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Theorem caonncan 6202
Description: Transfer nncan 9166-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 )
2 ffvelrn 5746 . . . . 5  |-  ( ( A : I --> S  /\  z  e.  I )  ->  ( A `  z
)  e.  S )
31, 2sylan 457 . . . 4  |-  ( (
ph  /\  z  e.  I )  ->  ( A `  z )  e.  S )
4 caonncan.b . . . . 5  |-  ( ph  ->  B : I --> S )
5 ffvelrn 5746 . . . . 5  |-  ( ( B : I --> S  /\  z  e.  I )  ->  ( B `  z
)  e.  S )
64, 5sylan 457 . . . 4  |-  ( (
ph  /\  z  e.  I )  ->  ( B `  z )  e.  S )
7 caonncan.z . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x M ( x M y ) )  =  y )
87ralrimivva 2711 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x M ( x M y ) )  =  y )
98adantr 451 . . . 4  |-  ( (
ph  /\  z  e.  I )  ->  A. x  e.  S  A. y  e.  S  ( x M ( x M y ) )  =  y )
10 id 19 . . . . . . 7  |-  ( x  =  ( A `  z )  ->  x  =  ( A `  z ) )
11 oveq1 5952 . . . . . . 7  |-  ( x  =  ( A `  z )  ->  (
x M y )  =  ( ( A `
 z ) M y ) )
1210, 11oveq12d 5963 . . . . . 6  |-  ( x  =  ( A `  z )  ->  (
x M ( x M y ) )  =  ( ( A `
 z ) M ( ( A `  z ) M y ) ) )
1312eqeq1d 2366 . . . . 5  |-  ( x  =  ( A `  z )  ->  (
( x M ( x M y ) )  =  y  <->  ( ( A `  z ) M ( ( A `
 z ) M y ) )  =  y ) )
14 oveq2 5953 . . . . . . 7  |-  ( y  =  ( B `  z )  ->  (
( A `  z
) M y )  =  ( ( A `
 z ) M ( B `  z
) ) )
1514oveq2d 5961 . . . . . 6  |-  ( y  =  ( B `  z )  ->  (
( A `  z
) M ( ( A `  z ) M y ) )  =  ( ( A `
 z ) M ( ( A `  z ) M ( B `  z ) ) ) )
16 id 19 . . . . . 6  |-  ( y  =  ( B `  z )  ->  y  =  ( B `  z ) )
1715, 16eqeq12d 2372 . . . . 5  |-  ( y  =  ( B `  z )  ->  (
( ( A `  z ) M ( ( A `  z
) M y ) )  =  y  <->  ( ( A `  z ) M ( ( A `
 z ) M ( B `  z
) ) )  =  ( B `  z
) ) )
1813, 17rspc2va 2967 . . . 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
) )
193, 6, 9, 18syl21anc 1181 . . 3  |-  ( (
ph  /\  z  e.  I )  ->  (
( A `  z
) M ( ( A `  z ) M ( B `  z ) ) )  =  ( B `  z ) )
2019mpteq2dva 4187 . 2  |-  ( ph  ->  ( z  e.  I  |->  ( ( A `  z ) M ( ( A `  z
) M ( B `
 z ) ) ) )  =  ( z  e.  I  |->  ( B `  z ) ) )
21 caonncan.i . . 3  |-  ( ph  ->  I  e.  V )
22 fvex 5622 . . . 4  |-  ( A `
 z )  e. 
_V
2322a1i 10 . . 3  |-  ( (
ph  /\  z  e.  I )  ->  ( A `  z )  e.  _V )
24 ovex 5970 . . . 4  |-  ( ( A `  z ) M ( B `  z ) )  e. 
_V
2524a1i 10 . . 3  |-  ( (
ph  /\  z  e.  I )  ->  (
( A `  z
) M ( B `
 z ) )  e.  _V )
261feqmptd 5658 . . 3  |-  ( ph  ->  A  =  ( z  e.  I  |->  ( A `
 z ) ) )
27 fvex 5622 . . . . 5  |-  ( B `
 z )  e. 
_V
2827a1i 10 . . . 4  |-  ( (
ph  /\  z  e.  I )  ->  ( B `  z )  e.  _V )
294feqmptd 5658 . . . 4  |-  ( ph  ->  B  =  ( z  e.  I  |->  ( B `
 z ) ) )
3021, 23, 28, 26, 29offval2 6182 . . 3  |-  ( ph  ->  ( A  o F M B )  =  ( z  e.  I  |->  ( ( A `  z ) M ( B `  z ) ) ) )
3121, 23, 25, 26, 30offval2 6182 . 2  |-  ( ph  ->  ( A  o F M ( A  o F M B ) )  =  ( z  e.  I  |->  ( ( A `
 z ) M ( ( A `  z ) M ( B `  z ) ) ) ) )
3220, 31, 293eqtr4d 2400 1  |-  ( ph  ->  ( A  o F M ( A  o F M B ) )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864    e. cmpt 4158   -->wf 5333   ` cfv 5337  (class class class)co 5945    o Fcof 6163
This theorem is referenced by:  psropprmul  16415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165
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