Theorem caovdilemd 6224

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 6186	. . 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 6186	. . 3 |- ( ph -> ( B G D ) e. S )
9		caovdilemd.h	. . 3 |- ( ph -> H e. S )
10	1, 5, 8, 9	caovdird 6211	. 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 6192	. . 3 |- ( ph -> ( ( A G C ) G H ) = ( A G ( C G H ) ) )
13	11, 6, 7, 9	caovassd 6192	. . 3 |- ( ph -> ( ( B G D ) G H ) = ( B G ( D G H ) ) )
14	12, 13	oveq12d 6046	. 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 2264	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 1005   = wceq 1398   e. wcel 2202  (class class class)co 6028

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031

This theorem is referenced by:  caovlem2d  6225  addassnqg  7662  addassnq0  7742  axmulass  8153