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Theorem oau 929
Description: Transformation lemma for studying the orthoarguesian law.
Hypothesis
Ref Expression
oau.1 (a ^ ((a ->1 c) v b)) =< c
Assertion
Ref Expression
oau b =< (a ->1 c)
Proof of Theorem oau
StepHypRef Expression
1 ax-a2 31 . . 3 (b v (a ->1 c)) = ((a ->1 c) v b)
2 lea 160 . . . . . . . 8 (a ^ ((a ->1 c) v b)) =< a
3 oau.1 . . . . . . . 8 (a ^ ((a ->1 c) v b)) =< c
42, 3ler2an 173 . . . . . . 7 (a ^ ((a ->1 c) v b)) =< (a ^ c)
5 u1lemaa 600 . . . . . . . 8 ((a ->1 c) ^ a) = (a ^ c)
65ax-r1 35 . . . . . . 7 (a ^ c) = ((a ->1 c) ^ a)
74, 6lbtr 139 . . . . . 6 (a ^ ((a ->1 c) v b)) =< ((a ->1 c) ^ a)
87lelor 166 . . . . 5 ((a ->1 c) v (a ^ ((a ->1 c) v b))) =< ((a ->1 c) v ((a ->1 c) ^ a))
9 u1lemc1 680 . . . . . . . 8 a C (a ->1 c)
109comcom 453 . . . . . . 7 (a ->1 c) C a
11 comorr 184 . . . . . . 7 (a ->1 c) C ((a ->1 c) v b)
1210, 11fh3 471 . . . . . 6 ((a ->1 c) v (a ^ ((a ->1 c) v b))) = (((a ->1 c) v a) ^ ((a ->1 c) v ((a ->1 c) v b)))
13 u1lemoa 620 . . . . . . 7 ((a ->1 c) v a) = 1
14 ax-a3 32 . . . . . . . . 9 (((a ->1 c) v (a ->1 c)) v b) = ((a ->1 c) v ((a ->1 c) v b))
1514ax-r1 35 . . . . . . . 8 ((a ->1 c) v ((a ->1 c) v b)) = (((a ->1 c) v (a ->1 c)) v b)
16 oridm 110 . . . . . . . . 9 ((a ->1 c) v (a ->1 c)) = (a ->1 c)
1716ax-r5 38 . . . . . . . 8 (((a ->1 c) v (a ->1 c)) v b) = ((a ->1 c) v b)
1815, 17ax-r2 36 . . . . . . 7 ((a ->1 c) v ((a ->1 c) v b)) = ((a ->1 c) v b)
1913, 182an 79 . . . . . 6 (((a ->1 c) v a) ^ ((a ->1 c) v ((a ->1 c) v b))) = (1 ^ ((a ->1 c) v b))
20 ancom 74 . . . . . . 7 (1 ^ ((a ->1 c) v b)) = (((a ->1 c) v b) ^ 1)
21 an1 106 . . . . . . 7 (((a ->1 c) v b) ^ 1) = ((a ->1 c) v b)
2220, 21ax-r2 36 . . . . . 6 (1 ^ ((a ->1 c) v b)) = ((a ->1 c) v b)
2312, 19, 223tr 65 . . . . 5 ((a ->1 c) v (a ^ ((a ->1 c) v b))) = ((a ->1 c) v b)
24 orabs 120 . . . . 5 ((a ->1 c) v ((a ->1 c) ^ a)) = (a ->1 c)
258, 23, 24le3tr2 141 . . . 4 ((a ->1 c) v b) =< (a ->1 c)
26 leo 158 . . . 4 (a ->1 c) =< ((a ->1 c) v b)
2725, 26lebi 145 . . 3 ((a ->1 c) v b) = (a ->1 c)
281, 27ax-r2 36 . 2 (b v (a ->1 c)) = (a ->1 c)
2928df-le1 130 1 b =< (a ->1 c)
Colors of variables: term
Syntax hints:   =< wle 2   v wo 6   ^ wa 7  1wt 8   ->1 wi1 12
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-r3 439
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  oa4uto4g  975
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