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Theorem caovdilemd 5744
Description: Lemma used by real number construction. (Contributed by Jim Kingdon, 16-Sep-2019.)
Hypotheses
Ref Expression
caovdilemd.com  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x G y )  =  ( y G x ) )
caovdilemd.distr  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y ) G z )  =  ( ( x G z ) F ( y G z ) ) )
caovdilemd.ass  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x G y ) G z )  =  ( x G ( y G z ) ) )
caovdilemd.cl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x G y )  e.  S )
caovdilemd.a  |-  ( ph  ->  A  e.  S )
caovdilemd.b  |-  ( ph  ->  B  e.  S )
caovdilemd.c  |-  ( ph  ->  C  e.  S )
caovdilemd.d  |-  ( ph  ->  D  e.  S )
caovdilemd.h  |-  ( ph  ->  H  e.  S )
Assertion
Ref Expression
caovdilemd  |-  ( ph  ->  ( ( ( A G C ) F ( B G D ) ) G H )  =  ( ( A G ( C G H ) ) F ( B G ( D G H ) ) ) )
Distinct variable groups:    x, y, z, A    x, B, y, z    x, C, y, z    x, D, y, z    ph, x, y, z   
x, F, y, z   
x, G, y, z   
x, H, y, z   
x, S, y, z

Proof of Theorem caovdilemd
StepHypRef Expression
1 caovdilemd.distr . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y ) G z )  =  ( ( x G z ) F ( y G z ) ) )
2 caovdilemd.cl . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x G y )  e.  S )
3 caovdilemd.a . . . 4  |-  ( ph  ->  A  e.  S )
4 caovdilemd.c . . . 4  |-  ( ph  ->  C  e.  S )
52, 3, 4caovcld 5706 . . 3  |-  ( ph  ->  ( A G C )  e.  S )
6 caovdilemd.b . . . 4  |-  ( ph  ->  B  e.  S )
7 caovdilemd.d . . . 4  |-  ( ph  ->  D  e.  S )
82, 6, 7caovcld 5706 . . 3  |-  ( ph  ->  ( B G D )  e.  S )
9 caovdilemd.h . . 3  |-  ( ph  ->  H  e.  S )
101, 5, 8, 9caovdird 5731 . 2  |-  ( ph  ->  ( ( ( A G C ) F ( B G D ) ) G H )  =  ( ( ( A G C ) G H ) F ( ( B G D ) G H ) ) )
11 caovdilemd.ass . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x G y ) G z )  =  ( x G ( y G z ) ) )
1211, 3, 4, 9caovassd 5712 . . 3  |-  ( ph  ->  ( ( A G C ) G H )  =  ( A G ( C G H ) ) )
1311, 6, 7, 9caovassd 5712 . . 3  |-  ( ph  ->  ( ( B G D ) G H )  =  ( B G ( D G H ) ) )
1412, 13oveq12d 5582 . 2  |-  ( ph  ->  ( ( ( A G C ) G H ) F ( ( B G D ) G H ) )  =  ( ( A G ( C G H ) ) F ( B G ( D G H ) ) ) )
1510, 14eqtrd 2115 1  |-  ( ph  ->  ( ( ( A G C ) F ( B G D ) ) G H )  =  ( ( A G ( C G H ) ) F ( B G ( D G H ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    = wceq 1285    e. wcel 1434  (class class class)co 5564
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-iota 4917  df-fv 4960  df-ov 5567
This theorem is referenced by:  caovlem2d  5745  addassnqg  6704  addassnq0  6784  axmulass  7171
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