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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 a' =< b
oa3to4lem.2 c' =< d
oa3to4lem.3 g = ((a ^ b) v (c ^ d))
Assertion
Ref Expression
oa3to4lem1 b =< (a ->1 g)
Proof of Theorem oa3to4lem1
StepHypRef Expression
1 leor 159 . . . 4 b =< (a' v b)
2 comid 187 . . . . . . . . 9 a C a
32comcom3 454 . . . . . . . 8 a' C a
4 oa3to4lem.1 . . . . . . . . 9 a' =< b
54lecom 180 . . . . . . . 8 a' C b
63, 5fh3 471 . . . . . . 7 (a' v (a ^ b)) = ((a' v a) ^ (a' v b))
7 ancom 74 . . . . . . . 8 (1 ^ (a' v b)) = ((a' v b) ^ 1)
8 df-t 41 . . . . . . . . . 10 1 = (a v a')
9 ax-a2 31 . . . . . . . . . 10 (a v a') = (a' v a)
108, 9ax-r2 36 . . . . . . . . 9 1 = (a' v a)
1110ran 78 . . . . . . . 8 (1 ^ (a' v b)) = ((a' v a) ^ (a' v b))
12 an1 106 . . . . . . . 8 ((a' v b) ^ 1) = (a' v b)
137, 11, 123tr2 64 . . . . . . 7 ((a' v a) ^ (a' v b)) = (a' v b)
146, 13ax-r2 36 . . . . . 6 (a' v (a ^ b)) = (a' v b)
1514ax-r1 35 . . . . 5 (a' v b) = (a' v (a ^ b))
16 anidm 111 . . . . . . . . 9 (a ^ a) = a
1716ran 78 . . . . . . . 8 ((a ^ a) ^ b) = (a ^ b)
1817ax-r1 35 . . . . . . 7 (a ^ b) = ((a ^ a) ^ b)
19 anass 76 . . . . . . 7 ((a ^ a) ^ b) = (a ^ (a ^ b))
2018, 19ax-r2 36 . . . . . 6 (a ^ b) = (a ^ (a ^ b))
2120lor 70 . . . . 5 (a' v (a ^ b)) = (a' v (a ^ (a ^ b)))
2215, 21ax-r2 36 . . . 4 (a' v b) = (a' v (a ^ (a ^ b)))
231, 22lbtr 139 . . 3 b =< (a' v (a ^ (a ^ b)))
24 leo 158 . . . . 5 (a ^ b) =< ((a ^ b) v (c ^ d))
2524lelan 167 . . . 4 (a ^ (a ^ b)) =< (a ^ ((a ^ b) v (c ^ d)))
2625lelor 166 . . 3 (a' v (a ^ (a ^ b))) =< (a' v (a ^ ((a ^ b) v (c ^ d))))
2723, 26letr 137 . 2 b =< (a' v (a ^ ((a ^ b) v (c ^ d))))
28 oa3to4lem.3 . . . . 5 g = ((a ^ b) v (c ^ d))
2928ud1lem0a 255 . . . 4 (a ->1 g) = (a ->1 ((a ^ b) v (c ^ d)))
30 df-i1 44 . . . 4 (a ->1 ((a ^ b) v (c ^ d))) = (a' v (a ^ ((a ^ b) v (c ^ d))))
3129, 30ax-r2 36 . . 3 (a ->1 g) = (a' v (a ^ ((a ^ b) v (c ^ d))))
3231ax-r1 35 . 2 (a' v (a ^ ((a ^ b) v (c ^ d)))) = (a ->1 g)
3327, 32lbtr 139 1 b =< (a ->1 g)
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   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:  oa3to4lem3  947
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