Theorem caopr4 4056

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 4053	. . 3 |- (BF(CFD)) = (CF(BFD))
7	6	opreq2i 3963	. 2 |- (AF(BF(CFD))) = (AF(CF(BFD)))
8		caopr.1	. . 3 |- A e. V
9		oprex 3974	. . 3 |- (CFD) e. V
10	8, 1, 9, 5	caoprass 4046	. 2 |- ((AFB)F(CFD)) = (AF(BF(CFD)))
11		oprex 3974	. . 3 |- (BFD) e. V
12	8, 2, 11, 5	caoprass 4046	. 2 |- ((AFC)F(BFD)) = (AF(CF(BFD)))
13	7, 10, 12	3eqtr4 1502	1 |- ((AFB)F(CFD)) = ((AFC)F(BFD))

Syntax hints:   = wceq 954   e. wcel 956  Vcvv 1807  (class class class)co 3954

This theorem is referenced by:  caopr42 4058  ecopoprtrn 4301  addcmpblnq 5032  mulcmpblnq 5033  ordpipq 5036  distrpq 5047  ltapq 5056  ltmpq 5057  reclem3pr 5138  addcmpblnr 5161  mulcmpblnrlem 5162  ltsrpr 5166  distrsr 5180  ltasr 5189  mulgt0sr 5194  sqgt0sr 5195  axdistr 5259

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fv 3193  df-opr 3956

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