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Theorem caonncan 6115
Description: Transfer nncan 9076-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 5663 . . . . 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 5663 . . . . 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 2635 . . . . 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 5865 . . . . . . 7  |-  ( x  =  ( A `  z )  ->  (
x M y )  =  ( ( A `
 z ) M y ) )
1210, 11oveq12d 5876 . . . . . 6  |-  ( x  =  ( A `  z )  ->  (
x M ( x M y ) )  =  ( ( A `
 z ) M ( ( A `  z ) M y ) ) )
1312eqeq1d 2291 . . . . 5  |-  ( x  =  ( A `  z )  ->  (
( x M ( x M y ) )  =  y  <->  ( ( A `  z ) M ( ( A `
 z ) M y ) )  =  y ) )
14 oveq2 5866 . . . . . . 7  |-  ( y  =  ( B `  z )  ->  (
( A `  z
) M y )  =  ( ( A `
 z ) M ( B `  z
) ) )
1514oveq2d 5874 . . . . . 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 2297 . . . . 5  |-  ( y  =  ( B `  z )  ->  (
( ( A `  z ) M ( ( A `  z
) M y ) )  =  y  <->  ( ( A `  z ) M ( ( A `
 z ) M ( B `  z
) ) )  =  ( B `  z
) ) )
1813, 17rspc2va 2891 . . . 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 4106 . 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 5539 . . . 4  |-  ( A `
 z )  e. 
_V
2322a1i 10 . . 3  |-  ( (
ph  /\  z  e.  I )  ->  ( A `  z )  e.  _V )
24 ovex 5883 . . . 4  |-  ( ( A `  z ) M ( B `  z ) )  e. 
_V
2524a1i 10 . . 3  |-  ( (
ph  /\  z  e.  I )  ->  (
( A `  z
) M ( B `
 z ) )  e.  _V )
261feqmptd 5575 . . 3  |-  ( ph  ->  A  =  ( z  e.  I  |->  ( A `
 z ) ) )
27 fvex 5539 . . . . 5  |-  ( B `
 z )  e. 
_V
2827a1i 10 . . . 4  |-  ( (
ph  /\  z  e.  I )  ->  ( B `  z )  e.  _V )
294feqmptd 5575 . . . 4  |-  ( ph  ->  B  =  ( z  e.  I  |->  ( B `
 z ) ) )
3021, 23, 28, 26, 29offval2 6095 . . 3  |-  ( ph  ->  ( A  o F M B )  =  ( z  e.  I  |->  ( ( A `  z ) M ( B `  z ) ) ) )
3121, 23, 25, 26, 30offval2 6095 . 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 2325 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 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076
This theorem is referenced by:  psropprmul  16316
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078
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