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Theorem genpassg 7477
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 7475 . 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 7476 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 2nd `  ( ( A F B ) F C ) )  =  ( 2nd `  ( A F ( B F C ) ) ) )
84caovcl 6005 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  e.  P. )
94caovcl 6005 . . . . 5  |-  ( ( ( A F B )  e.  P.  /\  C  e.  P. )  ->  ( ( A F B ) F C )  e.  P. )
108, 9sylan 281 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  C  e.  P. )  ->  ( ( A F B ) F C )  e.  P. )
11103impa 1189 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( A F B ) F C )  e.  P. )
124caovcl 6005 . . . . 5  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B F C )  e.  P. )
134caovcl 6005 . . . . 5  |-  ( ( A  e.  P.  /\  ( B F C )  e.  P. )  -> 
( A F ( B F C ) )  e.  P. )
1412, 13sylan2 284 . . . 4  |-  ( ( A  e.  P.  /\  ( B  e.  P.  /\  C  e.  P. )
)  ->  ( A F ( B F C ) )  e. 
P. )
15143impb 1194 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A F ( B F C ) )  e. 
P. )
16 preqlu 7423 . . 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 409 . 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 939 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 973    = wceq 1348    e. wcel 2141   E.wrex 2449   {crab 2452   <.cop 3584    X. cxp 4607   dom cdm 4609   ` cfv 5196  (class class class)co 5851    e. cmpo 5853   1stc1st 6115   2ndc2nd 6116   Q.cnq 7231   P.cnp 7242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-qs 6516  df-ni 7255  df-nqqs 7299  df-inp 7417
This theorem is referenced by:  addassprg  7530  mulassprg  7532
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