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Theorem caovdilemd 6060
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 6022 . . 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 6022 . . 3  |-  ( ph  ->  ( B G D )  e.  S )
9 caovdilemd.h . . 3  |-  ( ph  ->  H  e.  S )
101, 5, 8, 9caovdird 6047 . 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 6028 . . 3  |-  ( ph  ->  ( ( A G C ) G H )  =  ( A G ( C G H ) ) )
1311, 6, 7, 9caovassd 6028 . . 3  |-  ( ph  ->  ( ( B G D ) G H )  =  ( B G ( D G H ) ) )
1412, 13oveq12d 5887 . 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 2210 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 104    /\ w3a 978    = wceq 1353    e. wcel 2148  (class class class)co 5869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-iota 5174  df-fv 5220  df-ov 5872
This theorem is referenced by:  caovlem2d  6061  addassnqg  7369  addassnq0  7449  axmulass  7860
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