Theorem wledi 405

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

Assertion

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

Proof of Theorem wledi

Step	Hyp	Ref	Expression
1		anidm 111	. . . 4 (((a ∩ b) ∪ (a ∩ c)) ∩ ((a ∩ b) ∪ (a ∩ c))) = ((a ∩ b) ∪ (a ∩ c))
2	1	bi1 118	. . 3 ((((a ∩ b) ∪ (a ∩ c)) ∩ ((a ∩ b) ∪ (a ∩ c))) ≡ ((a ∩ b) ∪ (a ∩ c))) = 1
3	2	wr1 197	. 2 (((a ∩ b) ∪ (a ∩ c)) ≡ (((a ∩ b) ∪ (a ∩ c)) ∩ ((a ∩ b) ∪ (a ∩ c)))) = 1
4		wlea 388	. . . . 5 ((a ∩ b) ≤2 a) = 1
5		wlea 388	. . . . 5 ((a ∩ c) ≤2 a) = 1
6	4, 5	wle2or 403	. . . 4 (((a ∩ b) ∪ (a ∩ c)) ≤2 (a ∪ a)) = 1
7		oridm 110	. . . . 5 (a ∪ a) = a
8	7	bi1 118	. . . 4 ((a ∪ a) ≡ a) = 1
9	6, 8	wlbtr 398	. . 3 (((a ∩ b) ∪ (a ∩ c)) ≤2 a) = 1
10		ancom 74	. . . . . 6 (a ∩ b) = (b ∩ a)
11	10	bi1 118	. . . . 5 ((a ∩ b) ≡ (b ∩ a)) = 1
12		wlea 388	. . . . 5 ((b ∩ a) ≤2 b) = 1
13	11, 12	wbltr 397	. . . 4 ((a ∩ b) ≤2 b) = 1
14		ancom 74	. . . . . 6 (a ∩ c) = (c ∩ a)
15	14	bi1 118	. . . . 5 ((a ∩ c) ≡ (c ∩ a)) = 1
16		wlea 388	. . . . 5 ((c ∩ a) ≤2 c) = 1
17	15, 16	wbltr 397	. . . 4 ((a ∩ c) ≤2 c) = 1
18	13, 17	wle2or 403	. . 3 (((a ∩ b) ∪ (a ∩ c)) ≤2 (b ∪ c)) = 1
19	9, 18	wle2an 404	. 2 ((((a ∩ b) ∪ (a ∩ c)) ∩ ((a ∩ b) ∪ (a ∩ c))) ≤2 (a ∩ (b ∪ c))) = 1
20	3, 19	wbltr 397	1 (((a ∩ b) ∪ (a ∩ c)) ≤2 (a ∩ (b ∪ c))) = 1

This proof depends on syntax axioms:   = wb 1   ∪ wo 6   ∩ wa 7  1wt 8   ≤₂ wle2 10

This proof depends on 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 proof 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 used by:  wfh1  423  wfh2  424