Theorem ndmoprdistr 4982

Description: Any operation is distributive 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
ndmopr.6	|- dom G = (S X. S)

Assertion

Ref	Expression
ndmoprdistr	|- (-. (A e. S /\ B e. S /\ C e. S) -> (AG(BFC)) = ((AGB)F(AGC)))

Proof of Theorem ndmoprdistr

Step	Hyp	Ref	Expression
1		ndmopr.4	. . . . . . 7 |- C e. _V
2		ndmopr.2	. . . . . . 7 |- dom F = (S X. S)
3		ndmopr.5	. . . . . . 7 |- -. (/) e. S
4	1, 2, 3	ndmoprrcl 4979	. . . . . 6 |- ((BFC) e. S -> (B e. S /\ C e. S))
5	4	anim2i 362	. . . . 5 |- ((A e. S /\ (BFC) e. S) -> (A e. S /\ (B e. S /\ C e. S)))
6		3anass 862	. . . . 5 |- ((A e. S /\ B e. S /\ C e. S) <-> (A e. S /\ (B e. S /\ C e. S)))
7	5, 6	sylibr 217	. . . 4 |- ((A e. S /\ (BFC) e. S) -> (A e. S /\ B e. S /\ C e. S))
8	7	con3i 114	. . 3 |- (-. (A e. S /\ B e. S /\ C e. S) -> -. (A e. S /\ (BFC) e. S))
9		oprex 4907	. . . 4 |- (BFC) e. _V
10		ndmopr.6	. . . 4 |- dom G = (S X. S)
11	9, 10	ndmopr 4978	. . 3 |- (-. (A e. S /\ (BFC) e. S) -> (AG(BFC)) = (/))
12	8, 11	syl 12	. 2 |- (-. (A e. S /\ B e. S /\ C e. S) -> (AG(BFC)) = (/))
13		ndmopr.1	. . . . . . 7 |- B e. _V
14	13, 10, 3	ndmoprrcl 4979	. . . . . 6 |- ((AGB) e. S -> (A e. S /\ B e. S))
15	1, 10, 3	ndmoprrcl 4979	. . . . . 6 |- ((AGC) e. S -> (A e. S /\ C e. S))
16	14, 15	anim12i 360	. . . . 5 |- (((AGB) e. S /\ (AGC) e. S) -> ((A e. S /\ B e. S) /\ (A e. S /\ C e. S)))
17		anandi 568	. . . . . 6 |- ((A e. S /\ (B e. S /\ C e. S)) <-> ((A e. S /\ B e. S) /\ (A e. S /\ C e. S)))
18	6, 17	bitri 190	. . . . 5 |- ((A e. S /\ B e. S /\ C e. S) <-> ((A e. S /\ B e. S) /\ (A e. S /\ C e. S)))
19	16, 18	sylibr 217	. . . 4 |- (((AGB) e. S /\ (AGC) e. S) -> (A e. S /\ B e. S /\ C e. S))
20	19	con3i 114	. . 3 |- (-. (A e. S /\ B e. S /\ C e. S) -> -. ((AGB) e. S /\ (AGC) e. S))
21		oprex 4907	. . . 4 |- (AGC) e. _V
22	21, 2	ndmopr 4978	. . 3 |- (-. ((AGB) e. S /\ (AGC) e. S) -> ((AGB)F(AGC)) = (/))
23	20, 22	syl 12	. 2 |- (-. (A e. S /\ B e. S /\ C e. S) -> ((AGB)F(AGC)) = (/))
24	12, 23	eqtr4d 1928	1 |- (-. (A e. S /\ B e. S /\ C e. S) -> (AG(BFC)) = ((AGB)F(AGC)))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292  (/)c0 2875   X. cxp 3984  dom cdm 3986  (class class class)co 4884

This theorem is referenced by:  distrpi 6178  distrpq 6219  distrpr 6284  distrsr 6352

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-ral 2109  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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