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Mirrors > Home > QLE Home > Th. List > 4oadist | Unicode version |
Description: OA Distributive law. This is equivalent to the 6-variable OA law, as shown by theorem d6oa 997. |
Ref | Expression |
---|---|
4oa.1 | |
4oa.2 | |
4oadist.1 | |
4oadist.2 | |
4oadist.3 | |
4oadist.4 |
Ref | Expression |
---|---|
4oadist |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4oadist.2 | . . . . . . . . . 10 | |
2 | 4oadist.3 | . . . . . . . . . 10 | |
3 | 1, 2 | le2or 168 | . . . . . . . . 9 |
4 | oridm 110 | . . . . . . . . 9 | |
5 | 3, 4 | lbtr 139 | . . . . . . . 8 |
6 | 5 | lelan 167 | . . . . . . 7 |
7 | 6 | df2le2 136 | . . . . . 6 |
8 | 7 | ax-r1 35 | . . . . 5 |
9 | 4oa.2 | . . . . . . . . 9 | |
10 | or32 82 | . . . . . . . . 9 | |
11 | 9, 10 | ax-r2 36 | . . . . . . . 8 |
12 | 11 | lan 77 | . . . . . . 7 |
13 | 4oa.1 | . . . . . . . 8 | |
14 | leo 158 | . . . . . . . . 9 | |
15 | 11 | ax-r1 35 | . . . . . . . . 9 |
16 | 14, 15 | lbtr 139 | . . . . . . . 8 |
17 | 4oadist.1 | . . . . . . . 8 | |
18 | 13, 9, 16, 17 | 4oagen1b 1043 | . . . . . . 7 |
19 | 12, 18 | ax-r2 36 | . . . . . 6 |
20 | 19 | lan 77 | . . . . 5 |
21 | 8, 20 | ax-r2 36 | . . . 4 |
22 | lear 161 | . . . . 5 | |
23 | 4oadist.4 | . . . . . . . . 9 | |
24 | 23 | df2le2 136 | . . . . . . . 8 |
25 | 24 | ax-r1 35 | . . . . . . 7 |
26 | an32 83 | . . . . . . 7 | |
27 | 25, 26 | ax-r2 36 | . . . . . 6 |
28 | lea 160 | . . . . . 6 | |
29 | 27, 28 | bltr 138 | . . . . 5 |
30 | 22, 29 | letr 137 | . . . 4 |
31 | 21, 30 | bltr 138 | . . 3 |
32 | leor 159 | . . 3 | |
33 | 31, 32 | letr 137 | . 2 |
34 | ledi 174 | . 2 | |
35 | 33, 34 | lebi 145 | 1 |
Colors of variables: term |
Syntax hints: wb 1 wle 2 wo 6 wa 7 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 ax-4oa 1033 |
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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