Theorem ndmoprass 4034

Description: Any operation is associative outside its domain, if the domain doesn't contain the empty set.

Hypotheses

Ref	Expression
ndmopr.1	|- B e. V
ndmopr.2	|- dom F = (S X. S)
ndmopr.4	|- C e. V
ndmopr.5	|- -. (/) e. S

Assertion

Ref	Expression
ndmoprass	|- (-. (A e. S /\ B e. S /\ C e. S) -> ((AFB)FC) = (AF(BFC)))

Proof of Theorem ndmoprass

Step	Hyp	Ref	Expression
1		ndmopr.1	. . . . . . 7 |- B e. V
2		ndmopr.2	. . . . . . 7 |- dom F = (S X. S)
3		ndmopr.5	. . . . . . 7 |- -. (/) e. S
4	1, 2, 3	ndmoprrcl 4032	. . . . . 6 |- ((AFB) e. S -> (A e. S /\ B e. S))
5	4	anim1i 334	. . . . 5 |- (((AFB) e. S /\ C e. S) -> ((A e. S /\ B e. S) /\ C e. S))
6		df-3an 775	. . . . 5 |- ((A e. S /\ B e. S /\ C e. S) <-> ((A e. S /\ B e. S) /\ C e. S))
7	5, 6	sylibr 200	. . . 4 |- (((AFB) e. S /\ C e. S) -> (A e. S /\ B e. S /\ C e. S))
8	7	con3i 98	. . 3 |- (-. (A e. S /\ B e. S /\ C e. S) -> -. ((AFB) e. S /\ C e. S))
9		ndmopr.4	. . . 4 |- C e. V
10	9, 2	ndmopr 4031	. . 3 |- (-. ((AFB) e. S /\ C e. S) -> ((AFB)FC) = (/))
11	8, 10	syl 10	. 2 |- (-. (A e. S /\ B e. S /\ C e. S) -> ((AFB)FC) = (/))
12	9, 2, 3	ndmoprrcl 4032	. . . . . 6 |- ((BFC) e. S -> (B e. S /\ C e. S))
13	12	anim2i 335	. . . . 5 |- ((A e. S /\ (BFC) e. S) -> (A e. S /\ (B e. S /\ C e. S)))
14		3anass 777	. . . . 5 |- ((A e. S /\ B e. S /\ C e. S) <-> (A e. S /\ (B e. S /\ C e. S)))
15	13, 14	sylibr 200	. . . 4 |- ((A e. S /\ (BFC) e. S) -> (A e. S /\ B e. S /\ C e. S))
16	15	con3i 98	. . 3 |- (-. (A e. S /\ B e. S /\ C e. S) -> -. (A e. S /\ (BFC) e. S))
17		oprex 3968	. . . 4 |- (BFC) e. V
18	17, 2	ndmopr 4031	. . 3 |- (-. (A e. S /\ (BFC) e. S) -> (AF(BFC)) = (/))
19	16, 18	syl 10	. 2 |- (-. (A e. S /\ B e. S /\ C e. S) -> (AF(BFC)) = (/))
20	11, 19	eqtr4d 1502	1 |- (-. (A e. S /\ B e. S /\ C e. S) -> ((AFB)FC) = (AF(BFC)))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  Vcvv 1802  (/)c0 2270   X. cxp 3158  dom cdm 3160  (class class class)co 3948

This theorem is referenced by:  addasspi 4995  mulasspi 4997  addasspq 5035  mulasspq 5037  genpass 5084  addasssr 5169  mulasssr 5171

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-xp 3174  df-cnv 3176  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fv 3188  df-opr 3950

Copyright terms: Public domain