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Theorem caonncan 6345
Description: Transfer nncan 9335-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 5873 . . . 4  |-  ( (
ph  /\  z  e.  I )  ->  ( A `  z )  e.  S )
3 caonncan.b . . . . 5  |-  ( ph  ->  B : I --> S )
43ffvelrnda 5873 . . . 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 2800 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x M ( x M y ) )  =  y )
76adantr 453 . . . 4  |-  ( (
ph  /\  z  e.  I )  ->  A. x  e.  S  A. y  e.  S  ( x M ( x M y ) )  =  y )
8 id 21 . . . . . . 7  |-  ( x  =  ( A `  z )  ->  x  =  ( A `  z ) )
9 oveq1 6091 . . . . . . 7  |-  ( x  =  ( A `  z )  ->  (
x M y )  =  ( ( A `
 z ) M y ) )
108, 9oveq12d 6102 . . . . . 6  |-  ( x  =  ( A `  z )  ->  (
x M ( x M y ) )  =  ( ( A `
 z ) M ( ( A `  z ) M y ) ) )
1110eqeq1d 2446 . . . . 5  |-  ( x  =  ( A `  z )  ->  (
( x M ( x M y ) )  =  y  <->  ( ( A `  z ) M ( ( A `
 z ) M y ) )  =  y ) )
12 oveq2 6092 . . . . . . 7  |-  ( y  =  ( B `  z )  ->  (
( A `  z
) M y )  =  ( ( A `
 z ) M ( B `  z
) ) )
1312oveq2d 6100 . . . . . 6  |-  ( y  =  ( B `  z )  ->  (
( A `  z
) M ( ( A `  z ) M y ) )  =  ( ( A `
 z ) M ( ( A `  z ) M ( B `  z ) ) ) )
14 id 21 . . . . . 6  |-  ( y  =  ( B `  z )  ->  y  =  ( B `  z ) )
1513, 14eqeq12d 2452 . . . . 5  |-  ( y  =  ( B `  z )  ->  (
( ( A `  z ) M ( ( A `  z
) M y ) )  =  y  <->  ( ( A `  z ) M ( ( A `
 z ) M ( B `  z
) ) )  =  ( B `  z
) ) )
1611, 15rspc2va 3061 . . . 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 1184 . . 3  |-  ( (
ph  /\  z  e.  I )  ->  (
( A `  z
) M ( ( A `  z ) M ( B `  z ) ) )  =  ( B `  z ) )
1817mpteq2dva 4298 . 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 5745 . . . 4  |-  ( A `
 z )  e. 
_V
2120a1i 11 . . 3  |-  ( (
ph  /\  z  e.  I )  ->  ( A `  z )  e.  _V )
22 ovex 6109 . . . 4  |-  ( ( A `  z ) M ( B `  z ) )  e. 
_V
2322a1i 11 . . 3  |-  ( (
ph  /\  z  e.  I )  ->  (
( A `  z
) M ( B `
 z ) )  e.  _V )
241feqmptd 5782 . . 3  |-  ( ph  ->  A  =  ( z  e.  I  |->  ( A `
 z ) ) )
25 fvex 5745 . . . . 5  |-  ( B `
 z )  e. 
_V
2625a1i 11 . . . 4  |-  ( (
ph  /\  z  e.  I )  ->  ( B `  z )  e.  _V )
273feqmptd 5782 . . . 4  |-  ( ph  ->  B  =  ( z  e.  I  |->  ( B `
 z ) ) )
2819, 21, 26, 24, 27offval2 6325 . . 3  |-  ( ph  ->  ( A  o F M B )  =  ( z  e.  I  |->  ( ( A `  z ) M ( B `  z ) ) ) )
2919, 21, 23, 24, 28offval2 6325 . 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 2480 1  |-  ( ph  ->  ( A  o F M ( A  o F M B ) )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    e. cmpt 4269   -->wf 5453   ` cfv 5457  (class class class)co 6084    o Fcof 6306
This theorem is referenced by:  psropprmul  16637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308
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