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Theorem oa3to4lem2 946
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
oa3to4lem2 d =< (c ->1 g)
Proof of Theorem oa3to4lem2
StepHypRef Expression
1 leor 159 . . . 4 d =< (c' v d)
2 comid 187 . . . . . . . . 9 c C c
32comcom3 454 . . . . . . . 8 c' C c
4 oa3to4lem.2 . . . . . . . . 9 c' =< d
54lecom 180 . . . . . . . 8 c' C d
63, 5fh3 471 . . . . . . 7 (c' v (c ^ d)) = ((c' v c) ^ (c' v d))
7 ancom 74 . . . . . . . 8 (1 ^ (c' v d)) = ((c' v d) ^ 1)
8 df-t 41 . . . . . . . . . 10 1 = (c v c')
9 ax-a2 31 . . . . . . . . . 10 (c v c') = (c' v c)
108, 9ax-r2 36 . . . . . . . . 9 1 = (c' v c)
1110ran 78 . . . . . . . 8 (1 ^ (c' v d)) = ((c' v c) ^ (c' v d))
12 an1 106 . . . . . . . 8 ((c' v d) ^ 1) = (c' v d)
137, 11, 123tr2 64 . . . . . . 7 ((c' v c) ^ (c' v d)) = (c' v d)
146, 13ax-r2 36 . . . . . 6 (c' v (c ^ d)) = (c' v d)
1514ax-r1 35 . . . . 5 (c' v d) = (c' v (c ^ d))
16 anidm 111 . . . . . . . . 9 (c ^ c) = c
1716ran 78 . . . . . . . 8 ((c ^ c) ^ d) = (c ^ d)
1817ax-r1 35 . . . . . . 7 (c ^ d) = ((c ^ c) ^ d)
19 anass 76 . . . . . . 7 ((c ^ c) ^ d) = (c ^ (c ^ d))
2018, 19ax-r2 36 . . . . . 6 (c ^ d) = (c ^ (c ^ d))
2120lor 70 . . . . 5 (c' v (c ^ d)) = (c' v (c ^ (c ^ d)))
2215, 21ax-r2 36 . . . 4 (c' v d) = (c' v (c ^ (c ^ d)))
231, 22lbtr 139 . . 3 d =< (c' v (c ^ (c ^ d)))
24 leor 159 . . . . 5 (c ^ d) =< ((a ^ b) v (c ^ d))
2524lelan 167 . . . 4 (c ^ (c ^ d)) =< (c ^ ((a ^ b) v (c ^ d)))
2625lelor 166 . . 3 (c' v (c ^ (c ^ d))) =< (c' v (c ^ ((a ^ b) v (c ^ d))))
2723, 26letr 137 . 2 d =< (c' v (c ^ ((a ^ b) v (c ^ d))))
28 oa3to4lem.3 . . . . 5 g = ((a ^ b) v (c ^ d))
2928ud1lem0a 255 . . . 4 (c ->1 g) = (c ->1 ((a ^ b) v (c ^ d)))
30 df-i1 44 . . . 4 (c ->1 ((a ^ b) v (c ^ d))) = (c' v (c ^ ((a ^ b) v (c ^ d))))
3129, 30ax-r2 36 . . 3 (c ->1 g) = (c' v (c ^ ((a ^ b) v (c ^ d))))
3231ax-r1 35 . 2 (c' v (c ^ ((a ^ b) v (c ^ d)))) = (c ->1 g)
3327, 32lbtr 139 1 d =< (c ->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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