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Mirrors > Home > ILE Home > Th. List > eqord2 | Unicode version |
Description: A strictly decreasing real function on a subset of is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.) |
Ref | Expression |
---|---|
ltord.1 | |
ltord.2 | |
ltord.3 | |
ltord.4 | |
ltord.5 | |
ltord2.6 |
Ref | Expression |
---|---|
eqord2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltord.1 | . . . 4 | |
2 | 1 | negeqd 7957 | . . 3 |
3 | ltord.2 | . . . 4 | |
4 | 3 | negeqd 7957 | . . 3 |
5 | ltord.3 | . . . 4 | |
6 | 5 | negeqd 7957 | . . 3 |
7 | ltord.4 | . . 3 | |
8 | ltord.5 | . . . 4 | |
9 | 8 | renegcld 8142 | . . 3 |
10 | ltord2.6 | . . . 4 | |
11 | 8 | ralrimiva 2505 | . . . . . . 7 |
12 | 1 | eleq1d 2208 | . . . . . . . 8 |
13 | 12 | rspccva 2788 | . . . . . . 7 |
14 | 11, 13 | sylan 281 | . . . . . 6 |
15 | 14 | adantrl 469 | . . . . 5 |
16 | 8 | adantrr 470 | . . . . 5 |
17 | ltneg 8224 | . . . . 5 | |
18 | 15, 16, 17 | syl2anc 408 | . . . 4 |
19 | 10, 18 | sylibd 148 | . . 3 |
20 | 2, 4, 6, 7, 9, 19 | eqord1 8245 | . 2 |
21 | 3 | eleq1d 2208 | . . . . . . 7 |
22 | 21 | rspccva 2788 | . . . . . 6 |
23 | 11, 22 | sylan 281 | . . . . 5 |
24 | 23 | adantrr 470 | . . . 4 |
25 | 24 | recnd 7794 | . . 3 |
26 | 5 | eleq1d 2208 | . . . . . . 7 |
27 | 26 | rspccva 2788 | . . . . . 6 |
28 | 11, 27 | sylan 281 | . . . . 5 |
29 | 28 | adantrl 469 | . . . 4 |
30 | 29 | recnd 7794 | . . 3 |
31 | 25, 30 | neg11ad 8069 | . 2 |
32 | 20, 31 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 wss 3071 class class class wbr 3929 cr 7619 clt 7800 cneg 7934 |
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-in1 603 ax-in2 604 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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-apti 7735 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-sub 7935 df-neg 7936 |
This theorem is referenced by: (None) |
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