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Theorem oa4to6lem4 963
 Description: Lemma for orthoarguesian law (4-variable to 6-variable proof).
Hypotheses
Ref Expression
oa4to6lem.1 ab
oa4to6lem.2 cd
oa4to6lem.3 ef
oa4to6lem.4 g = (((ab) ∪ (cd)) ∪ (ef))
Assertion
Ref Expression
oa4to6lem4 (b ∩ (a ∪ (c ∩ (((ac) ∪ (bd)) ∪ (((ae) ∪ (bf)) ∩ ((ce) ∪ (df))))))) ≤ ((a1 g) ∩ (a ∪ (c ∩ (((ac) ∪ ((a1 g) ∩ (c1 g))) ∪ (((ae) ∪ ((a1 g) ∩ (e1 g))) ∩ ((ce) ∪ ((c1 g) ∩ (e1 g))))))))

Proof of Theorem oa4to6lem4
StepHypRef Expression
1 oa4to6lem.1 . . 3 ab
2 oa4to6lem.2 . . 3 cd
3 oa4to6lem.3 . . 3 ef
4 oa4to6lem.4 . . 3 g = (((ab) ∪ (cd)) ∪ (ef))
51, 2, 3, 4oa4to6lem1 960 . 2 b ≤ (a1 g)
61, 2, 3, 4oa4to6lem2 961 . . . . . . 7 d ≤ (c1 g)
75, 6le2an 169 . . . . . 6 (bd) ≤ ((a1 g) ∩ (c1 g))
87lelor 166 . . . . 5 ((ac) ∪ (bd)) ≤ ((ac) ∪ ((a1 g) ∩ (c1 g)))
91, 2, 3, 4oa4to6lem3 962 . . . . . . . 8 f ≤ (e1 g)
105, 9le2an 169 . . . . . . 7 (bf) ≤ ((a1 g) ∩ (e1 g))
1110lelor 166 . . . . . 6 ((ae) ∪ (bf)) ≤ ((ae) ∪ ((a1 g) ∩ (e1 g)))
126, 9le2an 169 . . . . . . 7 (df) ≤ ((c1 g) ∩ (e1 g))
1312lelor 166 . . . . . 6 ((ce) ∪ (df)) ≤ ((ce) ∪ ((c1 g) ∩ (e1 g)))
1411, 13le2an 169 . . . . 5 (((ae) ∪ (bf)) ∩ ((ce) ∪ (df))) ≤ (((ae) ∪ ((a1 g) ∩ (e1 g))) ∩ ((ce) ∪ ((c1 g) ∩ (e1 g))))
158, 14le2or 168 . . . 4 (((ac) ∪ (bd)) ∪ (((ae) ∪ (bf)) ∩ ((ce) ∪ (df)))) ≤ (((ac) ∪ ((a1 g) ∩ (c1 g))) ∪ (((ae) ∪ ((a1 g) ∩ (e1 g))) ∩ ((ce) ∪ ((c1 g) ∩ (e1 g)))))
1615lelan 167 . . 3 (c ∩ (((ac) ∪ (bd)) ∪ (((ae) ∪ (bf)) ∩ ((ce) ∪ (df))))) ≤ (c ∩ (((ac) ∪ ((a1 g) ∩ (c1 g))) ∪ (((ae) ∪ ((a1 g) ∩ (e1 g))) ∩ ((ce) ∪ ((c1 g) ∩ (e1 g))))))
1716lelor 166 . 2 (a ∪ (c ∩ (((ac) ∪ (bd)) ∪ (((ae) ∪ (bf)) ∩ ((ce) ∪ (df)))))) ≤ (a ∪ (c ∩ (((ac) ∪ ((a1 g) ∩ (c1 g))) ∪ (((ae) ∪ ((a1 g) ∩ (e1 g))) ∩ ((ce) ∪ ((c1 g) ∩ (e1 g)))))))
185, 17le2an 169 1 (b ∩ (a ∪ (c ∩ (((ac) ∪ (bd)) ∪ (((ae) ∪ (bf)) ∩ ((ce) ∪ (df))))))) ≤ ((a1 g) ∩ (a ∪ (c ∩ (((ac) ∪ ((a1 g) ∩ (c1 g))) ∪ (((ae) ∪ ((a1 g) ∩ (e1 g))) ∩ ((ce) ∪ ((c1 g) ∩ (e1 g))))))))
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ 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:  oa4to6dual  964
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