Theorem caovdilemd 6161

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

Step	Hyp	Ref	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 )
5	2, 3, 4	caovcld 6123	. . 3 |- ( ph -> ( A G C ) e. S )
6		caovdilemd.b	. . . 4 |- ( ph -> B e. S )
7		caovdilemd.d	. . . 4 |- ( ph -> D e. S )
8	2, 6, 7	caovcld 6123	. . 3 |- ( ph -> ( B G D ) e. S )
9		caovdilemd.h	. . 3 |- ( ph -> H e. S )
10	1, 5, 8, 9	caovdird 6148	. 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 ) ) )
12	11, 3, 4, 9	caovassd 6129	. . 3 |- ( ph -> ( ( A G C ) G H ) = ( A G ( C G H ) ) )
13	11, 6, 7, 9	caovassd 6129	. . 3 |- ( ph -> ( ( B G D ) G H ) = ( B G ( D G H ) ) )
14	12, 13	oveq12d 5985	. 2 |- ( ph -> ( ( ( A G C ) G H ) F ( ( B G D ) G H ) ) = ( ( A G ( C G H ) ) F ( B G ( D G H ) ) ) )
15	10, 14	eqtrd 2240	1 |- ( ph -> ( ( ( A G C ) F ( B G D ) ) G H ) = ( ( A G ( C G H ) ) F ( B G ( D G H ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2178  (class class class)co 5967

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298  df-ov 5970

This theorem is referenced by:  caovlem2d  6162  addassnqg  7530  addassnq0  7610  axmulass  8021