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Theorem oa3-u1lem 985
 Description: Lemma for a "universal" 3-OA. Equivalence with substitution into 6-OA dual. (Contributed by NM, 26-Dec-1998.)
Assertion
Ref Expression
oa3-u1lem (1 ∩ (c ∪ ((a1 c) ∩ (((c ∩ (a1 c)) ∪ (1 ∩ (a1 c))) ∪ (((c ∩ (b1 c)) ∪ (1 ∩ (b1 c))) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))))))) = (c ∪ ((a1 c) ∩ ((a1 c) ∪ ((b1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))))))

Proof of Theorem oa3-u1lem
StepHypRef Expression
1 ancom 74 . 2 (1 ∩ (c ∪ ((a1 c) ∩ (((c ∩ (a1 c)) ∪ (1 ∩ (a1 c))) ∪ (((c ∩ (b1 c)) ∪ (1 ∩ (b1 c))) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))))))) = ((c ∪ ((a1 c) ∩ (((c ∩ (a1 c)) ∪ (1 ∩ (a1 c))) ∪ (((c ∩ (b1 c)) ∪ (1 ∩ (b1 c))) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c))))))) ∩ 1)
2 an1 106 . 2 ((c ∪ ((a1 c) ∩ (((c ∩ (a1 c)) ∪ (1 ∩ (a1 c))) ∪ (((c ∩ (b1 c)) ∪ (1 ∩ (b1 c))) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c))))))) ∩ 1) = (c ∪ ((a1 c) ∩ (((c ∩ (a1 c)) ∪ (1 ∩ (a1 c))) ∪ (((c ∩ (b1 c)) ∪ (1 ∩ (b1 c))) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))))))
3 lea 160 . . . . . . . . 9 (ac) ≤ a
4 leo 158 . . . . . . . . 9 a ≤ (a ∪ (ac))
53, 4letr 137 . . . . . . . 8 (ac) ≤ (a ∪ (ac))
6 leor 159 . . . . . . . 8 (ac) ≤ (a ∪ (ac))
75, 6lel2or 170 . . . . . . 7 ((ac) ∪ (ac)) ≤ (a ∪ (ac))
87df-le2 131 . . . . . 6 (((ac) ∪ (ac)) ∪ (a ∪ (ac))) = (a ∪ (ac))
9 ancom 74 . . . . . . . 8 (c ∩ (a1 c)) = ((a1 c) ∩ c)
10 u1lemab 610 . . . . . . . 8 ((a1 c) ∩ c) = ((ac) ∪ (a c))
11 ax-a1 30 . . . . . . . . . . 11 a = a
1211ax-r1 35 . . . . . . . . . 10 a = a
1312ran 78 . . . . . . . . 9 (a c) = (ac)
1413lor 70 . . . . . . . 8 ((ac) ∪ (a c)) = ((ac) ∪ (ac))
159, 10, 143tr 65 . . . . . . 7 (c ∩ (a1 c)) = ((ac) ∪ (ac))
16 ancom 74 . . . . . . . 8 (1 ∩ (a1 c)) = ((a1 c) ∩ 1)
17 an1 106 . . . . . . . 8 ((a1 c) ∩ 1) = (a1 c)
18 df-i1 44 . . . . . . . 8 (a1 c) = (a ∪ (ac))
1916, 17, 183tr 65 . . . . . . 7 (1 ∩ (a1 c)) = (a ∪ (ac))
2015, 192or 72 . . . . . 6 ((c ∩ (a1 c)) ∪ (1 ∩ (a1 c))) = (((ac) ∪ (ac)) ∪ (a ∪ (ac)))
218, 20, 183tr1 63 . . . . 5 ((c ∩ (a1 c)) ∪ (1 ∩ (a1 c))) = (a1 c)
22 lea 160 . . . . . . . . . 10 (bc) ≤ b
23 leo 158 . . . . . . . . . 10 b ≤ (b ∪ (bc))
2422, 23letr 137 . . . . . . . . 9 (bc) ≤ (b ∪ (bc))
25 leor 159 . . . . . . . . 9 (bc) ≤ (b ∪ (bc))
2624, 25lel2or 170 . . . . . . . 8 ((bc) ∪ (bc)) ≤ (b ∪ (bc))
2726df-le2 131 . . . . . . 7 (((bc) ∪ (bc)) ∪ (b ∪ (bc))) = (b ∪ (bc))
28 ancom 74 . . . . . . . . 9 (c ∩ (b1 c)) = ((b1 c) ∩ c)
29 u1lemab 610 . . . . . . . . 9 ((b1 c) ∩ c) = ((bc) ∪ (b c))
30 ax-a1 30 . . . . . . . . . . . 12 b = b
3130ax-r1 35 . . . . . . . . . . 11 b = b
3231ran 78 . . . . . . . . . 10 (b c) = (bc)
3332lor 70 . . . . . . . . 9 ((bc) ∪ (b c)) = ((bc) ∪ (bc))
3428, 29, 333tr 65 . . . . . . . 8 (c ∩ (b1 c)) = ((bc) ∪ (bc))
35 ancom 74 . . . . . . . . 9 (1 ∩ (b1 c)) = ((b1 c) ∩ 1)
36 an1 106 . . . . . . . . 9 ((b1 c) ∩ 1) = (b1 c)
37 df-i1 44 . . . . . . . . 9 (b1 c) = (b ∪ (bc))
3835, 36, 373tr 65 . . . . . . . 8 (1 ∩ (b1 c)) = (b ∪ (bc))
3934, 382or 72 . . . . . . 7 ((c ∩ (b1 c)) ∪ (1 ∩ (b1 c))) = (((bc) ∪ (bc)) ∪ (b ∪ (bc)))
4027, 39, 373tr1 63 . . . . . 6 ((c ∩ (b1 c)) ∪ (1 ∩ (b1 c))) = (b1 c)
41 ax-a2 31 . . . . . 6 (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c))) = (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))
4240, 412an 79 . . . . 5 (((c ∩ (b1 c)) ∪ (1 ∩ (b1 c))) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))) = ((b1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c))))
4321, 422or 72 . . . 4 (((c ∩ (a1 c)) ∪ (1 ∩ (a1 c))) ∪ (((c ∩ (b1 c)) ∪ (1 ∩ (b1 c))) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c))))) = ((a1 c) ∪ ((b1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))))
4443lan 77 . . 3 ((a1 c) ∩ (((c ∩ (a1 c)) ∪ (1 ∩ (a1 c))) ∪ (((c ∩ (b1 c)) ∪ (1 ∩ (b1 c))) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))))) = ((a1 c) ∩ ((a1 c) ∪ ((b1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c))))))
4544lor 70 . 2 (c ∪ ((a1 c) ∩ (((c ∩ (a1 c)) ∪ (1 ∩ (a1 c))) ∪ (((c ∩ (b1 c)) ∪ (1 ∩ (b1 c))) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c))))))) = (c ∪ ((a1 c) ∩ ((a1 c) ∪ ((b1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))))))
461, 2, 453tr 65 1 (1 ∩ (c ∪ ((a1 c) ∩ (((c ∩ (a1 c)) ∪ (1 ∩ (a1 c))) ∪ (((c ∩ (b1 c)) ∪ (1 ∩ (b1 c))) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))))))) = (c ∪ ((a1 c) ∩ ((a1 c) ∪ ((b1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))))))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →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:  oa3-u1  991
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