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Theorem caonncan 6357
Description: Transfer nncan 9634-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  oF M ( A  oF 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 5840 . . . 4  |-  ( (
ph  /\  z  e.  I )  ->  ( A `  z )  e.  S )
3 caonncan.b . . . . 5  |-  ( ph  ->  B : I --> S )
43ffvelrnda 5840 . . . 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 2806 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x M ( x M y ) )  =  y )
76adantr 462 . . . 4  |-  ( (
ph  /\  z  e.  I )  ->  A. x  e.  S  A. y  e.  S  ( x M ( x M y ) )  =  y )
8 id 22 . . . . . . 7  |-  ( x  =  ( A `  z )  ->  x  =  ( A `  z ) )
9 oveq1 6097 . . . . . . 7  |-  ( x  =  ( A `  z )  ->  (
x M y )  =  ( ( A `
 z ) M y ) )
108, 9oveq12d 6108 . . . . . 6  |-  ( x  =  ( A `  z )  ->  (
x M ( x M y ) )  =  ( ( A `
 z ) M ( ( A `  z ) M y ) ) )
1110eqeq1d 2449 . . . . 5  |-  ( x  =  ( A `  z )  ->  (
( x M ( x M y ) )  =  y  <->  ( ( A `  z ) M ( ( A `
 z ) M y ) )  =  y ) )
12 oveq2 6098 . . . . . . 7  |-  ( y  =  ( B `  z )  ->  (
( A `  z
) M y )  =  ( ( A `
 z ) M ( B `  z
) ) )
1312oveq2d 6106 . . . . . 6  |-  ( y  =  ( B `  z )  ->  (
( A `  z
) M ( ( A `  z ) M y ) )  =  ( ( A `
 z ) M ( ( A `  z ) M ( B `  z ) ) ) )
14 id 22 . . . . . 6  |-  ( y  =  ( B `  z )  ->  y  =  ( B `  z ) )
1513, 14eqeq12d 2455 . . . . 5  |-  ( y  =  ( B `  z )  ->  (
( ( A `  z ) M ( ( A `  z
) M y ) )  =  y  <->  ( ( A `  z ) M ( ( A `
 z ) M ( B `  z
) ) )  =  ( B `  z
) ) )
1611, 15rspc2va 3077 . . . 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 1212 . . 3  |-  ( (
ph  /\  z  e.  I )  ->  (
( A `  z
) M ( ( A `  z ) M ( B `  z ) ) )  =  ( B `  z ) )
1817mpteq2dva 4375 . 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 5698 . . . 4  |-  ( A `
 z )  e. 
_V
2120a1i 11 . . 3  |-  ( (
ph  /\  z  e.  I )  ->  ( A `  z )  e.  _V )
22 ovex 6115 . . . 4  |-  ( ( A `  z ) M ( B `  z ) )  e. 
_V
2322a1i 11 . . 3  |-  ( (
ph  /\  z  e.  I )  ->  (
( A `  z
) M ( B `
 z ) )  e.  _V )
241feqmptd 5741 . . 3  |-  ( ph  ->  A  =  ( z  e.  I  |->  ( A `
 z ) ) )
25 fvex 5698 . . . . 5  |-  ( B `
 z )  e. 
_V
2625a1i 11 . . . 4  |-  ( (
ph  /\  z  e.  I )  ->  ( B `  z )  e.  _V )
273feqmptd 5741 . . . 4  |-  ( ph  ->  B  =  ( z  e.  I  |->  ( B `
 z ) ) )
2819, 21, 26, 24, 27offval2 6335 . . 3  |-  ( ph  ->  ( A  oF M B )  =  ( z  e.  I  |->  ( ( A `  z ) M ( B `  z ) ) ) )
2919, 21, 23, 24, 28offval2 6335 . 2  |-  ( ph  ->  ( A  oF M ( A  oF M B ) )  =  ( z  e.  I  |->  ( ( A `  z ) M ( ( A `
 z ) M ( B `  z
) ) ) ) )
3018, 29, 273eqtr4d 2483 1  |-  ( ph  ->  ( A  oF M ( A  oF M B ) )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970    e. cmpt 4347   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319
This theorem is referenced by:  psropprmul  17668
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