Theorem caopr4 4070

Description: Rearrange arguments in a commutative, associative operation.

Hypotheses

Ref	Expression
caopr.1	|- A e. V
caopr.2	|- B e. V
caopr.3	|- C e. V
caopr.com	|- (xFy) = (yFx)
caopr.ass	|- ((xFy)Fz) = (xF(yFz))
caopr.4	|- D e. V

Assertion

Ref	Expression
caopr4	|- ((AFB)F(CFD)) = ((AFC)F(BFD))

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

Proof of Theorem caopr4

Step	Hyp	Ref	Expression
1		caopr.2	. . . 4 |- B e. V
2		caopr.3	. . . 4 |- C e. V
3		caopr.4	. . . 4 |- D e. V
4		caopr.com	. . . 4 |- (xFy) = (yFx)
5		caopr.ass	. . . 4 |- ((xFy)Fz) = (xF(yFz))
6	1, 2, 3, 4, 5	caopr12 4067	. . 3 |- (BF(CFD)) = (CF(BFD))
7	6	opreq2i 3978	. 2 |- (AF(BF(CFD))) = (AF(CF(BFD)))
8		caopr.1	. . 3 |- A e. V
9		oprex 3989	. . 3 |- (CFD) e. V
10	8, 1, 9, 5	caoprass 4060	. 2 |- ((AFB)F(CFD)) = (AF(BF(CFD)))
11		oprex 3989	. . 3 |- (BFD) e. V
12	8, 2, 11, 5	caoprass 4060	. 2 |- ((AFC)F(BFD)) = (AF(CF(BFD)))
13	7, 10, 12	3eqtr4 1508	1 |- ((AFB)F(CFD)) = ((AFC)F(BFD))

Syntax hints:   = wceq 958   e. wcel 960  Vcvv 1814  (class class class)co 3969

This theorem is referenced by:  caopr42 4072  ecopoprtrn 4317  addcmpblnq 5064  mulcmpblnq 5065  ordpipq 5068  distrpq 5079  ltapq 5088  ltmpq 5089  reclem3pr 5170  addcmpblnr 5193  mulcmpblnrlem 5194  ltsrpr 5198  distrsr 5212  ltasr 5221  mulgt0sr 5226  sqgt0sr 5227  axdistr 5291

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-opr 3971

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