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Theorem caonncan 6301
Description: Transfer nncan 9286-shaped laws to vectors of numbers. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Hypotheses
Ref Expression
caonncan.i  |-  ( ph  ->  I  e.  V )
caonncan.a  |-  ( ph  ->  A : I --> S )
caonncan.b  |-  ( ph  ->  B : I --> S )
caonncan.z  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x M ( x M y ) )  =  y )
Assertion
Ref Expression
caonncan  |-  ( ph  ->  ( A  o F M ( A  o F M B ) )  =  B )
Distinct variable groups:    ph, x, y   
x, A, y    y, B    x, M, y    x, S, y
Allowed substitution hints:    B( x)    I( x, y)    V( x, y)

Proof of Theorem caonncan
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 caonncan.a . . . . 5  |-  ( ph  ->  A : I --> S )
21ffvelrnda 5829 . . . 4  |-  ( (
ph  /\  z  e.  I )  ->  ( A `  z )  e.  S )
3 caonncan.b . . . . 5  |-  ( ph  ->  B : I --> S )
43ffvelrnda 5829 . . . 4  |-  ( (
ph  /\  z  e.  I )  ->  ( B `  z )  e.  S )
5 caonncan.z . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x M ( x M y ) )  =  y )
65ralrimivva 2758 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x M ( x M y ) )  =  y )
76adantr 452 . . . 4  |-  ( (
ph  /\  z  e.  I )  ->  A. x  e.  S  A. y  e.  S  ( x M ( x M y ) )  =  y )
8 id 20 . . . . . . 7  |-  ( x  =  ( A `  z )  ->  x  =  ( A `  z ) )
9 oveq1 6047 . . . . . . 7  |-  ( x  =  ( A `  z )  ->  (
x M y )  =  ( ( A `
 z ) M y ) )
108, 9oveq12d 6058 . . . . . 6  |-  ( x  =  ( A `  z )  ->  (
x M ( x M y ) )  =  ( ( A `
 z ) M ( ( A `  z ) M y ) ) )
1110eqeq1d 2412 . . . . 5  |-  ( x  =  ( A `  z )  ->  (
( x M ( x M y ) )  =  y  <->  ( ( A `  z ) M ( ( A `
 z ) M y ) )  =  y ) )
12 oveq2 6048 . . . . . . 7  |-  ( y  =  ( B `  z )  ->  (
( A `  z
) M y )  =  ( ( A `
 z ) M ( B `  z
) ) )
1312oveq2d 6056 . . . . . 6  |-  ( y  =  ( B `  z )  ->  (
( A `  z
) M ( ( A `  z ) M y ) )  =  ( ( A `
 z ) M ( ( A `  z ) M ( B `  z ) ) ) )
14 id 20 . . . . . 6  |-  ( y  =  ( B `  z )  ->  y  =  ( B `  z ) )
1513, 14eqeq12d 2418 . . . . 5  |-  ( y  =  ( B `  z )  ->  (
( ( A `  z ) M ( ( A `  z
) M y ) )  =  y  <->  ( ( A `  z ) M ( ( A `
 z ) M ( B `  z
) ) )  =  ( B `  z
) ) )
1611, 15rspc2va 3019 . . . 4  |-  ( ( ( ( A `  z )  e.  S  /\  ( B `  z
)  e.  S )  /\  A. x  e.  S  A. y  e.  S  ( x M ( x M y ) )  =  y )  ->  ( ( A `  z ) M ( ( A `
 z ) M ( B `  z
) ) )  =  ( B `  z
) )
172, 4, 7, 16syl21anc 1183 . . 3  |-  ( (
ph  /\  z  e.  I )  ->  (
( A `  z
) M ( ( A `  z ) M ( B `  z ) ) )  =  ( B `  z ) )
1817mpteq2dva 4255 . 2  |-  ( ph  ->  ( z  e.  I  |->  ( ( A `  z ) M ( ( A `  z
) M ( B `
 z ) ) ) )  =  ( z  e.  I  |->  ( B `  z ) ) )
19 caonncan.i . . 3  |-  ( ph  ->  I  e.  V )
20 fvex 5701 . . . 4  |-  ( A `
 z )  e. 
_V
2120a1i 11 . . 3  |-  ( (
ph  /\  z  e.  I )  ->  ( A `  z )  e.  _V )
22 ovex 6065 . . . 4  |-  ( ( A `  z ) M ( B `  z ) )  e. 
_V
2322a1i 11 . . 3  |-  ( (
ph  /\  z  e.  I )  ->  (
( A `  z
) M ( B `
 z ) )  e.  _V )
241feqmptd 5738 . . 3  |-  ( ph  ->  A  =  ( z  e.  I  |->  ( A `
 z ) ) )
25 fvex 5701 . . . . 5  |-  ( B `
 z )  e. 
_V
2625a1i 11 . . . 4  |-  ( (
ph  /\  z  e.  I )  ->  ( B `  z )  e.  _V )
273feqmptd 5738 . . . 4  |-  ( ph  ->  B  =  ( z  e.  I  |->  ( B `
 z ) ) )
2819, 21, 26, 24, 27offval2 6281 . . 3  |-  ( ph  ->  ( A  o F M B )  =  ( z  e.  I  |->  ( ( A `  z ) M ( B `  z ) ) ) )
2919, 21, 23, 24, 28offval2 6281 . 2  |-  ( ph  ->  ( A  o F M ( A  o F M B ) )  =  ( z  e.  I  |->  ( ( A `
 z ) M ( ( A `  z ) M ( B `  z ) ) ) ) )
3018, 29, 273eqtr4d 2446 1  |-  ( ph  ->  ( A  o F M ( A  o F M B ) )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    e. cmpt 4226   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262
This theorem is referenced by:  psropprmul  16587
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264
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