Theorem wledio 388

Description: Half of distributive law.

Assertion

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

Proof of Theorem wledio

Step	Hyp	Ref	Expression
1		anidm 103	. . . . . 6 (a ∩ a) = a
2	1	bi1 110	. . . . 5 ((a ∩ a) ≡ a) = 1
3	2	wr1 189	. . . 4 (a ≡ (a ∩ a)) = 1
4		wleo 369	. . . . 5 (a ≤2 (a ∪ b)) = 1
5		wleo 369	. . . . 5 (a ≤2 (a ∪ c)) = 1
6	4, 5	wle2an 386	. . . 4 ((a ∩ a) ≤2 ((a ∪ b) ∩ (a ∪ c))) = 1
7	3, 6	wbltr 379	. . 3 (a ≤2 ((a ∪ b) ∩ (a ∪ c))) = 1
8		wleo 369	. . . . 5 (b ≤2 (b ∪ a)) = 1
9		ax-a2 30	. . . . . 6 (b ∪ a) = (a ∪ b)
10	9	bi1 110	. . . . 5 ((b ∪ a) ≡ (a ∪ b)) = 1
11	8, 10	wlbtr 380	. . . 4 (b ≤2 (a ∪ b)) = 1
12		wleo 369	. . . . 5 (c ≤2 (c ∪ a)) = 1
13		ax-a2 30	. . . . . 6 (c ∪ a) = (a ∪ c)
14	13	bi1 110	. . . . 5 ((c ∪ a) ≡ (a ∪ c)) = 1
15	12, 14	wlbtr 380	. . . 4 (c ≤2 (a ∪ c)) = 1
16	11, 15	wle2an 386	. . 3 ((b ∩ c) ≤2 ((a ∪ b) ∩ (a ∪ c))) = 1
17	7, 16	wle2or 385	. 2 ((a ∪ (b ∩ c)) ≤2 (((a ∪ b) ∩ (a ∪ c)) ∪ ((a ∪ b) ∩ (a ∪ c)))) = 1
18		oridm 102	. . 3 (((a ∪ b) ∩ (a ∪ c)) ∪ ((a ∪ b) ∩ (a ∪ c))) = ((a ∪ b) ∩ (a ∪ c))
19	18	bi1 110	. 2 ((((a ∪ b) ∩ (a ∪ c)) ∪ ((a ∪ b) ∩ (a ∪ c))) ≡ ((a ∪ b) ∩ (a ∪ c))) = 1
20	17, 19	wlbtr 380	1 ((a ∪ (b ∩ c)) ≤2 ((a ∪ b) ∩ (a ∪ c))) = 1

Syntax hints:   = wb 1   ∪ wo 6   ∩ wa 7  1wt 9   ≤₂ wle2 11

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-wom 343

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le 121  df-le1 122  df-le2 123

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