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Theorem d4oa 996
 Description: Variant of proper 4-OA proved from OA distributive law.
Hypotheses
Ref Expression
d4oa.2 e = ((ab) ∪ ((a1 d) ∩ (b1 d)))
d4oa.1 f = (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d))))
Assertion
Ref Expression
d4oa ((a1 d) ∩ (ef)) ≤ (b1 d)

Proof of Theorem d4oa
StepHypRef Expression
1 ax-a2 31 . . . 4 (ef) = (fe)
21lan 77 . . 3 ((a1 d) ∩ (ef)) = ((a1 d) ∩ (fe))
3 id 59 . . . 4 (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d)))) = (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d))))
4 d4oa.2 . . . . 5 e = ((ab) ∪ ((a1 d) ∩ (b1 d)))
5 d4oa.1 . . . . 5 f = (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d))))
64, 52or 72 . . . 4 (ef) = (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d)))))
7 leid 148 . . . 4 (a1 d) ≤ (a1 d)
8 leor 159 . . . 4 f ≤ (ef)
9 leo 158 . . . 4 e ≤ (ef)
10 leor 159 . . . . 5 ((a1 d) ∩ (b1 d)) ≤ ((ab) ∪ ((a1 d) ∩ (b1 d)))
114ax-r1 35 . . . . 5 ((ab) ∪ ((a1 d) ∩ (b1 d))) = e
1210, 11lbtr 139 . . . 4 ((a1 d) ∩ (b1 d)) ≤ e
133, 6, 7, 8, 9, 12ax-oadist 994 . . 3 ((a1 d) ∩ (fe)) = (((a1 d) ∩ f) ∪ ((a1 d) ∩ e))
142, 13ax-r2 36 . 2 ((a1 d) ∩ (ef)) = (((a1 d) ∩ f) ∪ ((a1 d) ∩ e))
155lan 77 . . . . . 6 ((a1 d) ∩ f) = ((a1 d) ∩ (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d)))))
16 anass 76 . . . . . . 7 (((a1 d) ∩ ((ac) ∪ ((a1 d) ∩ (c1 d)))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d)))) = ((a1 d) ∩ (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d)))))
1716ax-r1 35 . . . . . 6 ((a1 d) ∩ (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d))))) = (((a1 d) ∩ ((ac) ∪ ((a1 d) ∩ (c1 d)))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d))))
1815, 17ax-r2 36 . . . . 5 ((a1 d) ∩ f) = (((a1 d) ∩ ((ac) ∪ ((a1 d) ∩ (c1 d)))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d))))
19 id 59 . . . . . . 7 ((ac) ∪ ((a1 d) ∩ (c1 d))) = ((ac) ∪ ((a1 d) ∩ (c1 d)))
2019d3oa 995 . . . . . 6 ((a1 d) ∩ ((ac) ∪ ((a1 d) ∩ (c1 d)))) ≤ (c1 d)
2120leran 153 . . . . 5 (((a1 d) ∩ ((ac) ∪ ((a1 d) ∩ (c1 d)))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d)))) ≤ ((c1 d) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d))))
2218, 21bltr 138 . . . 4 ((a1 d) ∩ f) ≤ ((c1 d) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d))))
23 ancom 74 . . . . . 6 (bc) = (cb)
24 ancom 74 . . . . . 6 ((b1 d) ∩ (c1 d)) = ((c1 d) ∩ (b1 d))
2523, 242or 72 . . . . 5 ((bc) ∪ ((b1 d) ∩ (c1 d))) = ((cb) ∪ ((c1 d) ∩ (b1 d)))
2625d3oa 995 . . . 4 ((c1 d) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d)))) ≤ (b1 d)
2722, 26letr 137 . . 3 ((a1 d) ∩ f) ≤ (b1 d)
284d3oa 995 . . 3 ((a1 d) ∩ e) ≤ (b1 d)
2927, 28lel2or 170 . 2 (((a1 d) ∩ f) ∪ ((a1 d) ∩ e)) ≤ (b1 d)
3014, 29bltr 138 1 ((a1 d) ∩ (ef)) ≤ (b1 d)
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2   ∪ 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  ax-oadist 994 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  d6oa  997
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