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Theorem slesolvec 18627
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 18447 . . . . . 6  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
43simpld 459 . . . . 5  |-  ( X  e.  B  ->  N  e.  Fin )
5 simpr 461 . . . . . . . 8  |-  ( ( N  =/=  (/)  /\  N  e.  Fin )  ->  N  e.  Fin )
6 simpl 457 . . . . . . . 8  |-  ( ( N  =/=  (/)  /\  N  e.  Fin )  ->  N  =/=  (/) )
75, 5, 63jca 1168 . . . . . . 7  |-  ( ( N  =/=  (/)  /\  N  e.  Fin )  ->  ( N  e.  Fin  /\  N  e.  Fin  /\  N  =/=  (/) ) )
87ex 434 . . . . . 6  |-  ( N  =/=  (/)  ->  ( N  e.  Fin  ->  ( N  e.  Fin  /\  N  e. 
Fin  /\  N  =/=  (/) ) ) )
98adantr 465 . . . . 5  |-  ( ( N  =/=  (/)  /\  R  e.  Ring )  ->  ( N  e.  Fin  ->  ( N  e.  Fin  /\  N  e.  Fin  /\  N  =/=  (/) ) ) )
104, 9syl5com 30 . . . 4  |-  ( X  e.  B  ->  (
( N  =/=  (/)  /\  R  e.  Ring )  ->  ( N  e.  Fin  /\  N  e.  Fin  /\  N  =/=  (/) ) ) )
1110adantr 465 . . 3  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  ( ( N  =/=  (/)  /\  R  e.  Ring )  ->  ( N  e. 
Fin  /\  N  e.  Fin  /\  N  =/=  (/) ) ) )
1211impcom 430 . 2  |-  ( ( ( N  =/=  (/)  /\  R  e.  Ring )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( N  e.  Fin  /\  N  e. 
Fin  /\  N  =/=  (/) ) )
13 simpr 461 . . 3  |-  ( ( N  =/=  (/)  /\  R  e.  Ring )  ->  R  e.  Ring )
14 simpr 461 . . 3  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  Y  e.  V )
1513, 14anim12i 566 . 2  |-  ( ( ( N  =/=  (/)  /\  R  e.  Ring )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( R  e.  Ring  /\  Y  e.  V ) )
16 eqid 2454 . . 3  |-  ( Base `  R )  =  (
Base `  R )
17 eqid 2454 . . 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 18499 . 2  |-  ( ( ( N  e.  Fin  /\  N  e.  Fin  /\  N  =/=  (/) )  /\  ( R  e.  Ring  /\  Y  e.  V ) )  -> 
( ( X  .x.  Z )  =  Y  ->  Z  e.  V
) )
2112, 15, 20syl2anc 661 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   _Vcvv 3078   (/)c0 3748   <.cop 3994    X. cxp 4949   ` cfv 5529  (class class class)co 6203    ^m cmap 7327   Fincfn 7423   Basecbs 14296   Ringcrg 16778   Mat cmat 18415   maVecMul cmvmul 18488
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 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-map 7329  df-slot 14300  df-base 14301  df-mat 18417  df-mvmul 18489
This theorem is referenced by:  slesolinv  18628  cramerimplem3  18633  cramerimp  18634  cramer  18639
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