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Mirrors > Home > ILE Home > Th. List > infxrnegsupex | Unicode version |
Description: The infimum of a set of extended reals is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 2-May-2023.) |
Ref | Expression |
---|---|
infxrnegsupex.ex | |
infxrnegsupex.ss |
Ref | Expression |
---|---|
infxrnegsupex | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrlttri3 9583 | . . . . 5 | |
2 | 1 | adantl 275 | . . . 4 |
3 | infxrnegsupex.ex | . . . 4 | |
4 | 2, 3 | infclti 6910 | . . 3 inf |
5 | xnegneg 9616 | . . 3 inf inf inf | |
6 | 4, 5 | syl 14 | . 2 inf inf |
7 | xnegeq 9610 | . . . . . . . . 9 | |
8 | 7 | cbvmptv 4024 | . . . . . . . 8 |
9 | 8 | mptpreima 5032 | . . . . . . 7 |
10 | eqid 2139 | . . . . . . . . . 10 | |
11 | 10 | xrnegiso 11031 | . . . . . . . . 9 |
12 | 11 | simpri 112 | . . . . . . . 8 |
13 | 12 | imaeq1i 4878 | . . . . . . 7 |
14 | 9, 13 | eqtr3i 2162 | . . . . . 6 |
15 | 14 | supeq1i 6875 | . . . . 5 |
16 | 11 | simpli 110 | . . . . . . . . 9 |
17 | isocnv 5712 | . . . . . . . . 9 | |
18 | 16, 17 | ax-mp 5 | . . . . . . . 8 |
19 | isoeq1 5702 | . . . . . . . . 9 | |
20 | 12, 19 | ax-mp 5 | . . . . . . . 8 |
21 | 18, 20 | mpbi 144 | . . . . . . 7 |
22 | 21 | a1i 9 | . . . . . 6 |
23 | infxrnegsupex.ss | . . . . . 6 | |
24 | 3 | cnvinfex 6905 | . . . . . 6 |
25 | 2 | cnvti 6906 | . . . . . 6 |
26 | 22, 23, 24, 25 | supisoti 6897 | . . . . 5 |
27 | 15, 26 | syl5eq 2184 | . . . 4 |
28 | df-inf 6872 | . . . . . . 7 inf | |
29 | 28 | eqcomi 2143 | . . . . . 6 inf |
30 | 29 | fveq2i 5424 | . . . . 5 inf |
31 | eqidd 2140 | . . . . . 6 | |
32 | xnegeq 9610 | . . . . . . 7 inf inf | |
33 | 32 | adantl 275 | . . . . . 6 inf inf |
34 | 4 | xnegcld 9638 | . . . . . 6 inf |
35 | 31, 33, 4, 34 | fvmptd 5502 | . . . . 5 inf inf |
36 | 30, 35 | syl5eq 2184 | . . . 4 inf |
37 | 27, 36 | eqtr2d 2173 | . . 3 inf |
38 | xnegeq 9610 | . . 3 inf inf | |
39 | 37, 38 | syl 14 | . 2 inf |
40 | 6, 39 | eqtr3d 2174 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 wrex 2417 crab 2420 wss 3071 class class class wbr 3929 cmpt 3989 ccnv 4538 cima 4542 cfv 5123 wiso 5124 csup 6869 infcinf 6870 cxr 7799 clt 7800 cxne 9556 |
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-3or 963 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-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sup 6871 df-inf 6872 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-sub 7935 df-neg 7936 df-xneg 9559 |
This theorem is referenced by: xrminmax 11034 |
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