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Theorem oa3to4lem1 945
 Description: Lemma for orthoarguesian law (Godowski/Greechie 3-variable to 4-variable proof).
Hypotheses
Ref Expression
oa3to4lem.1 ab
oa3to4lem.2 cd
oa3to4lem.3 g = ((ab) ∪ (cd))
Assertion
Ref Expression
oa3to4lem1 b ≤ (a1 g)

Proof of Theorem oa3to4lem1
StepHypRef Expression
1 leor 159 . . . 4 b ≤ (ab)
2 comid 187 . . . . . . . . 9 a C a
32comcom3 454 . . . . . . . 8 a C a
4 oa3to4lem.1 . . . . . . . . 9 ab
54lecom 180 . . . . . . . 8 a C b
63, 5fh3 471 . . . . . . 7 (a ∪ (ab)) = ((aa) ∩ (ab))
7 ancom 74 . . . . . . . 8 (1 ∩ (ab)) = ((ab) ∩ 1)
8 df-t 41 . . . . . . . . . 10 1 = (aa )
9 ax-a2 31 . . . . . . . . . 10 (aa ) = (aa)
108, 9ax-r2 36 . . . . . . . . 9 1 = (aa)
1110ran 78 . . . . . . . 8 (1 ∩ (ab)) = ((aa) ∩ (ab))
12 an1 106 . . . . . . . 8 ((ab) ∩ 1) = (ab)
137, 11, 123tr2 64 . . . . . . 7 ((aa) ∩ (ab)) = (ab)
146, 13ax-r2 36 . . . . . 6 (a ∪ (ab)) = (ab)
1514ax-r1 35 . . . . 5 (ab) = (a ∪ (ab))
16 anidm 111 . . . . . . . . 9 (aa) = a
1716ran 78 . . . . . . . 8 ((aa) ∩ b) = (ab)
1817ax-r1 35 . . . . . . 7 (ab) = ((aa) ∩ b)
19 anass 76 . . . . . . 7 ((aa) ∩ b) = (a ∩ (ab))
2018, 19ax-r2 36 . . . . . 6 (ab) = (a ∩ (ab))
2120lor 70 . . . . 5 (a ∪ (ab)) = (a ∪ (a ∩ (ab)))
2215, 21ax-r2 36 . . . 4 (ab) = (a ∪ (a ∩ (ab)))
231, 22lbtr 139 . . 3 b ≤ (a ∪ (a ∩ (ab)))
24 leo 158 . . . . 5 (ab) ≤ ((ab) ∪ (cd))
2524lelan 167 . . . 4 (a ∩ (ab)) ≤ (a ∩ ((ab) ∪ (cd)))
2625lelor 166 . . 3 (a ∪ (a ∩ (ab))) ≤ (a ∪ (a ∩ ((ab) ∪ (cd))))
2723, 26letr 137 . 2 b ≤ (a ∪ (a ∩ ((ab) ∪ (cd))))
28 oa3to4lem.3 . . . . 5 g = ((ab) ∪ (cd))
2928ud1lem0a 255 . . . 4 (a1 g) = (a1 ((ab) ∪ (cd)))
30 df-i1 44 . . . 4 (a1 ((ab) ∪ (cd))) = (a ∪ (a ∩ ((ab) ∪ (cd))))
3129, 30ax-r2 36 . . 3 (a1 g) = (a ∪ (a ∩ ((ab) ∪ (cd))))
3231ax-r1 35 . 2 (a ∪ (a ∩ ((ab) ∪ (cd)))) = (a1 g)
3327, 32lbtr 139 1 b ≤ (a1 g)
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ 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:  oa3to4lem3  947
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