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Theorem sa5 836
 Description: Possible axiom for a "Sasaki algebra" for orthoarguesian lattices.
Hypothesis
Ref Expression
sa5.1 (a1 c) ≤ (b1 c)
Assertion
Ref Expression
sa5 (b1 c) ≤ ((a1 c) ∪ c)

Proof of Theorem sa5
StepHypRef Expression
1 leor 159 . . . . 5 b ≤ (cb)
2 ax-a2 31 . . . . . . . . . 10 (bc ) = (cb )
32lan 77 . . . . . . . . 9 (b ∩ (bc )) = (b ∩ (cb ))
43ax-r5 38 . . . . . . . 8 ((b ∩ (bc )) ∪ c) = ((b ∩ (cb )) ∪ c)
5 ax-a2 31 . . . . . . . 8 ((b ∩ (cb )) ∪ c) = (c ∪ (b ∩ (cb )))
6 oml6 488 . . . . . . . 8 (c ∪ (b ∩ (cb ))) = (cb)
74, 5, 63tr 65 . . . . . . 7 ((b ∩ (bc )) ∪ c) = (cb)
87ax-r1 35 . . . . . 6 (cb) = ((b ∩ (bc )) ∪ c)
9 sa5.1 . . . . . . . . . 10 (a1 c) ≤ (b1 c)
109lecon 154 . . . . . . . . 9 (b1 c) ≤ (a1 c)
11 ud1lem0c 277 . . . . . . . . 9 (b1 c) = (b ∩ (bc ))
12 ud1lem0c 277 . . . . . . . . 9 (a1 c) = (a ∩ (ac ))
1310, 11, 12le3tr2 141 . . . . . . . 8 (b ∩ (bc )) ≤ (a ∩ (ac ))
14 lea 160 . . . . . . . 8 (a ∩ (ac )) ≤ a
1513, 14letr 137 . . . . . . 7 (b ∩ (bc )) ≤ a
1615leror 152 . . . . . 6 ((b ∩ (bc )) ∪ c) ≤ (ac)
178, 16bltr 138 . . . . 5 (cb) ≤ (ac)
181, 17letr 137 . . . 4 b ≤ (ac)
19 ax-a1 30 . . . 4 b = b
20 ax-a1 30 . . . . . 6 a = a
21 ax-a2 31 . . . . . . 7 (c ∪ (ca )) = ((ca ) ∪ c)
22 orabs 120 . . . . . . 7 (c ∪ (ca )) = c
23 ancom 74 . . . . . . . 8 (ca ) = (ac)
2423ax-r5 38 . . . . . . 7 ((ca ) ∪ c) = ((ac) ∪ c)
2521, 22, 243tr2 64 . . . . . 6 c = ((ac) ∪ c)
2620, 252or 72 . . . . 5 (ac) = (a ∪ ((ac) ∪ c))
27 ax-a3 32 . . . . . 6 ((a ∪ (ac)) ∪ c) = (a ∪ ((ac) ∪ c))
2827ax-r1 35 . . . . 5 (a ∪ ((ac) ∪ c)) = ((a ∪ (ac)) ∪ c)
2926, 28ax-r2 36 . . . 4 (ac) = ((a ∪ (ac)) ∪ c)
3018, 19, 29le3tr2 141 . . 3 b ≤ ((a ∪ (ac)) ∪ c)
31 lear 161 . . . 4 (bc) ≤ c
32 leor 159 . . . 4 c ≤ ((a ∪ (ac)) ∪ c)
3331, 32letr 137 . . 3 (bc) ≤ ((a ∪ (ac)) ∪ c)
3430, 33lel2or 170 . 2 (b ∪ (bc)) ≤ ((a ∪ (ac)) ∪ c)
35 df-i1 44 . 2 (b1 c) = (b ∪ (bc))
36 df-i1 44 . . 3 (a1 c) = (a ∪ (ac))
3736ax-r5 38 . 2 ((a1 c) ∪ c) = ((a ∪ (ac)) ∪ c)
3834, 35, 37le3tr1 140 1 (b1 c) ≤ ((a1 c) ∪ c)
 Colors of variables: term Syntax hints:   ≤ 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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