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Theorem lmod1lem4 32049
Description: Lemma 4 for lmod1 32051. (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
lmod1lem4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
( q ( .r
`  (Scalar `  M )
) r ) ( .s `  M ) I )  =  ( q ( .s `  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 lmod1lem4
StepHypRef Expression
1 fvex 5869 . . . . . . 7  |-  ( Base `  R )  e.  _V
2 snex 4683 . . . . . . 7  |-  { I }  e.  _V
31, 2pm3.2i 455 . . . . . 6  |-  ( (
Base `  R )  e.  _V  /\  { I }  e.  _V )
43a1i 11 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
( Base `  R )  e.  _V  /\  { I }  e.  _V )
)
5 mpt2exga 6851 . . . . 5  |-  ( ( ( Base `  R
)  e.  _V  /\  { I }  e.  _V )  ->  ( x  e.  ( Base `  R
) ,  y  e. 
{ I }  |->  y )  e.  _V )
6 lmod1.m . . . . . 6  |-  M  =  ( { <. ( Base `  ndx ) ,  { I } >. , 
<. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
>. ,  <. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  ( x  e.  (
Base `  R ) ,  y  e.  { I }  |->  y ) >. } )
76lmodvsca 14614 . . . . 5  |-  ( ( x  e.  ( Base `  R ) ,  y  e.  { I }  |->  y )  e.  _V  ->  ( x  e.  (
Base `  R ) ,  y  e.  { I }  |->  y )  =  ( .s `  M
) )
84, 5, 73syl 20 . . . 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 ) )
98eqcomd 2470 . . 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 ) )
10 simprr 756 . . 3  |-  ( ( ( ( I  e.  V  /\  R  e. 
Ring )  /\  (
q  e.  ( Base `  R )  /\  r  e.  ( Base `  R
) ) )  /\  ( x  =  (
q ( .r `  (Scalar `  M ) ) r )  /\  y  =  I ) )  -> 
y  =  I )
11 simplr 754 . . . . . . 7  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  R  e.  Ring )
126lmodsca 14613 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  =  (Scalar `  M )
)
1312fveq2d 5863 . . . . . . 7  |-  ( R  e.  Ring  ->  ( .r
`  R )  =  ( .r `  (Scalar `  M ) ) )
1411, 13syl 16 . . . . . 6  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  ( .r `  R )  =  ( .r `  (Scalar `  M ) ) )
1514eqcomd 2470 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  ( .r `  (Scalar `  M
) )  =  ( .r `  R ) )
1615oveqd 6294 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
q ( .r `  (Scalar `  M ) ) r )  =  ( q ( .r `  R ) r ) )
17 simprl 755 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  q  e.  ( Base `  R
) )
18 simprr 756 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  r  e.  ( Base `  R
) )
19 eqid 2462 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
20 eqid 2462 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
2119, 20rngcl 16994 . . . . 5  |-  ( ( R  e.  Ring  /\  q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
)  ->  ( q
( .r `  R
) r )  e.  ( Base `  R
) )
2211, 17, 18, 21syl3anc 1223 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
q ( .r `  R ) r )  e.  ( Base `  R
) )
2316, 22eqeltrd 2550 . . 3  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
q ( .r `  (Scalar `  M ) ) r )  e.  (
Base `  R )
)
24 snidg 4048 . . . 4  |-  ( I  e.  V  ->  I  e.  { I } )
2524ad2antrr 725 . . 3  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  I  e.  { I } )
269, 10, 23, 25, 25ovmpt2d 6407 . 2  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
( q ( .r
`  (Scalar `  M )
) r ) ( .s `  M ) I )  =  I )
27 simprr 756 . . . . 5  |-  ( ( ( ( I  e.  V  /\  R  e. 
Ring )  /\  (
q  e.  ( Base `  R )  /\  r  e.  ( Base `  R
) ) )  /\  ( x  =  r  /\  y  =  I
) )  ->  y  =  I )
289, 27, 18, 25, 25ovmpt2d 6407 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
r ( .s `  M ) I )  =  I )
2928oveq2d 6293 . . 3  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
q ( .s `  M ) ( r ( .s `  M
) I ) )  =  ( q ( .s `  M ) I ) )
30 simprr 756 . . . 4  |-  ( ( ( ( I  e.  V  /\  R  e. 
Ring )  /\  (
q  e.  ( Base `  R )  /\  r  e.  ( Base `  R
) ) )  /\  ( x  =  q  /\  y  =  I
) )  ->  y  =  I )
319, 30, 17, 25, 25ovmpt2d 6407 . . 3  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
q ( .s `  M ) I )  =  I )
3229, 31eqtrd 2503 . 2  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
q ( .s `  M ) ( r ( .s `  M
) I ) )  =  I )
3326, 32eqtr4d 2506 1  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
( q ( .r
`  (Scalar `  M )
) r ) ( .s `  M ) I )  =  ( q ( .s `  M ) ( r ( .s `  M
) I ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3108    u. cun 3469   {csn 4022   {ctp 4026   <.cop 4028   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   ndxcnx 14478   Basecbs 14481   +g cplusg 14546   .rcmulr 14547  Scalarcsca 14549   .scvsca 14550   Ringcrg 16981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-plusg 14559  df-sca 14562  df-vsca 14563  df-mnd 15723  df-mgp 16927  df-rng 16983
This theorem is referenced by:  lmod1  32051
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