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Mirrors > Home > QLE Home > Th. List > oi3oa3lem1 | GIF version |
Description: An attempt at the OA3 conjecture, which is true if (a ≡ b) = 1. (Contributed by Josiah Burroughs, 27-May-2004.) |
Ref | Expression |
---|---|
oi3oa3lem1.1 | 1 = (b ≡ a) |
Ref | Expression |
---|---|
oi3oa3lem1 | (((a →1 c) ∩ (b →1 c)) ∪ (a ∩ b)) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oi3oa3lem1.1 | . . . . . 6 1 = (b ≡ a) | |
2 | 1 | r3a 440 | . . . . 5 b = a |
3 | 2 | ud1lem0b 256 | . . . 4 (b →1 c) = (a →1 c) |
4 | 3 | lan 77 | . . 3 ((a →1 c) ∩ (b →1 c)) = ((a →1 c) ∩ (a →1 c)) |
5 | 2 | lan 77 | . . 3 (a ∩ b) = (a ∩ a) |
6 | 4, 5 | 2or 72 | . 2 (((a →1 c) ∩ (b →1 c)) ∪ (a ∩ b)) = (((a →1 c) ∩ (a →1 c)) ∪ (a ∩ a)) |
7 | anidm 111 | . . 3 ((a →1 c) ∩ (a →1 c)) = (a →1 c) | |
8 | anidm 111 | . . 3 (a ∩ a) = a | |
9 | 7, 8 | 2or 72 | . 2 (((a →1 c) ∩ (a →1 c)) ∪ (a ∩ a)) = ((a →1 c) ∪ a) |
10 | u1lemoa 620 | . 2 ((a →1 c) ∪ a) = 1 | |
11 | 6, 9, 10 | 3tr 65 | 1 (((a →1 c) ∩ (b →1 c)) ∪ (a ∩ b)) = 1 |
Colors of variables: term |
Syntax hints: = wb 1 ≡ tb 5 ∪ 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 |
This theorem is referenced by: oi3oa3 733 |
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