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Mirrors > Home > QLE Home > Th. List > oa3to4lem4 | GIF version |
Description: Lemma for orthoarguesian law (Godowski/Greechie 3-variable to 4-variable proof). (Contributed by NM, 19-Dec-1998.) |
Ref | Expression |
---|---|
oa3to4lem.1 | a⊥ ≤ b |
oa3to4lem.2 | c⊥ ≤ d |
oa3to4lem.3 | g = ((a ∩ b) ∪ (c ∩ d)) |
oa3to4lem.oa3 | (a ∩ ((a →1 g) ∪ ((c →1 g) ∩ ((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g)))))) ≤ ((a ∩ g) ∪ (c ∩ g)) |
Ref | Expression |
---|---|
oa3to4lem4 | (a ∩ (b ∪ (d ∩ ((a ∩ c) ∪ (b ∩ d))))) ≤ g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oa3to4lem.1 | . . 3 a⊥ ≤ b | |
2 | oa3to4lem.2 | . . 3 c⊥ ≤ d | |
3 | oa3to4lem.3 | . . 3 g = ((a ∩ b) ∪ (c ∩ d)) | |
4 | 1, 2, 3 | oa3to4lem3 947 | . 2 (a ∩ (b ∪ (d ∩ ((a ∩ c) ∪ (b ∩ d))))) ≤ (a ∩ ((a →1 g) ∪ ((c →1 g) ∩ ((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g)))))) |
5 | oa3to4lem.oa3 | . . 3 (a ∩ ((a →1 g) ∪ ((c →1 g) ∩ ((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g)))))) ≤ ((a ∩ g) ∪ (c ∩ g)) | |
6 | lear 161 | . . . 4 (a ∩ g) ≤ g | |
7 | lear 161 | . . . 4 (c ∩ g) ≤ g | |
8 | 6, 7 | lel2or 170 | . . 3 ((a ∩ g) ∪ (c ∩ g)) ≤ g |
9 | 5, 8 | letr 137 | . 2 (a ∩ ((a →1 g) ∪ ((c →1 g) ∩ ((a ∩ c) ∪ ((a →1 g) ∩ (c →1 g)))))) ≤ g |
10 | 4, 9 | letr 137 | 1 (a ∩ (b ∪ (d ∩ ((a ∩ c) ∪ (b ∩ d))))) ≤ g |
Colors of variables: term |
Syntax hints: = wb 1 ≤ wle 2 ⊥ wn 4 ∪ 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 |
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: oa3to4lem6 950 |
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