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Theorem oa3to4lem3 947
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
oa3to4lem3 (a ^ (b v (d ^ ((a ^ c) v (b ^ d))))) =< (a ^ ((a ->1 g) v ((c ->1 g) ^ ((a ^ c) v ((a ->1 g) ^ (c ->1 g))))))
Proof of Theorem oa3to4lem3
StepHypRef Expression
1 oa3to4lem.1 . . . 4 a' =< b
2 oa3to4lem.2 . . . 4 c' =< d
3 oa3to4lem.3 . . . 4 g = ((a ^ b) v (c ^ d))
41, 2, 3oa3to4lem1 945 . . 3 b =< (a ->1 g)
51, 2, 3oa3to4lem2 946 . . . 4 d =< (c ->1 g)
64, 5le2an 169 . . . . 5 (b ^ d) =< ((a ->1 g) ^ (c ->1 g))
76lelor 166 . . . 4 ((a ^ c) v (b ^ d)) =< ((a ^ c) v ((a ->1 g) ^ (c ->1 g)))
85, 7le2an 169 . . 3 (d ^ ((a ^ c) v (b ^ d))) =< ((c ->1 g) ^ ((a ^ c) v ((a ->1 g) ^ (c ->1 g))))
94, 8le2or 168 . 2 (b v (d ^ ((a ^ c) v (b ^ d)))) =< ((a ->1 g) v ((c ->1 g) ^ ((a ^ c) v ((a ->1 g) ^ (c ->1 g)))))
109lelan 167 1 (a ^ (b v (d ^ ((a ^ c) v (b ^ d))))) =< (a ^ ((a ->1 g) v ((c ->1 g) ^ ((a ^ c) v ((a ->1 g) ^ (c ->1 g))))))
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->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:  oa3to4lem4  948
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