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Theorem genpassg 7146
Description: Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-Dec-2019.)
Hypotheses
Ref Expression
genpelvl.1  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v
)  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) ) } >. )
genpelvl.2  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
genpassg.4  |-  dom  F  =  ( P.  X.  P. )
genpassg.5  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f F g )  e.  P. )
genpassg.6  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f G g ) G h )  =  ( f G ( g G h ) ) )
Assertion
Ref Expression
genpassg  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( A F B ) F C )  =  ( A F ( B F C ) ) )
Distinct variable groups:    x, y, z, f, g, h, w, v, A    x, B, y, z, f, g, h, w, v    x, G, y, z, f, g, h, w, v    f, F, g    C, f, g, h, v, w, x, y, z    h, F, v, w, x, y, z

Proof of Theorem genpassg
StepHypRef Expression
1 genpelvl.1 . . 3  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v
)  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) ) } >. )
2 genpelvl.2 . . 3  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
3 genpassg.4 . . 3  |-  dom  F  =  ( P.  X.  P. )
4 genpassg.5 . . 3  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f F g )  e.  P. )
5 genpassg.6 . . 3  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f G g ) G h )  =  ( f G ( g G h ) ) )
61, 2, 3, 4, 5genpassl 7144 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 1st `  ( ( A F B ) F C ) )  =  ( 1st `  ( A F ( B F C ) ) ) )
71, 2, 3, 4, 5genpassu 7145 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 2nd `  ( ( A F B ) F C ) )  =  ( 2nd `  ( A F ( B F C ) ) ) )
84caovcl 5813 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  e.  P. )
94caovcl 5813 . . . . 5  |-  ( ( ( A F B )  e.  P.  /\  C  e.  P. )  ->  ( ( A F B ) F C )  e.  P. )
108, 9sylan 278 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  C  e.  P. )  ->  ( ( A F B ) F C )  e.  P. )
11103impa 1139 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( A F B ) F C )  e.  P. )
124caovcl 5813 . . . . 5  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B F C )  e.  P. )
134caovcl 5813 . . . . 5  |-  ( ( A  e.  P.  /\  ( B F C )  e.  P. )  -> 
( A F ( B F C ) )  e.  P. )
1412, 13sylan2 281 . . . 4  |-  ( ( A  e.  P.  /\  ( B  e.  P.  /\  C  e.  P. )
)  ->  ( A F ( B F C ) )  e. 
P. )
15143impb 1140 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A F ( B F C ) )  e. 
P. )
16 preqlu 7092 . . 3  |-  ( ( ( ( A F B ) F C )  e.  P.  /\  ( A F ( B F C ) )  e.  P. )  -> 
( ( ( A F B ) F C )  =  ( A F ( B F C ) )  <-> 
( ( 1st `  (
( A F B ) F C ) )  =  ( 1st `  ( A F ( B F C ) ) )  /\  ( 2nd `  ( ( A F B ) F C ) )  =  ( 2nd `  ( A F ( B F C ) ) ) ) ) )
1711, 15, 16syl2anc 404 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( ( A F B ) F C )  =  ( A F ( B F C ) )  <->  ( ( 1st `  ( ( A F B ) F C ) )  =  ( 1st `  ( A F ( B F C ) ) )  /\  ( 2nd `  (
( A F B ) F C ) )  =  ( 2nd `  ( A F ( B F C ) ) ) ) ) )
186, 7, 17mpbir2and 891 1  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( A F B ) F C )  =  ( A F ( B F C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 925    = wceq 1290    e. wcel 1439   E.wrex 2361   {crab 2364   <.cop 3453    X. cxp 4450   dom cdm 4452   ` cfv 5028  (class class class)co 5666    |-> cmpt2 5668   1stc1st 5923   2ndc2nd 5924   Q.cnq 6900   P.cnp 6911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-qs 6312  df-ni 6924  df-nqqs 6968  df-inp 7086
This theorem is referenced by:  addassprg  7199  mulassprg  7201
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