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Theorem oau 929
 Description: Transformation lemma for studying the orthoarguesian law. (Contributed by NM, 28-Dec-1998.)
Hypothesis
Ref Expression
oau.1 (a ∩ ((a1 c) ∪ b)) ≤ c
Assertion
Ref Expression
oau b ≤ (a1 c)

Proof of Theorem oau
StepHypRef Expression
1 ax-a2 31 . . 3 (b ∪ (a1 c)) = ((a1 c) ∪ b)
2 lea 160 . . . . . . . 8 (a ∩ ((a1 c) ∪ b)) ≤ a
3 oau.1 . . . . . . . 8 (a ∩ ((a1 c) ∪ b)) ≤ c
42, 3ler2an 173 . . . . . . 7 (a ∩ ((a1 c) ∪ b)) ≤ (ac)
5 u1lemaa 600 . . . . . . . 8 ((a1 c) ∩ a) = (ac)
65ax-r1 35 . . . . . . 7 (ac) = ((a1 c) ∩ a)
74, 6lbtr 139 . . . . . 6 (a ∩ ((a1 c) ∪ b)) ≤ ((a1 c) ∩ a)
87lelor 166 . . . . 5 ((a1 c) ∪ (a ∩ ((a1 c) ∪ b))) ≤ ((a1 c) ∪ ((a1 c) ∩ a))
9 u1lemc1 680 . . . . . . . 8 a C (a1 c)
109comcom 453 . . . . . . 7 (a1 c) C a
11 comorr 184 . . . . . . 7 (a1 c) C ((a1 c) ∪ b)
1210, 11fh3 471 . . . . . 6 ((a1 c) ∪ (a ∩ ((a1 c) ∪ b))) = (((a1 c) ∪ a) ∩ ((a1 c) ∪ ((a1 c) ∪ b)))
13 u1lemoa 620 . . . . . . 7 ((a1 c) ∪ a) = 1
14 ax-a3 32 . . . . . . . . 9 (((a1 c) ∪ (a1 c)) ∪ b) = ((a1 c) ∪ ((a1 c) ∪ b))
1514ax-r1 35 . . . . . . . 8 ((a1 c) ∪ ((a1 c) ∪ b)) = (((a1 c) ∪ (a1 c)) ∪ b)
16 oridm 110 . . . . . . . . 9 ((a1 c) ∪ (a1 c)) = (a1 c)
1716ax-r5 38 . . . . . . . 8 (((a1 c) ∪ (a1 c)) ∪ b) = ((a1 c) ∪ b)
1815, 17ax-r2 36 . . . . . . 7 ((a1 c) ∪ ((a1 c) ∪ b)) = ((a1 c) ∪ b)
1913, 182an 79 . . . . . 6 (((a1 c) ∪ a) ∩ ((a1 c) ∪ ((a1 c) ∪ b))) = (1 ∩ ((a1 c) ∪ b))
20 ancom 74 . . . . . . 7 (1 ∩ ((a1 c) ∪ b)) = (((a1 c) ∪ b) ∩ 1)
21 an1 106 . . . . . . 7 (((a1 c) ∪ b) ∩ 1) = ((a1 c) ∪ b)
2220, 21ax-r2 36 . . . . . 6 (1 ∩ ((a1 c) ∪ b)) = ((a1 c) ∪ b)
2312, 19, 223tr 65 . . . . 5 ((a1 c) ∪ (a ∩ ((a1 c) ∪ b))) = ((a1 c) ∪ b)
24 orabs 120 . . . . 5 ((a1 c) ∪ ((a1 c) ∩ a)) = (a1 c)
258, 23, 24le3tr2 141 . . . 4 ((a1 c) ∪ b) ≤ (a1 c)
26 leo 158 . . . 4 (a1 c) ≤ ((a1 c) ∪ b)
2725, 26lebi 145 . . 3 ((a1 c) ∪ b) = (a1 c)
281, 27ax-r2 36 . 2 (b ∪ (a1 c)) = (a1 c)
2928df-le1 130 1 b ≤ (a1 c)
 Colors of variables: term Syntax hints:   ≤ wle 2   ∪ 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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