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Theorem oa3to4lem4 948
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))
oa3to4lem.oa3 (a ^ ((a ->1 g) v ((c ->1 g) ^ ((a ^ c) v ((a ->1 g) ^ (c ->1 g)))))) =< ((a ^ g) v (c ^ g))
Assertion
Ref Expression
oa3to4lem4 (a ^ (b v (d ^ ((a ^ c) v (b ^ d))))) =< g
Proof of Theorem oa3to4lem4
StepHypRef Expression
1 oa3to4lem.1 . . 3 a' =< b
2 oa3to4lem.2 . . 3 c' =< d
3 oa3to4lem.3 . . 3 g = ((a ^ b) v (c ^ d))
41, 2, 3oa3to4lem3 947 . 2 (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))))))
5 oa3to4lem.oa3 . . 3 (a ^ ((a ->1 g) v ((c ->1 g) ^ ((a ^ c) v ((a ->1 g) ^ (c ->1 g)))))) =< ((a ^ g) v (c ^ g))
6 lear 161 . . . 4 (a ^ g) =< g
7 lear 161 . . . 4 (c ^ g) =< g
86, 7lel2or 170 . . 3 ((a ^ g) v (c ^ g)) =< g
95, 8letr 137 . 2 (a ^ ((a ->1 g) v ((c ->1 g) ^ ((a ^ c) v ((a ->1 g) ^ (c ->1 g)))))) =< g
104, 9letr 137 1 (a ^ (b v (d ^ ((a ^ c) v (b ^ d))))) =< 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:  oa3to4lem6  950
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