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Mirrors > Home > MPE Home > Th. List > infmxrgelb | Unicode version |
Description: The infimum of a set of extended reals is greater than or equal to a lower bound. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by Mario Carneiro, 6-Sep-2014.) |
Ref | Expression |
---|---|
infmxrgelb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltso 11376 | . . . . . . . 8 | |
2 | cnvso 5551 | . . . . . . . 8 | |
3 | 1, 2 | mpbi 208 | . . . . . . 7 |
4 | 3 | a1i 11 | . . . . . 6 |
5 | xrinfmss2 11531 | . . . . . 6 | |
6 | id 22 | . . . . . 6 | |
7 | 4, 5, 6 | suplub2 7941 | . . . . 5 |
8 | simpr 461 | . . . . . 6 | |
9 | 3 | supex 7943 | . . . . . 6 |
10 | brcnvg 5188 | . . . . . 6 | |
11 | 8, 9, 10 | sylancl 662 | . . . . 5 |
12 | vex 3112 | . . . . . . 7 | |
13 | brcnvg 5188 | . . . . . . 7 | |
14 | 8, 12, 13 | sylancl 662 | . . . . . 6 |
15 | 14 | rexbidv 2968 | . . . . 5 |
16 | 7, 11, 15 | 3bitr3d 283 | . . . 4 |
17 | 16 | notbid 294 | . . 3 |
18 | ralnex 2903 | . . 3 | |
19 | 17, 18 | syl6bbr 263 | . 2 |
20 | id 22 | . . 3 | |
21 | infmxrcl 11537 | . . 3 | |
22 | xrlenlt 9673 | . . 3 | |
23 | 20, 21, 22 | syl2anr 478 | . 2 |
24 | simplr 755 | . . . 4 | |
25 | simpl 457 | . . . . 5 | |
26 | 25 | sselda 3503 | . . . 4 |
27 | xrlenlt 9673 | . . . 4 | |
28 | 24, 26, 27 | syl2anc 661 | . . 3 |
29 | 28 | ralbidva 2893 | . 2 |
30 | 19, 23, 29 | 3bitr4d 285 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -. wn 3 -> wi 4
<-> wb 184 /\ wa 369 e. wcel 1818
A. wral 2807 E. wrex 2808 cvv 3109
C_ wss 3475 class class class wbr 4452
Or wor 4804 `' ccnv 5003 sup csup 7920
cxr 9648
clt 9649 cle 9650 |
This theorem is referenced by: infmxrre 11556 ixxlb 11580 limsuple 13301 limsupval2 13303 imasdsf1olem 20876 nmogelb 21223 metdsf 21352 metdsge 21353 ovolgelb 21891 ovolge0 21892 ovolsslem 21895 ovolicc2 21933 ismblfin 30055 infmxrss 31492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 ax-cnex 9569 ax-resscn 9570 ax-1cn 9571 ax-icn 9572 ax-addcl 9573 ax-addrcl 9574 ax-mulcl 9575 ax-mulrcl 9576 ax-mulcom 9577 ax-addass 9578 ax-mulass 9579 ax-distr 9580 ax-i2m1 9581 ax-1ne0 9582 ax-1rid 9583 ax-rnegex 9584 ax-rrecex 9585 ax-cnre 9586 ax-pre-lttri 9587 ax-pre-lttrn 9588 ax-pre-ltadd 9589 ax-pre-mulgt0 9590 ax-pre-sup 9591 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-nel 2655 df-ral 2812 df-rex 2813 df-reu 2814 df-rmo 2815 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-op 4036 df-uni 4250 df-br 4453 df-opab 4511 df-mpt 4512 df-id 4800 df-po 4805 df-so 4806 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 df-riota 6257 df-ov 6299 df-oprab 6300 df-mpt2 6301 df-er 7330 df-en 7537 df-dom 7538 df-sdom 7539 df-sup 7921 df-pnf 9651 df-mnf 9652 df-xr 9653 df-ltxr 9654 df-le 9655 df-sub 9830 df-neg 9831 |
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