Theorem ndmoprdistr 3063

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

Hypotheses

Ref	Expression
ndmopr.1	⊢ B ∈ V
ndmopr.2	⊢ dom F = (S × S)
ndmopr.4	⊢ C ∈ V
ndmopr.5	⊢ ¬ ∅ ∈ S
ndmopr.6	⊢ dom G = (S × S)

Assertion

Ref	Expression
ndmoprdistr	⊢ (¬ (A ∈ S ∧ B ∈ S ∧ C ∈ S) → (AG(BFC)) = ((AGB)F(AGC)))

Proof of Theorem ndmoprdistr

Step	Hyp	Ref	Expression
1		ndmopr.4	. . . . . . 7 ⊢ C ∈ V
2		ndmopr.2	. . . . . . 7 ⊢ dom F = (S × S)
3		ndmopr.5	. . . . . . 7 ⊢ ¬ ∅ ∈ S
4	1, 2, 3	ndmoprrcl 3060	. . . . . 6 ⊢ ((BFC) ∈ S → (B ∈ S ∧ C ∈ S))
5	4	anim2i 270	. . . . 5 ⊢ ((A ∈ S ∧ (BFC) ∈ S) → (A ∈ S ∧ (B ∈ S ∧ C ∈ S)))
6		3anass 585	. . . . 5 ⊢ ((A ∈ S ∧ B ∈ S ∧ C ∈ S) ↔ (A ∈ S ∧ (B ∈ S ∧ C ∈ S)))
7	5, 6	sylibr 175	. . . 4 ⊢ ((A ∈ S ∧ (BFC) ∈ S) → (A ∈ S ∧ B ∈ S ∧ C ∈ S))
8	7	con3i 90	. . 3 ⊢ (¬ (A ∈ S ∧ B ∈ S ∧ C ∈ S) → ¬ (A ∈ S ∧ (BFC) ∈ S))
9		oprex 3018	. . . 4 ⊢ (BFC) ∈ V
10		ndmopr.6	. . . 4 ⊢ dom G = (S × S)
11	9, 10	ndmopr 3059	. . 3 ⊢ (¬ (A ∈ S ∧ (BFC) ∈ S) → (AG(BFC)) = ∅)
12	8, 11	syl 12	. 2 ⊢ (¬ (A ∈ S ∧ B ∈ S ∧ C ∈ S) → (AG(BFC)) = ∅)
13		ndmopr.1	. . . . . . 7 ⊢ B ∈ V
14	13, 10, 3	ndmoprrcl 3060	. . . . . 6 ⊢ ((AGB) ∈ S → (A ∈ S ∧ B ∈ S))
15	1, 10, 3	ndmoprrcl 3060	. . . . . 6 ⊢ ((AGC) ∈ S → (A ∈ S ∧ C ∈ S))
16	14, 15	anim12i 268	. . . . 5 ⊢ (((AGB) ∈ S ∧ (AGC) ∈ S) → ((A ∈ S ∧ B ∈ S) ∧ (A ∈ S ∧ C ∈ S)))
17		anandi 392	. . . . . 6 ⊢ ((A ∈ S ∧ (B ∈ S ∧ C ∈ S)) ↔ ((A ∈ S ∧ B ∈ S) ∧ (A ∈ S ∧ C ∈ S)))
18	6, 17	bitr 151	. . . . 5 ⊢ ((A ∈ S ∧ B ∈ S ∧ C ∈ S) ↔ ((A ∈ S ∧ B ∈ S) ∧ (A ∈ S ∧ C ∈ S)))
19	16, 18	sylibr 175	. . . 4 ⊢ (((AGB) ∈ S ∧ (AGC) ∈ S) → (A ∈ S ∧ B ∈ S ∧ C ∈ S))
20	19	con3i 90	. . 3 ⊢ (¬ (A ∈ S ∧ B ∈ S ∧ C ∈ S) → ¬ ((AGB) ∈ S ∧ (AGC) ∈ S))
21		oprex 3018	. . . 4 ⊢ (AGC) ∈ V
22	21, 2	ndmopr 3059	. . 3 ⊢ (¬ ((AGB) ∈ S ∧ (AGC) ∈ S) → ((AGB)F(AGC)) = ∅)
23	20, 22	syl 12	. 2 ⊢ (¬ (A ∈ S ∧ B ∈ S ∧ C ∈ S) → ((AGB)F(AGC)) = ∅)
24	12, 23	eqtr4d 1131	1 ⊢ (¬ (A ∈ S ∧ B ∈ S ∧ C ∈ S) → (AG(BFC)) = ((AGB)F(AGC)))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∧ w3a 581   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707   × cxp 2408  dom cdm 2410  (class class class)co 3001

This theorem is referenced by:  distrpi 3820  distrpq 3861  distrpr 3926  distrsr 3994

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-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  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-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  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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