Theorem 2oai1u 804

Description: Orthoarguesian-like OM law.

Assertion

Ref	Expression
2oai1u	((a ->1 c) ^ (((a ->1 c) ^ (b ->1 c))_|_ ->2 ((a_|_ ->1 c) ^ (b_|_ ->1 c)))) =< (b ->1 c)

Proof of Theorem 2oai1u

Step	Hyp	Ref	Expression
1		1oai1 803	. 2 (((a_|_ ->1 c) ->1 c) ^ (((a_|_ ->1 c) ^ (b_|_ ->1 c))_|_ ->1 (((a_|_ ->1 c) ->1 c) ^ ((b_|_ ->1 c) ->1 c)))) =< ((b_|_ ->1 c) ->1 c)
2		u1lem11 762	. . 3 ((a_|_ ->1 c) ->1 c) = (a ->1 c)
3		u1lem11 762	. . . . . . . 8 ((b_|_ ->1 c) ->1 c) = (b ->1 c)
4	2, 3	2an 72	. . . . . . 7 (((a_|_ ->1 c) ->1 c) ^ ((b_|_ ->1 c) ->1 c)) = ((a ->1 c) ^ (b ->1 c))
5	4	ax-r1 34	. . . . . 6 ((a ->1 c) ^ (b ->1 c)) = (((a_|_ ->1 c) ->1 c) ^ ((b_|_ ->1 c) ->1 c))
6	5	ud1lem0a 247	. . . . 5 (((a_|_ ->1 c) ^ (b_|_ ->1 c))_|_ ->1 ((a ->1 c) ^ (b ->1 c))) = (((a_|_ ->1 c) ^ (b_|_ ->1 c))_|_ ->1 (((a_|_ ->1 c) ->1 c) ^ ((b_|_ ->1 c) ->1 c)))
7	6	ax-r1 34	. . . 4 (((a_|_ ->1 c) ^ (b_|_ ->1 c))_|_ ->1 (((a_|_ ->1 c) ->1 c) ^ ((b_|_ ->1 c) ->1 c))) = (((a_|_ ->1 c) ^ (b_|_ ->1 c))_|_ ->1 ((a ->1 c) ^ (b ->1 c)))
8		i1i2con2 261	. . . 4 (((a_|_ ->1 c) ^ (b_|_ ->1 c))_|_ ->1 ((a ->1 c) ^ (b ->1 c))) = (((a ->1 c) ^ (b ->1 c))_|_ ->2 ((a_|_ ->1 c) ^ (b_|_ ->1 c)))
9	7, 8	ax-r2 35	. . 3 (((a_|_ ->1 c) ^ (b_|_ ->1 c))_|_ ->1 (((a_|_ ->1 c) ->1 c) ^ ((b_|_ ->1 c) ->1 c))) = (((a ->1 c) ^ (b ->1 c))_|_ ->2 ((a_|_ ->1 c) ^ (b_|_ ->1 c)))
10	2, 9	2an 72	. 2 (((a_|_ ->1 c) ->1 c) ^ (((a_|_ ->1 c) ^ (b_|_ ->1 c))_|_ ->1 (((a_|_ ->1 c) ->1 c) ^ ((b_|_ ->1 c) ->1 c)))) = ((a ->1 c) ^ (((a ->1 c) ^ (b ->1 c))_|_ ->2 ((a_|_ ->1 c) ^ (b_|_ ->1 c))))
11	1, 10, 3	le3tr2 133	1 ((a ->1 c) ^ (((a ->1 c) ^ (b ->1 c))_|_ ->2 ((a_|_ ->1 c) ^ (b_|_ ->1 c)))) =< (b ->1 c)

Syntax hints:   =< wle 2  _|_wn 4   ^ wa 7   ->1 wi1 13   ->2 wi2 14

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-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

metamath.org