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Mirrors > Home > ILE Home > Th. List > smoel | Unicode version |
Description: If is less than then a strictly monotone function's value will be strictly less at than at . (Contributed by Andrew Salmon, 22-Nov-2011.) |
Ref | Expression |
---|---|
smoel |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smodm 6188 | . . . . 5 | |
2 | ordtr1 4310 | . . . . . . 7 | |
3 | 2 | ancomsd 267 | . . . . . 6 |
4 | 3 | expdimp 257 | . . . . 5 |
5 | 1, 4 | sylan 281 | . . . 4 |
6 | df-smo 6183 | . . . . . 6 | |
7 | eleq1 2202 | . . . . . . . . . . 11 | |
8 | fveq2 5421 | . . . . . . . . . . . 12 | |
9 | 8 | eleq1d 2208 | . . . . . . . . . . 11 |
10 | 7, 9 | imbi12d 233 | . . . . . . . . . 10 |
11 | eleq2 2203 | . . . . . . . . . . 11 | |
12 | fveq2 5421 | . . . . . . . . . . . 12 | |
13 | 12 | eleq2d 2209 | . . . . . . . . . . 11 |
14 | 11, 13 | imbi12d 233 | . . . . . . . . . 10 |
15 | 10, 14 | rspc2v 2802 | . . . . . . . . 9 |
16 | 15 | ancoms 266 | . . . . . . . 8 |
17 | 16 | com12 30 | . . . . . . 7 |
18 | 17 | 3ad2ant3 1004 | . . . . . 6 |
19 | 6, 18 | sylbi 120 | . . . . 5 |
20 | 19 | expdimp 257 | . . . 4 |
21 | 5, 20 | syld 45 | . . 3 |
22 | 21 | pm2.43d 50 | . 2 |
23 | 22 | 3impia 1178 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 wral 2416 word 4284 con0 4285 cdm 4539 wf 5119 cfv 5123 wsmo 6182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-tr 4027 df-iord 4288 df-iota 5088 df-fv 5131 df-smo 6183 |
This theorem is referenced by: smoiun 6198 smoel2 6200 |
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