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Theorem oas 925
 Description: "Strengthening" lemma for studying the orthoarguesian law. (Contributed by NM, 25-Dec-1998.)
Hypothesis
Ref Expression
oas.1 (a ∩ (ab)) ≤ c
Assertion
Ref Expression
oas ((a1 c) ∩ (ab)) ≤ c

Proof of Theorem oas
StepHypRef Expression
1 oml 445 . . . . . 6 (a ∪ (a ∩ (ab))) = (ab)
21ax-r1 35 . . . . 5 (ab) = (a ∪ (a ∩ (ab)))
3 lea 160 . . . . . . 7 (a ∩ (ab)) ≤ a
4 oas.1 . . . . . . 7 (a ∩ (ab)) ≤ c
53, 4ler2an 173 . . . . . 6 (a ∩ (ab)) ≤ (ac)
65lelor 166 . . . . 5 (a ∪ (a ∩ (ab))) ≤ (a ∪ (ac))
72, 6bltr 138 . . . 4 (ab) ≤ (a ∪ (ac))
87lelan 167 . . 3 ((a1 c) ∩ (ab)) ≤ ((a1 c) ∩ (a ∪ (ac)))
9 u1lemc1 680 . . . . 5 a C (a1 c)
10 comanr1 464 . . . . . 6 a C (ac)
1110comcom6 459 . . . . 5 a C (ac)
129, 11fh2 470 . . . 4 ((a1 c) ∩ (a ∪ (ac))) = (((a1 c) ∩ a) ∪ ((a1 c) ∩ (ac)))
13 u1lemaa 600 . . . . 5 ((a1 c) ∩ a) = (ac)
14 ancom 74 . . . . . 6 ((a1 c) ∩ (ac)) = ((ac) ∩ (a1 c))
15 lea 160 . . . . . . . 8 (ac) ≤ a
16 leo 158 . . . . . . . . 9 a ≤ (a ∪ (ac))
17 df-i1 44 . . . . . . . . . 10 (a1 c) = (a ∪ (ac))
1817ax-r1 35 . . . . . . . . 9 (a ∪ (ac)) = (a1 c)
1916, 18lbtr 139 . . . . . . . 8 a ≤ (a1 c)
2015, 19letr 137 . . . . . . 7 (ac) ≤ (a1 c)
2120df2le2 136 . . . . . 6 ((ac) ∩ (a1 c)) = (ac)
2214, 21ax-r2 36 . . . . 5 ((a1 c) ∩ (ac)) = (ac)
2313, 222or 72 . . . 4 (((a1 c) ∩ a) ∪ ((a1 c) ∩ (ac))) = ((ac) ∪ (ac))
2412, 23ax-r2 36 . . 3 ((a1 c) ∩ (a ∪ (ac))) = ((ac) ∪ (ac))
258, 24lbtr 139 . 2 ((a1 c) ∩ (ab)) ≤ ((ac) ∪ (ac))
26 lear 161 . . 3 (ac) ≤ c
27 lear 161 . . 3 (ac) ≤ c
2826, 27lel2or 170 . 2 ((ac) ∪ (ac)) ≤ c
2925, 28letr 137 1 ((a1 c) ∩ (ab)) ≤ 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:  oa4ctob  967  oa3-2wto2  989
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