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Theorem lmod1lem3 31164
Description: Lemma 3 for lmod1 31167. (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 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 14425 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  =  (Scalar `  M )
)
76fveq2d 5804 . . . . . . 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 2462 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  ( +g  `  (Scalar `  M
) )  =  ( +g  `  R ) )
109oveqd 6218 . . . 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 2454 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
14 eqid 2454 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
1513, 14rngacl 16796 . . . . 5  |-  ( ( R  e.  Ring  /\  q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
)  ->  ( q
( +g  `  R ) r )  e.  (
Base `  R )
)
164, 11, 12, 15syl3anc 1219 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
q ( +g  `  R
) r )  e.  ( Base `  R
) )
1710, 16eqeltrd 2542 . . 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 4012 . . . . 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 6329 . 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 5810 . . . . . . 7  |-  ( Base `  R )  e.  _V
25 snex 4642 . . . . . . 7  |-  { I }  e.  _V
2624, 25pm3.2i 455 . . . . . 6  |-  ( (
Base `  R )  e.  _V  /\  { I }  e.  _V )
27 eqid 2454 . . . . . . 7  |-  ( x  e.  ( Base `  R
) ,  y  e. 
{ I }  |->  y )  =  ( x  e.  ( Base `  R
) ,  y  e. 
{ I }  |->  y )
2827mpt2exg 6759 . . . . . 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 14426 . . . . 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 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 ) )
3332oveqd 6218 . 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 6218 . . . . 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 6329 . . . . 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 2495 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
q ( .s `  M ) I )  =  I )
3832oveqd 6218 . . . . 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 6329 . . . . 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 2495 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
r ( .s `  M ) I )  =  I )
4237, 41oveq12d 6219 . . 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 4642 . . . . . . 7  |-  { <. <.
I ,  I >. ,  I >. }  e.  _V
445lmodplusg 14424 . . . . . . 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 2462 . . . . 5  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  ( +g  `  M )  =  { <. <. I ,  I >. ,  I >. } )
4746oveqd 6218 . . . 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 6204 . . . . 5  |-  ( I { <. <. I ,  I >. ,  I >. } I
)  =  ( {
<. <. I ,  I >. ,  I >. } `  <. I ,  I >. )
49 opex 4665 . . . . . . . 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 6022 . . . . . 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 2507 . . . 4  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
I { <. <. I ,  I >. ,  I >. } I )  =  I )
5547, 54eqtrd 2495 . . 3  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
I ( +g  `  M
) I )  =  I )
5642, 55eqtrd 2495 . 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 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 369    = wceq 1370    e. wcel 1758   _Vcvv 3078    u. cun 3435   {csn 3986   {ctp 3990   <.cop 3992   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203   ndxcnx 14290   Basecbs 14293   +g cplusg 14358  Scalarcsca 14361   .scvsca 14362   Ringcrg 16769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-plusg 14371  df-sca 14374  df-vsca 14375  df-mnd 15535  df-grp 15665  df-rng 16771
This theorem is referenced by:  lmod1  31167
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