.  _V  /\  { I }  e.  _V )
26 mpt2exga 6849 . . . . . 6  |-  ( ( ( Base `  R
)  e.  _V  /\  { I }  e.  _V )  ->  ( x  e.  ( Base `  R
) ,  y  e. 
{ I }  |->  y )  e.  _V )
2725, 26mp1i 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 )
284lmodvsca 14856 . . . . 5  |-  ( ( x  e.  ( Base `  R ) ,  y  e.  { I }  |->  y )  e.  _V  ->  ( x  e.  (
Base `  R ) ,  y  e.  { I }  |->  y )  =  ( .s `  M
) )
2927, 28syl 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 ) )
3029eqcomd 2462 . . 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 ) )
3130oveqd 6287 . 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 755 . . . . 5  |-  ( ( ( ( I  e.  V  /\  R  e. 
Ring )  /\  (
q  e.  ( Base `  R )  /\  r  e.  ( Base `  R
) ) )  /\  ( x  =  q  /\  y  =  I
) )  ->  y  =  I )
3330, 32, 10, 19, 19ovmpt2d 6403 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
q ( .s `  M ) I )  =  I )
34 simprr 755 . . . . 5  |-  ( ( ( ( I  e.  V  /\  R  e. 
Ring )  /\  (
q  e.  ( Base `  R )  /\  r  e.  ( Base `  R
) ) )  /\  ( x  =  r  /\  y  =  I
) )  ->  y  =  I )
3530, 34, 11, 19, 19ovmpt2d 6403 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
r ( .s `  M ) I )  =  I )
3633, 35oveq12d 6288 . . 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 4678 . . . . . 6  |-  { <. <.
I ,  I >. ,  I >. }  e.  _V
384lmodplusg 14854 . . . . . 6  |-  ( {
<. <. I ,  I >. ,  I >. }  e.  _V  ->  { <. <. I ,  I >. ,  I >. }  =  ( +g  `  M
) )
3937, 38mp1i 12 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  { <. <.
I ,  I >. ,  I >. }  =  ( +g  `  M ) )
4039eqcomd 2462 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  ( +g  `  M )  =  { <. <. I ,  I >. ,  I >. } )
4140oveqd 6287 . . 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 6273 . . . 4  |-  ( I { <. <. I ,  I >. ,  I >. } I
)  =  ( {
<. <. I ,  I >. ,  I >. } `  <. I ,  I >. )
43 opex 4701 . . . . . . 7  |-  <. I ,  I >.  e.  _V
4420, 43jctil 535 . . . . . 6  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( <. I ,  I >.  e.  _V  /\  I  e.  V ) )
4544adantr 463 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  ( <. I ,  I >.  e. 
_V  /\  I  e.  V ) )
46 fvsng 6081 . . . . 5  |-  ( (
<. I ,  I >.  e. 
_V  /\  I  e.  V )  ->  ( { <. <. I ,  I >. ,  I >. } `  <. I ,  I >. )  =  I )
4745, 46syl 16 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  ( { <. <. I ,  I >. ,  I >. } `  <. I ,  I >. )  =  I )
4842, 47syl5eq 2507 . . 3  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
I { <. <. I ,  I >. ,  I >. } I )  =  I )
4936, 41, 483eqtrd 2499 . 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 )
5022, 31, 493eqtr4d 2505 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 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    u. cun 3459   {csn 4016   {ctp 4020   <.cop 4022   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   ndxcnx 14713   Basecbs 14716   +g cplusg 14784  Scalarcsca 14787   .scvsca 14788   Ringcrg 17393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-plusg 14797  df-sca 14800  df-vsca 14801  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-ring 17395
This theorem is referenced by:  lmod1  33347
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