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Theorem oa8to5 972
 Description: Orthoarguesian law 5OA converted from 8 to 5 variables.
Hypotheses
Ref Expression
oa8to5.1 (((ab ) ∩ (cd )) ∩ ((ef ) ∩ (gh ))) ≤ (b ∪ (a ∩ (c ∪ ((((ac ) ∩ (bd )) ∩ (((ag ) ∩ (bh )) ∪ ((cg ) ∩ (dh )))) ∩ ((((ae ) ∩ (bf )) ∩ (((ag ) ∩ (bh )) ∪ ((eg ) ∩ (fh )))) ∪ (((ce ) ∩ (df )) ∩ (((cg ) ∩ (dh )) ∪ ((eg ) ∩ (fh )))))))))
oa8to5.2 b = (a1 j)
oa8to5.3 d = (c1 j)
oa8to5.4 f = (e1 j)
oa8to5.5 h = (g1 j)
Assertion
Ref Expression
oa8to5 ((a1 j) ∩ (a ∪ (c ∩ ((((ac) ∪ ((a1 j) ∩ (c1 j))) ∪ (((ag) ∪ ((a1 j) ∩ (g1 j))) ∩ ((cg) ∪ ((c1 j) ∩ (g1 j))))) ∪ ((((ae) ∪ ((a1 j) ∩ (e1 j))) ∪ (((ag) ∪ ((a1 j) ∩ (g1 j))) ∩ ((eg) ∪ ((e1 j) ∩ (g1 j))))) ∩ (((ce) ∪ ((c1 j) ∩ (e1 j))) ∪ (((cg) ∪ ((c1 j) ∩ (g1 j))) ∩ ((eg) ∪ ((e1 j) ∩ (g1 j)))))))))) ≤ (((aj) ∪ (cj)) ∪ ((ej) ∪ (gj)))

Proof of Theorem oa8to5
StepHypRef Expression
1 oa8to5.1 . . 3 (((ab ) ∩ (cd )) ∩ ((ef ) ∩ (gh ))) ≤ (b ∪ (a ∩ (c ∪ ((((ac ) ∩ (bd )) ∩ (((ag ) ∩ (bh )) ∪ ((cg ) ∩ (dh )))) ∩ ((((ae ) ∩ (bf )) ∩ (((ag ) ∩ (bh )) ∪ ((eg ) ∩ (fh )))) ∪ (((ce ) ∩ (df )) ∩ (((cg ) ∩ (dh )) ∪ ((eg ) ∩ (fh )))))))))
21oa8todual 971 . 2 (b ∩ (a ∪ (c ∩ ((((ac) ∪ (bd)) ∪ (((ag) ∪ (bh)) ∩ ((cg) ∪ (dh)))) ∪ ((((ae) ∪ (bf)) ∪ (((ag) ∪ (bh)) ∩ ((eg) ∪ (fh)))) ∩ (((ce) ∪ (df)) ∪ (((cg) ∪ (dh)) ∩ ((eg) ∪ (fh))))))))) ≤ (((ab) ∪ (cd)) ∪ ((ef) ∪ (gh)))
3 oa8to5.2 . . . 4 b = (a1 j)
43con1 66 . . 3 b = (a1 j)
5 oa8to5.3 . . . . . . . . . 10 d = (c1 j)
65con1 66 . . . . . . . . 9 d = (c1 j)
74, 62an 79 . . . . . . . 8 (bd) = ((a1 j) ∩ (c1 j))
87lor 70 . . . . . . 7 ((ac) ∪ (bd)) = ((ac) ∪ ((a1 j) ∩ (c1 j)))
9 oa8to5.5 . . . . . . . . . . 11 h = (g1 j)
109con1 66 . . . . . . . . . 10 h = (g1 j)
114, 102an 79 . . . . . . . . 9 (bh) = ((a1 j) ∩ (g1 j))
1211lor 70 . . . . . . . 8 ((ag) ∪ (bh)) = ((ag) ∪ ((a1 j) ∩ (g1 j)))
136, 102an 79 . . . . . . . . 9 (dh) = ((c1 j) ∩ (g1 j))
1413lor 70 . . . . . . . 8 ((cg) ∪ (dh)) = ((cg) ∪ ((c1 j) ∩ (g1 j)))
1512, 142an 79 . . . . . . 7 (((ag) ∪ (bh)) ∩ ((cg) ∪ (dh))) = (((ag) ∪ ((a1 j) ∩ (g1 j))) ∩ ((cg) ∪ ((c1 j) ∩ (g1 j))))
168, 152or 72 . . . . . 6 (((ac) ∪ (bd)) ∪ (((ag) ∪ (bh)) ∩ ((cg) ∪ (dh)))) = (((ac) ∪ ((a1 j) ∩ (c1 j))) ∪ (((ag) ∪ ((a1 j) ∩ (g1 j))) ∩ ((cg) ∪ ((c1 j) ∩ (g1 j)))))
17 oa8to5.4 . . . . . . . . . . 11 f = (e1 j)
1817con1 66 . . . . . . . . . 10 f = (e1 j)
194, 182an 79 . . . . . . . . 9 (bf) = ((a1 j) ∩ (e1 j))
2019lor 70 . . . . . . . 8 ((ae) ∪ (bf)) = ((ae) ∪ ((a1 j) ∩ (e1 j)))
2118, 102an 79 . . . . . . . . . 10 (fh) = ((e1 j) ∩ (g1 j))
2221lor 70 . . . . . . . . 9 ((eg) ∪ (fh)) = ((eg) ∪ ((e1 j) ∩ (g1 j)))
2312, 222an 79 . . . . . . . 8 (((ag) ∪ (bh)) ∩ ((eg) ∪ (fh))) = (((ag) ∪ ((a1 j) ∩ (g1 j))) ∩ ((eg) ∪ ((e1 j) ∩ (g1 j))))
2420, 232or 72 . . . . . . 7 (((ae) ∪ (bf)) ∪ (((ag) ∪ (bh)) ∩ ((eg) ∪ (fh)))) = (((ae) ∪ ((a1 j) ∩ (e1 j))) ∪ (((ag) ∪ ((a1 j) ∩ (g1 j))) ∩ ((eg) ∪ ((e1 j) ∩ (g1 j)))))
256, 182an 79 . . . . . . . . 9 (df) = ((c1 j) ∩ (e1 j))
2625lor 70 . . . . . . . 8 ((ce) ∪ (df)) = ((ce) ∪ ((c1 j) ∩ (e1 j)))
2714, 222an 79 . . . . . . . 8 (((cg) ∪ (dh)) ∩ ((eg) ∪ (fh))) = (((cg) ∪ ((c1 j) ∩ (g1 j))) ∩ ((eg) ∪ ((e1 j) ∩ (g1 j))))
2826, 272or 72 . . . . . . 7 (((ce) ∪ (df)) ∪ (((cg) ∪ (dh)) ∩ ((eg) ∪ (fh)))) = (((ce) ∪ ((c1 j) ∩ (e1 j))) ∪ (((cg) ∪ ((c1 j) ∩ (g1 j))) ∩ ((eg) ∪ ((e1 j) ∩ (g1 j)))))
2924, 282an 79 . . . . . 6 ((((ae) ∪ (bf)) ∪ (((ag) ∪ (bh)) ∩ ((eg) ∪ (fh)))) ∩ (((ce) ∪ (df)) ∪ (((cg) ∪ (dh)) ∩ ((eg) ∪ (fh))))) = ((((ae) ∪ ((a1 j) ∩ (e1 j))) ∪ (((ag) ∪ ((a1 j) ∩ (g1 j))) ∩ ((eg) ∪ ((e1 j) ∩ (g1 j))))) ∩ (((ce) ∪ ((c1 j) ∩ (e1 j))) ∪ (((cg) ∪ ((c1 j) ∩ (g1 j))) ∩ ((eg) ∪ ((e1 j) ∩ (g1 j))))))
3016, 292or 72 . . . . 5 ((((ac) ∪ (bd)) ∪ (((ag) ∪ (bh)) ∩ ((cg) ∪ (dh)))) ∪ ((((ae) ∪ (bf)) ∪ (((ag) ∪ (bh)) ∩ ((eg) ∪ (fh)))) ∩ (((ce) ∪ (df)) ∪ (((cg) ∪ (dh)) ∩ ((eg) ∪ (fh)))))) = ((((ac) ∪ ((a1 j) ∩ (c1 j))) ∪ (((ag) ∪ ((a1 j) ∩ (g1 j))) ∩ ((cg) ∪ ((c1 j) ∩ (g1 j))))) ∪ ((((ae) ∪ ((a1 j) ∩ (e1 j))) ∪ (((ag) ∪ ((a1 j) ∩ (g1 j))) ∩ ((eg) ∪ ((e1 j) ∩ (g1 j))))) ∩ (((ce) ∪ ((c1 j) ∩ (e1 j))) ∪ (((cg) ∪ ((c1 j) ∩ (g1 j))) ∩ ((eg) ∪ ((e1 j) ∩ (g1 j)))))))
3130lan 77 . . . 4 (c ∩ ((((ac) ∪ (bd)) ∪ (((ag) ∪ (bh)) ∩ ((cg) ∪ (dh)))) ∪ ((((ae) ∪ (bf)) ∪ (((ag) ∪ (bh)) ∩ ((eg) ∪ (fh)))) ∩ (((ce) ∪ (df)) ∪ (((cg) ∪ (dh)) ∩ ((eg) ∪ (fh))))))) = (c ∩ ((((ac) ∪ ((a1 j) ∩ (c1 j))) ∪ (((ag) ∪ ((a1 j) ∩ (g1 j))) ∩ ((cg) ∪ ((c1 j) ∩ (g1 j))))) ∪ ((((ae) ∪ ((a1 j) ∩ (e1 j))) ∪ (((ag) ∪ ((a1 j) ∩ (g1 j))) ∩ ((eg) ∪ ((e1 j) ∩ (g1 j))))) ∩ (((ce) ∪ ((c1 j) ∩ (e1 j))) ∪ (((cg) ∪ ((c1 j) ∩ (g1 j))) ∩ ((eg) ∪ ((e1 j) ∩ (g1 j))))))))
3231lor 70 . . 3 (a ∪ (c ∩ ((((ac) ∪ (bd)) ∪ (((ag) ∪ (bh)) ∩ ((cg) ∪ (dh)))) ∪ ((((ae) ∪ (bf)) ∪ (((ag) ∪ (bh)) ∩ ((eg) ∪ (fh)))) ∩ (((ce) ∪ (df)) ∪ (((cg) ∪ (dh)) ∩ ((eg) ∪ (fh)))))))) = (a ∪ (c ∩ ((((ac) ∪ ((a1 j) ∩ (c1 j))) ∪ (((ag) ∪ ((a1 j) ∩ (g1 j))) ∩ ((cg) ∪ ((c1 j) ∩ (g1 j))))) ∪ ((((ae) ∪ ((a1 j) ∩ (e1 j))) ∪ (((ag) ∪ ((a1 j) ∩ (g1 j))) ∩ ((eg) ∪ ((e1 j) ∩ (g1 j))))) ∩ (((ce) ∪ ((c1 j) ∩ (e1 j))) ∪ (((cg) ∪ ((c1 j) ∩ (g1 j))) ∩ ((eg) ∪ ((e1 j) ∩ (g1 j)))))))))
334, 322an 79 . 2 (b ∩ (a ∪ (c ∩ ((((ac) ∪ (bd)) ∪ (((ag) ∪ (bh)) ∩ ((cg) ∪ (dh)))) ∪ ((((ae) ∪ (bf)) ∪ (((ag) ∪ (bh)) ∩ ((eg) ∪ (fh)))) ∩ (((ce) ∪ (df)) ∪ (((cg) ∪ (dh)) ∩ ((eg) ∪ (fh))))))))) = ((a1 j) ∩ (a ∪ (c ∩ ((((ac) ∪ ((a1 j) ∩ (c1 j))) ∪ (((ag) ∪ ((a1 j) ∩ (g1 j))) ∩ ((cg) ∪ ((c1 j) ∩ (g1 j))))) ∪ ((((ae) ∪ ((a1 j) ∩ (e1 j))) ∪ (((ag) ∪ ((a1 j) ∩ (g1 j))) ∩ ((eg) ∪ ((e1 j) ∩ (g1 j))))) ∩ (((ce) ∪ ((c1 j) ∩ (e1 j))) ∪ (((cg) ∪ ((c1 j) ∩ (g1 j))) ∩ ((eg) ∪ ((e1 j) ∩ (g1 j))))))))))
344lan 77 . . . . 5 (ab) = (a ∩ (a1 j))
35 ancom 74 . . . . 5 (a ∩ (a1 j)) = ((a1 j) ∩ a)
36 u1lemaa 600 . . . . 5 ((a1 j) ∩ a) = (aj)
3734, 35, 363tr 65 . . . 4 (ab) = (aj)
386lan 77 . . . . 5 (cd) = (c ∩ (c1 j))
39 ancom 74 . . . . 5 (c ∩ (c1 j)) = ((c1 j) ∩ c)
40 u1lemaa 600 . . . . 5 ((c1 j) ∩ c) = (cj)
4138, 39, 403tr 65 . . . 4 (cd) = (cj)
4237, 412or 72 . . 3 ((ab) ∪ (cd)) = ((aj) ∪ (cj))
4318lan 77 . . . . 5 (ef) = (e ∩ (e1 j))
44 ancom 74 . . . . 5 (e ∩ (e1 j)) = ((e1 j) ∩ e)
45 u1lemaa 600 . . . . 5 ((e1 j) ∩ e) = (ej)
4643, 44, 453tr 65 . . . 4 (ef) = (ej)
4710lan 77 . . . . 5 (gh) = (g ∩ (g1 j))
48 ancom 74 . . . . 5 (g ∩ (g1 j)) = ((g1 j) ∩ g)
49 u1lemaa 600 . . . . 5 ((g1 j) ∩ g) = (gj)
5047, 48, 493tr 65 . . . 4 (gh) = (gj)
5146, 502or 72 . . 3 ((ef) ∪ (gh)) = ((ej) ∪ (gj))
5242, 512or 72 . 2 (((ab) ∪ (cd)) ∪ ((ef) ∪ (gh))) = (((aj) ∪ (cj)) ∪ ((ej) ∪ (gj)))
532, 33, 52le3tr2 141 1 ((a1 j) ∩ (a ∪ (c ∩ ((((ac) ∪ ((a1 j) ∩ (c1 j))) ∪ (((ag) ∪ ((a1 j) ∩ (g1 j))) ∩ ((cg) ∪ ((c1 j) ∩ (g1 j))))) ∪ ((((ae) ∪ ((a1 j) ∩ (e1 j))) ∪ (((ag) ∪ ((a1 j) ∩ (g1 j))) ∩ ((eg) ∪ ((e1 j) ∩ (g1 j))))) ∩ (((ce) ∪ ((c1 j) ∩ (e1 j))) ∪ (((cg) ∪ ((c1 j) ∩ (g1 j))) ∩ ((eg) ∪ ((e1 j) ∩ (g1 j)))))))))) ≤ (((aj) ∪ (cj)) ∪ ((ej) ∪ (gj)))
 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: (None)
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