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Theorem sadm3 838
 Description: Weak DeMorgan's law for attempt at Sasaki algebra.
Assertion
Ref Expression
sadm3 (((a1 c) ∩ (b1 c)) →1 c) ≤ ((a1 c) ∪ (b1 c))

Proof of Theorem sadm3
StepHypRef Expression
1 oran3 93 . . . . . . 7 ((a1 c) ∪ (b1 c) ) = ((a1 c) ∩ (b1 c))
21ax-r1 35 . . . . . 6 ((a1 c) ∩ (b1 c)) = ((a1 c) ∪ (b1 c) )
3 u1lem9a 777 . . . . . . 7 (a1 c)a
4 u1lem9a 777 . . . . . . 7 (b1 c)b
53, 4le2or 168 . . . . . 6 ((a1 c) ∪ (b1 c) ) ≤ (ab )
62, 5bltr 138 . . . . 5 ((a1 c) ∩ (b1 c)) ≤ (ab )
7 an32 83 . . . . . 6 (((a1 c) ∩ (b1 c)) ∩ c) = (((a1 c) ∩ c) ∩ (b1 c))
8 lea 160 . . . . . 6 (((a1 c) ∩ c) ∩ (b1 c)) ≤ ((a1 c) ∩ c)
97, 8bltr 138 . . . . 5 (((a1 c) ∩ (b1 c)) ∩ c) ≤ ((a1 c) ∩ c)
106, 9le2or 168 . . . 4 (((a1 c) ∩ (b1 c)) ∪ (((a1 c) ∩ (b1 c)) ∩ c)) ≤ ((ab ) ∪ ((a1 c) ∩ c))
11 leo 158 . . . . . 6 (ab ) ≤ ((ab ) ∪ (ac))
12 or32 82 . . . . . 6 ((ab ) ∪ (ac)) = ((a ∪ (ac)) ∪ b )
1311, 12lbtr 139 . . . . 5 (ab ) ≤ ((a ∪ (ac)) ∪ b )
14 u1lemab 610 . . . . . . 7 ((a1 c) ∩ c) = ((ac) ∪ (a c))
15 lea 160 . . . . . . . 8 (ac) ≤ a
16 ax-a1 30 . . . . . . . . . . 11 a = a
1716ax-r1 35 . . . . . . . . . 10 a = a
1817bile 142 . . . . . . . . 9 a a
1918leran 153 . . . . . . . 8 (a c) ≤ (ac)
2015, 19le2or 168 . . . . . . 7 ((ac) ∪ (a c)) ≤ (a ∪ (ac))
2114, 20bltr 138 . . . . . 6 ((a1 c) ∩ c) ≤ (a ∪ (ac))
22 leo 158 . . . . . 6 (a ∪ (ac)) ≤ ((a ∪ (ac)) ∪ b )
2321, 22letr 137 . . . . 5 ((a1 c) ∩ c) ≤ ((a ∪ (ac)) ∪ b )
2413, 23lel2or 170 . . . 4 ((ab ) ∪ ((a1 c) ∩ c)) ≤ ((a ∪ (ac)) ∪ b )
2510, 24letr 137 . . 3 (((a1 c) ∩ (b1 c)) ∪ (((a1 c) ∩ (b1 c)) ∩ c)) ≤ ((a ∪ (ac)) ∪ b )
26 leo 158 . . . 4 b ≤ (b ∪ (bc))
2726lelor 166 . . 3 ((a ∪ (ac)) ∪ b ) ≤ ((a ∪ (ac)) ∪ (b ∪ (bc)))
2825, 27letr 137 . 2 (((a1 c) ∩ (b1 c)) ∪ (((a1 c) ∩ (b1 c)) ∩ c)) ≤ ((a ∪ (ac)) ∪ (b ∪ (bc)))
29 df-i1 44 . 2 (((a1 c) ∩ (b1 c)) →1 c) = (((a1 c) ∩ (b1 c)) ∪ (((a1 c) ∩ (b1 c)) ∩ c))
30 df-i1 44 . . 3 (a1 c) = (a ∪ (ac))
31 df-i1 44 . . 3 (b1 c) = (b ∪ (bc))
3230, 312or 72 . 2 ((a1 c) ∪ (b1 c)) = ((a ∪ (ac)) ∪ (b ∪ (bc)))
3328, 29, 32le3tr1 140 1 (((a1 c) ∩ (b1 c)) →1 c) ≤ ((a1 c) ∪ (b1 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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