Theorem caoprdilem 5001

Description: Lemma used by real number construction.

Hypotheses

Ref	Expression
caoprd.1	|- A e. _V
caoprd.2	|- B e. _V
caoprd.3	|- C e. _V
caoprd.com	|- (xGy) = (yGx)
caoprd.distr	|- (xG(yFz)) = ((xGy)F(xGz))
caoprdl.4	|- D e. _V
caoprdl.5	|- H e. _V
caoprdl.ass	|- ((xGy)Gz) = (xG(yGz))

Assertion

Ref	Expression
caoprdilem	|- (((AGC)F(BGD))GH) = ((AG(CGH))F(BG(DGH)))

Distinct variable groups:   x,y,z,F   x,A,y,z   x,B,y,z   x,C,y,z   x,D,y,z   x,G,y,z   x,H,y,z

Proof of Theorem caoprdilem

Step	Hyp	Ref	Expression
1		oprex 4907	. . 3 |- (AGC) e. _V
2		oprex 4907	. . 3 |- (BGD) e. _V
3		caoprdl.5	. . 3 |- H e. _V
4		caoprd.com	. . 3 |- (xGy) = (yGx)
5		caoprd.distr	. . 3 |- (xG(yFz)) = ((xGy)F(xGz))
6	1, 2, 3, 4, 5	caoprdistrr 5000	. 2 |- (((AGC)F(BGD))GH) = (((AGC)GH)F((BGD)GH))
7		caoprd.1	. . . 4 |- A e. _V
8		caoprd.3	. . . 4 |- C e. _V
9		caoprdl.ass	. . . 4 |- ((xGy)Gz) = (xG(yGz))
10	7, 8, 3, 9	caoprass 4987	. . 3 |- ((AGC)GH) = (AG(CGH))
11		caoprd.2	. . . 4 |- B e. _V
12		caoprdl.4	. . . 4 |- D e. _V
13	11, 12, 3, 9	caoprass 4987	. . 3 |- ((BGD)GH) = (BG(DGH))
14	10, 13	opreq12i 4894	. 2 |- (((AGC)GH)F((BGD)GH)) = ((AG(CGH))F(BG(DGH)))
15	6, 14	eqtri 1908	1 |- (((AGC)F(BGD))GH) = ((AG(CGH))F(BG(DGH)))

Syntax hints:   = wceq 1298   e. wcel 1300  _Vcvv 2292  (class class class)co 4884

This theorem is referenced by:  caoprlem2 5002  addasspq 6215  axmulass 6431

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886

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