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Theorem oa3-2lema 978
 Description: Lemma for 3-OA(2). Equivalence with substitution into 4-OA.
Assertion
Ref Expression
oa3-2lema ((a1 c) ∩ (a ∪ (b ∩ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((a ∩ 0) ∪ ((a1 c) ∩ (0 →1 c))) ∩ ((b ∩ 0) ∪ ((b1 c) ∩ (0 →1 c)))))))) = ((a1 c) ∩ (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c))))))

Proof of Theorem oa3-2lema
StepHypRef Expression
1 ax-a3 32 . . . . 5 (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((a ∩ 0) ∪ ((a1 c) ∩ (0 →1 c))) ∩ ((b ∩ 0) ∪ ((b1 c) ∩ (0 →1 c))))) = ((ab) ∪ (((a1 c) ∩ (b1 c)) ∪ (((a ∩ 0) ∪ ((a1 c) ∩ (0 →1 c))) ∩ ((b ∩ 0) ∪ ((b1 c) ∩ (0 →1 c))))))
2 an0 108 . . . . . . . . . . 11 (a ∩ 0) = 0
32ax-r5 38 . . . . . . . . . 10 ((a ∩ 0) ∪ ((a1 c) ∩ (0 →1 c))) = (0 ∪ ((a1 c) ∩ (0 →1 c)))
4 ax-a2 31 . . . . . . . . . 10 (0 ∪ ((a1 c) ∩ (0 →1 c))) = (((a1 c) ∩ (0 →1 c)) ∪ 0)
5 or0 102 . . . . . . . . . . 11 (((a1 c) ∩ (0 →1 c)) ∪ 0) = ((a1 c) ∩ (0 →1 c))
6 0i1 273 . . . . . . . . . . . 12 (0 →1 c) = 1
76lan 77 . . . . . . . . . . 11 ((a1 c) ∩ (0 →1 c)) = ((a1 c) ∩ 1)
8 an1 106 . . . . . . . . . . 11 ((a1 c) ∩ 1) = (a1 c)
95, 7, 83tr 65 . . . . . . . . . 10 (((a1 c) ∩ (0 →1 c)) ∪ 0) = (a1 c)
103, 4, 93tr 65 . . . . . . . . 9 ((a ∩ 0) ∪ ((a1 c) ∩ (0 →1 c))) = (a1 c)
11 an0 108 . . . . . . . . . . 11 (b ∩ 0) = 0
1211ax-r5 38 . . . . . . . . . 10 ((b ∩ 0) ∪ ((b1 c) ∩ (0 →1 c))) = (0 ∪ ((b1 c) ∩ (0 →1 c)))
13 ax-a2 31 . . . . . . . . . 10 (0 ∪ ((b1 c) ∩ (0 →1 c))) = (((b1 c) ∩ (0 →1 c)) ∪ 0)
14 or0 102 . . . . . . . . . . 11 (((b1 c) ∩ (0 →1 c)) ∪ 0) = ((b1 c) ∩ (0 →1 c))
156lan 77 . . . . . . . . . . 11 ((b1 c) ∩ (0 →1 c)) = ((b1 c) ∩ 1)
16 an1 106 . . . . . . . . . . 11 ((b1 c) ∩ 1) = (b1 c)
1714, 15, 163tr 65 . . . . . . . . . 10 (((b1 c) ∩ (0 →1 c)) ∪ 0) = (b1 c)
1812, 13, 173tr 65 . . . . . . . . 9 ((b ∩ 0) ∪ ((b1 c) ∩ (0 →1 c))) = (b1 c)
1910, 182an 79 . . . . . . . 8 (((a ∩ 0) ∪ ((a1 c) ∩ (0 →1 c))) ∩ ((b ∩ 0) ∪ ((b1 c) ∩ (0 →1 c)))) = ((a1 c) ∩ (b1 c))
2019lor 70 . . . . . . 7 (((a1 c) ∩ (b1 c)) ∪ (((a ∩ 0) ∪ ((a1 c) ∩ (0 →1 c))) ∩ ((b ∩ 0) ∪ ((b1 c) ∩ (0 →1 c))))) = (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))
21 oridm 110 . . . . . . 7 (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c))) = ((a1 c) ∩ (b1 c))
2220, 21ax-r2 36 . . . . . 6 (((a1 c) ∩ (b1 c)) ∪ (((a ∩ 0) ∪ ((a1 c) ∩ (0 →1 c))) ∩ ((b ∩ 0) ∪ ((b1 c) ∩ (0 →1 c))))) = ((a1 c) ∩ (b1 c))
2322lor 70 . . . . 5 ((ab) ∪ (((a1 c) ∩ (b1 c)) ∪ (((a ∩ 0) ∪ ((a1 c) ∩ (0 →1 c))) ∩ ((b ∩ 0) ∪ ((b1 c) ∩ (0 →1 c)))))) = ((ab) ∪ ((a1 c) ∩ (b1 c)))
241, 23ax-r2 36 . . . 4 (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((a ∩ 0) ∪ ((a1 c) ∩ (0 →1 c))) ∩ ((b ∩ 0) ∪ ((b1 c) ∩ (0 →1 c))))) = ((ab) ∪ ((a1 c) ∩ (b1 c)))
2524lan 77 . . 3 (b ∩ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((a ∩ 0) ∪ ((a1 c) ∩ (0 →1 c))) ∩ ((b ∩ 0) ∪ ((b1 c) ∩ (0 →1 c)))))) = (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c))))
2625lor 70 . 2 (a ∪ (b ∩ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((a ∩ 0) ∪ ((a1 c) ∩ (0 →1 c))) ∩ ((b ∩ 0) ∪ ((b1 c) ∩ (0 →1 c))))))) = (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c)))))
2726lan 77 1 ((a1 c) ∩ (a ∪ (b ∩ (((ab) ∪ ((a1 c) ∩ (b1 c))) ∪ (((a ∩ 0) ∪ ((a1 c) ∩ (0 →1 c))) ∩ ((b ∩ 0) ∪ ((b1 c) ∩ (0 →1 c)))))))) = ((a1 c) ∩ (a ∪ (b ∩ ((ab) ∪ ((a1 c) ∩ (b1 c))))))
 Colors of variables: term Syntax hints:   = wb 1   ∪ 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 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44 This theorem is referenced by:  oa3-2to2s  990
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