Theorem wledio 406

Description: Half of distributive law. (Contributed by NM, 13-Oct-1997.)

Assertion

Ref	Expression
wledio	((a ∪ (b ∩ c)) ≤2 ((a ∪ b) ∩ (a ∪ c))) = 1

Proof of Theorem wledio

Step	Hyp	Ref	Expression
1		anidm 111	. . . . . 6 (a ∩ a) = a
2	1	bi1 118	. . . . 5 ((a ∩ a) ≡ a) = 1
3	2	wr1 197	. . . 4 (a ≡ (a ∩ a)) = 1
4		wleo 387	. . . . 5 (a ≤2 (a ∪ b)) = 1
5		wleo 387	. . . . 5 (a ≤2 (a ∪ c)) = 1
6	4, 5	wle2an 404	. . . 4 ((a ∩ a) ≤2 ((a ∪ b) ∩ (a ∪ c))) = 1
7	3, 6	wbltr 397	. . 3 (a ≤2 ((a ∪ b) ∩ (a ∪ c))) = 1
8		wleo 387	. . . . 5 (b ≤2 (b ∪ a)) = 1
9		ax-a2 31	. . . . . 6 (b ∪ a) = (a ∪ b)
10	9	bi1 118	. . . . 5 ((b ∪ a) ≡ (a ∪ b)) = 1
11	8, 10	wlbtr 398	. . . 4 (b ≤2 (a ∪ b)) = 1
12		wleo 387	. . . . 5 (c ≤2 (c ∪ a)) = 1
13		ax-a2 31	. . . . . 6 (c ∪ a) = (a ∪ c)
14	13	bi1 118	. . . . 5 ((c ∪ a) ≡ (a ∪ c)) = 1
15	12, 14	wlbtr 398	. . . 4 (c ≤2 (a ∪ c)) = 1
16	11, 15	wle2an 404	. . 3 ((b ∩ c) ≤2 ((a ∪ b) ∩ (a ∪ c))) = 1
17	7, 16	wle2or 403	. 2 ((a ∪ (b ∩ c)) ≤2 (((a ∪ b) ∩ (a ∪ c)) ∪ ((a ∪ b) ∩ (a ∪ c)))) = 1
18		oridm 110	. . 3 (((a ∪ b) ∩ (a ∪ c)) ∪ ((a ∪ b) ∩ (a ∪ c))) = ((a ∪ b) ∩ (a ∪ c))
19	18	bi1 118	. 2 ((((a ∪ b) ∩ (a ∪ c)) ∪ ((a ∪ b) ∩ (a ∪ c))) ≡ ((a ∪ b) ∩ (a ∪ c))) = 1
20	17, 19	wlbtr 398	1 ((a ∪ (b ∩ c)) ≤2 ((a ∪ b) ∩ (a ∪ c))) = 1

Syntax hints:   = wb 1   ∪ wo 6   ∩ wa 7  1wt 8   ≤₂ wle2 10

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-wom 361

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131

This theorem is referenced by: (None)