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Theorem slesolvec 19473
Description: Every solution of a system of linear equations represented by a matrix and a vector is a vector. (Contributed by AV, 10-Feb-2019.) (Revised by AV, 27-Feb-2019.)
Hypotheses
Ref Expression
slesolex.a  |-  A  =  ( N Mat  R )
slesolex.b  |-  B  =  ( Base `  A
)
slesolex.v  |-  V  =  ( ( Base `  R
)  ^m  N )
slesolex.x  |-  .x.  =  ( R maVecMul  <. N ,  N >. )
Assertion
Ref Expression
slesolvec  |-  ( ( ( N  =/=  (/)  /\  R  e.  Ring )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( ( X  .x.  Z )  =  Y  ->  Z  e.  V ) )

Proof of Theorem slesolvec
StepHypRef Expression
1 slesolex.a . . . . . . 7  |-  A  =  ( N Mat  R )
2 slesolex.b . . . . . . 7  |-  B  =  ( Base `  A
)
31, 2matrcl 19206 . . . . . 6  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
43simpld 457 . . . . 5  |-  ( X  e.  B  ->  N  e.  Fin )
5 simpr 459 . . . . . . . 8  |-  ( ( N  =/=  (/)  /\  N  e.  Fin )  ->  N  e.  Fin )
6 simpl 455 . . . . . . . 8  |-  ( ( N  =/=  (/)  /\  N  e.  Fin )  ->  N  =/=  (/) )
75, 5, 63jca 1177 . . . . . . 7  |-  ( ( N  =/=  (/)  /\  N  e.  Fin )  ->  ( N  e.  Fin  /\  N  e.  Fin  /\  N  =/=  (/) ) )
87ex 432 . . . . . 6  |-  ( N  =/=  (/)  ->  ( N  e.  Fin  ->  ( N  e.  Fin  /\  N  e. 
Fin  /\  N  =/=  (/) ) ) )
98adantr 463 . . . . 5  |-  ( ( N  =/=  (/)  /\  R  e.  Ring )  ->  ( N  e.  Fin  ->  ( N  e.  Fin  /\  N  e.  Fin  /\  N  =/=  (/) ) ) )
104, 9syl5com 28 . . . 4  |-  ( X  e.  B  ->  (
( N  =/=  (/)  /\  R  e.  Ring )  ->  ( N  e.  Fin  /\  N  e.  Fin  /\  N  =/=  (/) ) ) )
1110adantr 463 . . 3  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  ( ( N  =/=  (/)  /\  R  e.  Ring )  ->  ( N  e. 
Fin  /\  N  e.  Fin  /\  N  =/=  (/) ) ) )
1211impcom 428 . 2  |-  ( ( ( N  =/=  (/)  /\  R  e.  Ring )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( N  e.  Fin  /\  N  e. 
Fin  /\  N  =/=  (/) ) )
13 simpr 459 . . 3  |-  ( ( N  =/=  (/)  /\  R  e.  Ring )  ->  R  e.  Ring )
14 simpr 459 . . 3  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  Y  e.  V )
1513, 14anim12i 564 . 2  |-  ( ( ( N  =/=  (/)  /\  R  e.  Ring )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( R  e.  Ring  /\  Y  e.  V ) )
16 eqid 2402 . . 3  |-  ( Base `  R )  =  (
Base `  R )
17 eqid 2402 . . 3  |-  ( (
Base `  R )  ^m  ( N  X.  N
) )  =  ( ( Base `  R
)  ^m  ( N  X.  N ) )
18 slesolex.v . . 3  |-  V  =  ( ( Base `  R
)  ^m  N )
19 slesolex.x . . 3  |-  .x.  =  ( R maVecMul  <. N ,  N >. )
2016, 17, 18, 19, 18mavmulsolcl 19345 . 2  |-  ( ( ( N  e.  Fin  /\  N  e.  Fin  /\  N  =/=  (/) )  /\  ( R  e.  Ring  /\  Y  e.  V ) )  -> 
( ( X  .x.  Z )  =  Y  ->  Z  e.  V
) )
2112, 15, 20syl2anc 659 1  |-  ( ( ( N  =/=  (/)  /\  R  e.  Ring )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( ( X  .x.  Z )  =  Y  ->  Z  e.  V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   _Vcvv 3059   (/)c0 3738   <.cop 3978    X. cxp 4821   ` cfv 5569  (class class class)co 6278    ^m cmap 7457   Fincfn 7554   Basecbs 14841   Ringcrg 17518   Mat cmat 19201   maVecMul cmvmul 19334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-map 7459  df-slot 14845  df-base 14846  df-mat 19202  df-mvmul 19335
This theorem is referenced by:  slesolinv  19474  cramerimplem3  19479  cramerimp  19480  cramer  19485
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