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Theorem caovdilemd 5962
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 5924 . . 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 5924 . . 3  |-  ( ph  ->  ( B G D )  e.  S )
9 caovdilemd.h . . 3  |-  ( ph  ->  H  e.  S )
101, 5, 8, 9caovdird 5949 . 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 5930 . . 3  |-  ( ph  ->  ( ( A G C ) G H )  =  ( A G ( C G H ) ) )
1311, 6, 7, 9caovassd 5930 . . 3  |-  ( ph  ->  ( ( B G D ) G H )  =  ( B G ( D G H ) ) )
1412, 13oveq12d 5792 . 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 2172 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 103    /\ w3a 962    = wceq 1331    e. wcel 1480  (class class class)co 5774
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777
This theorem is referenced by:  caovlem2d  5963  addassnqg  7197  addassnq0  7277  axmulass  7688
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