Theorem oa4to6lem4 943

Description: Lemma for orthoarguesian law (4-variable to 6-variable proof).

Hypotheses

Ref	Expression
oa4to6lem.1	a_|_ =< b
oa4to6lem.2	c_|_ =< d
oa4to6lem.3	e_|_ =< f
oa4to6lem.4	g = (((a ^ b) v (c ^ d)) v (e ^ f))

Assertion

Ref	Expression
oa4to6lem4	(b ^ (a v (c ^ (((a ^ c) v (b ^ d)) v (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f))))))) =< ((a ->1 g) ^ (a v (c ^ (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g))))))))

Proof of Theorem oa4to6lem4

Step	Hyp	Ref	Expression
1		oa4to6lem.1	. . 3 a_|_ =< b
2		oa4to6lem.2	. . 3 c_|_ =< d
3		oa4to6lem.3	. . 3 e_|_ =< f
4		oa4to6lem.4	. . 3 g = (((a ^ b) v (c ^ d)) v (e ^ f))
5	1, 2, 3, 4	oa4to6lem1 940	. 2 b =< (a ->1 g)
6	1, 2, 3, 4	oa4to6lem2 941	. . . . . . 7 d =< (c ->1 g)
7	5, 6	le2an 161	. . . . . 6 (b ^ d) =< ((a ->1 g) ^ (c ->1 g))
8	7	lelor 158	. . . . 5 ((a ^ c) v (b ^ d)) =< ((a ^ c) v ((a ->1 g) ^ (c ->1 g)))
9	1, 2, 3, 4	oa4to6lem3 942	. . . . . . . 8 f =< (e ->1 g)
10	5, 9	le2an 161	. . . . . . 7 (b ^ f) =< ((a ->1 g) ^ (e ->1 g))
11	10	lelor 158	. . . . . 6 ((a ^ e) v (b ^ f)) =< ((a ^ e) v ((a ->1 g) ^ (e ->1 g)))
12	6, 9	le2an 161	. . . . . . 7 (d ^ f) =< ((c ->1 g) ^ (e ->1 g))
13	12	lelor 158	. . . . . 6 ((c ^ e) v (d ^ f)) =< ((c ^ e) v ((c ->1 g) ^ (e ->1 g)))
14	11, 13	le2an 161	. . . . 5 (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f))) =< (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g))))
15	8, 14	le2or 160	. . . 4 (((a ^ c) v (b ^ d)) v (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f)))) =< (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g)))))
16	15	lelan 159	. . 3 (c ^ (((a ^ c) v (b ^ d)) v (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f))))) =< (c ^ (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g))))))
17	16	lelor 158	. 2 (a v (c ^ (((a ^ c) v (b ^ d)) v (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f)))))) =< (a v (c ^ (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g)))))))
18	5, 17	le2an 161	1 (b ^ (a v (c ^ (((a ^ c) v (b ^ d)) v (((a ^ e) v (b ^ f)) ^ ((c ^ e) v (d ^ f))))))) =< ((a ->1 g) ^ (a v (c ^ (((a ^ c) v ((a ->1 g) ^ (c ->1 g))) v (((a ^ e) v ((a ->1 g) ^ (e ->1 g))) ^ ((c ^ e) v ((c ->1 g) ^ (e ->1 g))))))))

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

This theorem is referenced by:  oa4to6dual 944

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

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