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Theorem oa4to6lem3 962
 Description: Lemma for orthoarguesian law (4-variable to 6-variable proof). (Contributed by NM, 18-Dec-1998.)
Hypotheses
Ref Expression
oa4to6lem.1 ab
oa4to6lem.2 cd
oa4to6lem.3 ef
oa4to6lem.4 g = (((ab) ∪ (cd)) ∪ (ef))
Assertion
Ref Expression
oa4to6lem3 f ≤ (e1 g)

Proof of Theorem oa4to6lem3
StepHypRef Expression
1 leor 159 . . . 4 f ≤ (ef)
2 comid 187 . . . . . . . . 9 e C e
32comcom3 454 . . . . . . . 8 e C e
4 oa4to6lem.3 . . . . . . . . 9 ef
54lecom 180 . . . . . . . 8 e C f
63, 5fh3 471 . . . . . . 7 (e ∪ (ef)) = ((ee) ∩ (ef))
7 ancom 74 . . . . . . . 8 (1 ∩ (ef)) = ((ef) ∩ 1)
8 df-t 41 . . . . . . . . . 10 1 = (ee )
9 ax-a2 31 . . . . . . . . . 10 (ee ) = (ee)
108, 9ax-r2 36 . . . . . . . . 9 1 = (ee)
1110ran 78 . . . . . . . 8 (1 ∩ (ef)) = ((ee) ∩ (ef))
12 an1 106 . . . . . . . 8 ((ef) ∩ 1) = (ef)
137, 11, 123tr2 64 . . . . . . 7 ((ee) ∩ (ef)) = (ef)
146, 13ax-r2 36 . . . . . 6 (e ∪ (ef)) = (ef)
1514ax-r1 35 . . . . 5 (ef) = (e ∪ (ef))
16 anidm 111 . . . . . . . . 9 (ee) = e
1716ran 78 . . . . . . . 8 ((ee) ∩ f) = (ef)
1817ax-r1 35 . . . . . . 7 (ef) = ((ee) ∩ f)
19 anass 76 . . . . . . 7 ((ee) ∩ f) = (e ∩ (ef))
2018, 19ax-r2 36 . . . . . 6 (ef) = (e ∩ (ef))
2120lor 70 . . . . 5 (e ∪ (ef)) = (e ∪ (e ∩ (ef)))
2215, 21ax-r2 36 . . . 4 (ef) = (e ∪ (e ∩ (ef)))
231, 22lbtr 139 . . 3 f ≤ (e ∪ (e ∩ (ef)))
24 leor 159 . . . . 5 (ef) ≤ (((ab) ∪ (cd)) ∪ (ef))
2524lelan 167 . . . 4 (e ∩ (ef)) ≤ (e ∩ (((ab) ∪ (cd)) ∪ (ef)))
2625lelor 166 . . 3 (e ∪ (e ∩ (ef))) ≤ (e ∪ (e ∩ (((ab) ∪ (cd)) ∪ (ef))))
2723, 26letr 137 . 2 f ≤ (e ∪ (e ∩ (((ab) ∪ (cd)) ∪ (ef))))
28 oa4to6lem.4 . . . . 5 g = (((ab) ∪ (cd)) ∪ (ef))
2928ud1lem0a 255 . . . 4 (e1 g) = (e1 (((ab) ∪ (cd)) ∪ (ef)))
30 df-i1 44 . . . 4 (e1 (((ab) ∪ (cd)) ∪ (ef))) = (e ∪ (e ∩ (((ab) ∪ (cd)) ∪ (ef))))
3129, 30ax-r2 36 . . 3 (e1 g) = (e ∪ (e ∩ (((ab) ∪ (cd)) ∪ (ef))))
3231ax-r1 35 . 2 (e ∪ (e ∩ (((ab) ∪ (cd)) ∪ (ef)))) = (e1 g)
3327, 32lbtr 139 1 f ≤ (e1 g)
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →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:  oa4to6lem4  963
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