Theorem caoprass 3068

Description: Convert an operation associative law to class notation.

Hypotheses

Ref	Expression
caoprass.1	⊢ A ∈ V
caoprass.2	⊢ B ∈ V
caoprass.3	⊢ C ∈ V
caoprass.4	⊢ ((xFy)Fz) = (xF(yFz))

Assertion

Ref	Expression
caoprass	⊢ ((AFB)FC) = (AF(BFC))

Distinct variable group(s):   x,y,z,F   x,A,y,z   x,B,y,z   x,C,y,z

Proof of Theorem caoprass

Step	Hyp	Ref	Expression
1		caoprass.1	. 2 ⊢ A ∈ V
2		caoprass.2	. 2 ⊢ B ∈ V
3		caoprass.3	. 2 ⊢ C ∈ V
4		opreq1 3006	. . . . 5 ⊢ (x = A → (xFy) = (AFy))
5	4	opreq1d 3012	. . . 4 ⊢ (x = A → ((xFy)Fz) = ((AFy)Fz))
6		opreq1 3006	. . . 4 ⊢ (x = A → (xF(yFz)) = (AF(yFz)))
7	5, 6	cleq12d 1115	. . 3 ⊢ (x = A → (((xFy)Fz) = (xF(yFz)) ↔ ((AFy)Fz) = (AF(yFz))))
8		opreq2 3007	. . . . 5 ⊢ (y = B → (AFy) = (AFB))
9	8	opreq1d 3012	. . . 4 ⊢ (y = B → ((AFy)Fz) = ((AFB)Fz))
10		opreq1 3006	. . . . 5 ⊢ (y = B → (yFz) = (BFz))
11	10	opreq2d 3013	. . . 4 ⊢ (y = B → (AF(yFz)) = (AF(BFz)))
12	9, 11	cleq12d 1115	. . 3 ⊢ (y = B → (((AFy)Fz) = (AF(yFz)) ↔ ((AFB)Fz) = (AF(BFz))))
13		opreq2 3007	. . . 4 ⊢ (z = C → ((AFB)Fz) = ((AFB)FC))
14		opreq2 3007	. . . . 5 ⊢ (z = C → (BFz) = (BFC))
15	14	opreq2d 3013	. . . 4 ⊢ (z = C → (AF(BFz)) = (AF(BFC)))
16	13, 15	cleq12d 1115	. . 3 ⊢ (z = C → (((AFB)Fz) = (AF(BFz)) ↔ ((AFB)FC) = (AF(BFC))))
17	7, 12, 16	syl3an9b 634	. 2 ⊢ ((x = A ∧ y = B ∧ z = C) → (((xFy)Fz) = (xF(yFz)) ↔ ((AFB)FC) = (AF(BFC))))
18		caoprass.4	. 2 ⊢ ((xFy)Fz) = (xF(yFz))
19	1, 2, 3, 17, 18	vtocl3 1380	1 ⊢ ((AFB)FC) = (AF(BFC))

Syntax hints:   = wceq 1091   ∈ wcel 1092  Vcvv 1348  (class class class)co 3001

This theorem is referenced by:  caopr32 3074  caopr12 3075  caopr31 3076  caopr13 3077  caopr4 3078  caopr411 3079  caoprdilem 3082  caoprmo 3084

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 <799  ax-10 800  ax-11 801  ax-12N/A> 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

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