Theorem oa3to4lem4 928

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

Step	Hyp	Ref	Expression
1		oa3to4lem.1	. . 3 a_|_ =< b
2		oa3to4lem.2	. . 3 c_|_ =< d
3		oa3to4lem.3	. . 3 g = ((a ^ b) v (c ^ d))
4	1, 2, 3	oa3to4lem3 927	. 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 153	. . . 4 (a ^ g) =< g
7		lear 153	. . . 4 (c ^ g) =< g
8	6, 7	lel2or 162	. . 3 ((a ^ g) v (c ^ g)) =< g
9	5, 8	letr 129	. 2 (a ^ ((a ->1 g) v ((c ->1 g) ^ ((a ^ c) v ((a ->1 g) ^ (c ->1 g)))))) =< g
10	4, 9	letr 129	1 (a ^ (b v (d ^ ((a ^ c) v (b ^ d))))) =< g

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13

This theorem is referenced by:  oa3to4lem6 930

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

metamath.org