Theorem oadist2a 987

Description: Distributive inference derived from OA.

Hypothesis

Ref	Expression
oadist2a.1	(d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))

Assertion

Ref	Expression
oadist2a	((a ->2 b) ^ (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))) = (((a ->2 b) ^ d) v ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))))

Proof of Theorem oadist2a

Step	Hyp	Ref	Expression
1		ax-a2 30	. . 3 (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) = (((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) v d)
2	1	lan 70	. 2 ((a ->2 b) ^ (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))) = ((a ->2 b) ^ (((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) v d))
3		ax-a2 30	. . . . . . 7 (((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) v d) = (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))
4		oadist2a.1	. . . . . . 7 (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
5	3, 4	bltr 130	. . . . . 6 (((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) v d) =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
6	5	lelan 159	. . . . 5 ((a ->2 b) ^ (((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) v d)) =< ((a ->2 b) ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))))
7		df-i0 42	. . . . . . . 8 ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) = ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
8	7	lan 70	. . . . . . 7 ((a ->2 b) ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))
9		oath1 984	. . . . . . 7 ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
10	8, 9	ax-r2 35	. . . . . 6 ((a ->2 b) ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
11		leo 150	. . . . . . 7 ((a ->2 b) ^ (a ->2 c)) =< (((a ->2 b) ^ (a ->2 c)) v ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_))
12		df-i2 44	. . . . . . . 8 ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) = (((a ->2 b) ^ (a ->2 c)) v ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_))
13	12	ax-r1 34	. . . . . . 7 (((a ->2 b) ^ (a ->2 c)) v ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_)) = ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))
14	11, 13	lbtr 131	. . . . . 6 ((a ->2 b) ^ (a ->2 c)) =< ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))
15	10, 14	bltr 130	. . . . 5 ((a ->2 b) ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) =< ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))
16	6, 15	letr 129	. . . 4 ((a ->2 b) ^ (((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) v d)) =< ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))
17	16	distlem 180	. . 3 ((a ->2 b) ^ (((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) v d)) = (((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) v ((a ->2 b) ^ d))
18		ax-a2 30	. . 3 (((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) v ((a ->2 b) ^ d)) = (((a ->2 b) ^ d) v ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))))
19	17, 18	ax-r2 35	. 2 ((a ->2 b) ^ (((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) v d)) = (((a ->2 b) ^ d) v ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))))
20	2, 19	ax-r2 35	1 ((a ->2 b) ^ (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))) = (((a ->2 b) ^ d) v ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))))

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->0 wi0 12   ->2 wi2 14

This theorem is referenced by:  oadist2b 988  oadist2 989

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-3oa 978

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i0 42  df-i1 43  df-i2 44  df-le1 122  df-le2 123

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