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Theorem oa3-6to3 987
 Description: Derivation of 3-OA variant (3) from (6). (Contributed by NM, 24-Dec-1998.)
Hypothesis
Ref Expression
oa3-6to3.1 ((a1 c) ∩ (a ∪ (b ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))))) ≤ c
Assertion
Ref Expression
oa3-6to3 (a ∩ (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c)))))) ≤ c

Proof of Theorem oa3-6to3
StepHypRef Expression
1 oa3-3lem 981 . . 3 (a ∩ (a ∪ (b ∩ (((ab) ∪ (ab )) ∪ (((a ∩ 1) ∪ (ac)) ∩ ((b ∩ 1) ∪ (bc))))))) = (a ∩ (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c))))))
21ax-r1 35 . 2 (a ∩ (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c)))))) = (a ∩ (a ∪ (b ∩ (((ab) ∪ (ab )) ∪ (((a ∩ 1) ∪ (ac)) ∩ ((b ∩ 1) ∪ (bc)))))))
3 leid 148 . . 3 aa
4 leid 148 . . 3 bb
5 df-f 42 . . . . 5 0 = 1
65ax-r1 35 . . . 4 1 = 0
7 le0 147 . . . 4 0 ≤ c
86, 7bltr 138 . . 3 1c
9 ancom 74 . . . . . . . 8 (1 ∩ c) = (c ∩ 1)
10 an1 106 . . . . . . . 8 (c ∩ 1) = c
119, 10ax-r2 36 . . . . . . 7 (1 ∩ c) = c
12 dff 101 . . . . . . . . . 10 0 = (aa )
13 dff 101 . . . . . . . . . 10 0 = (bb )
1412, 132or 72 . . . . . . . . 9 (0 ∪ 0) = ((aa ) ∪ (bb ))
1514ax-r1 35 . . . . . . . 8 ((aa ) ∪ (bb )) = (0 ∪ 0)
16 or0 102 . . . . . . . 8 (0 ∪ 0) = 0
1715, 16ax-r2 36 . . . . . . 7 ((aa ) ∪ (bb )) = 0
1811, 172or 72 . . . . . 6 ((1 ∩ c) ∪ ((aa ) ∪ (bb ))) = (c ∪ 0)
19 or0 102 . . . . . 6 (c ∪ 0) = c
2018, 19ax-r2 36 . . . . 5 ((1 ∩ c) ∪ ((aa ) ∪ (bb ))) = c
2120ax-r1 35 . . . 4 c = ((1 ∩ c) ∪ ((aa ) ∪ (bb )))
22 ax-a2 31 . . . 4 ((1 ∩ c) ∪ ((aa ) ∪ (bb ))) = (((aa ) ∪ (bb )) ∪ (1 ∩ c))
2321, 22ax-r2 36 . . 3 c = (((aa ) ∪ (bb )) ∪ (1 ∩ c))
24 oa3-6lem 980 . . . 4 ((a1 c) ∩ (a ∪ (b ∩ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((a ∩ 1) ∪ ((a1 c) ∩ (1 →1 c))) ∩ ((b ∩ 1) ∪ ((b1 c) ∩ (1 →1 c)))))))) = ((a1 c) ∩ (a ∪ (b ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c))))))
25 oa3-6to3.1 . . . 4 ((a1 c) ∩ (a ∪ (b ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))))) ≤ c
2624, 25bltr 138 . . 3 ((a1 c) ∩ (a ∪ (b ∩ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((a ∩ 1) ∪ ((a1 c) ∩ (1 →1 c))) ∩ ((b ∩ 1) ∪ ((b1 c) ∩ (1 →1 c)))))))) ≤ c
273, 4, 8, 23, 26oa4to6dual 964 . 2 (a ∩ (a ∪ (b ∩ (((ab) ∪ (ab )) ∪ (((a ∩ 1) ∪ (ac)) ∩ ((b ∩ 1) ∪ (bc))))))) ≤ c
282, 27bltr 138 1 (a ∩ (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c)))))) ≤ c
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →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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