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Mirrors > Home > MPE Home > Th. List > ioorinv2 | Structured version Visualization version GIF version |
Description: The function 𝐹 is an "inverse" of sorts to the open interval function. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.) |
Ref | Expression |
---|---|
ioorf.1 | ⊢ 𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, 〈0, 0〉, 〈inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )〉)) |
Ref | Expression |
---|---|
ioorinv2 | ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐹‘(𝐴(,)𝐵)) = 〈𝐴, 𝐵〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioorebas 12275 | . . 3 ⊢ (𝐴(,)𝐵) ∈ ran (,) | |
2 | ioorf.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, 〈0, 0〉, 〈inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )〉)) | |
3 | 2 | ioorval 23342 | . . 3 ⊢ ((𝐴(,)𝐵) ∈ ran (,) → (𝐹‘(𝐴(,)𝐵)) = if((𝐴(,)𝐵) = ∅, 〈0, 0〉, 〈inf((𝐴(,)𝐵), ℝ*, < ), sup((𝐴(,)𝐵), ℝ*, < )〉)) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ (𝐹‘(𝐴(,)𝐵)) = if((𝐴(,)𝐵) = ∅, 〈0, 0〉, 〈inf((𝐴(,)𝐵), ℝ*, < ), sup((𝐴(,)𝐵), ℝ*, < )〉) |
5 | ifnefalse 4098 | . . 3 ⊢ ((𝐴(,)𝐵) ≠ ∅ → if((𝐴(,)𝐵) = ∅, 〈0, 0〉, 〈inf((𝐴(,)𝐵), ℝ*, < ), sup((𝐴(,)𝐵), ℝ*, < )〉) = 〈inf((𝐴(,)𝐵), ℝ*, < ), sup((𝐴(,)𝐵), ℝ*, < )〉) | |
6 | n0 3931 | . . . . . . 7 ⊢ ((𝐴(,)𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴(,)𝐵)) | |
7 | eliooxr 12232 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) | |
8 | 7 | exlimiv 1858 | . . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
9 | 6, 8 | sylbi 207 | . . . . . 6 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
10 | 9 | simpld 475 | . . . . 5 ⊢ ((𝐴(,)𝐵) ≠ ∅ → 𝐴 ∈ ℝ*) |
11 | 9 | simprd 479 | . . . . 5 ⊢ ((𝐴(,)𝐵) ≠ ∅ → 𝐵 ∈ ℝ*) |
12 | id 22 | . . . . 5 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴(,)𝐵) ≠ ∅) | |
13 | df-ioo 12179 | . . . . . 6 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
14 | idd 24 | . . . . . 6 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 < 𝐵 → 𝑤 < 𝐵)) | |
15 | xrltle 11982 | . . . . . 6 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵)) | |
16 | idd 24 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 < 𝑤 → 𝐴 < 𝑤)) | |
17 | xrltle 11982 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 < 𝑤 → 𝐴 ≤ 𝑤)) | |
18 | 13, 14, 15, 16, 17 | ixxlb 12197 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝐴(,)𝐵) ≠ ∅) → inf((𝐴(,)𝐵), ℝ*, < ) = 𝐴) |
19 | 10, 11, 12, 18 | syl3anc 1326 | . . . 4 ⊢ ((𝐴(,)𝐵) ≠ ∅ → inf((𝐴(,)𝐵), ℝ*, < ) = 𝐴) |
20 | 13, 14, 15, 16, 17 | ixxub 12196 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝐴(,)𝐵) ≠ ∅) → sup((𝐴(,)𝐵), ℝ*, < ) = 𝐵) |
21 | 10, 11, 12, 20 | syl3anc 1326 | . . . 4 ⊢ ((𝐴(,)𝐵) ≠ ∅ → sup((𝐴(,)𝐵), ℝ*, < ) = 𝐵) |
22 | 19, 21 | opeq12d 4410 | . . 3 ⊢ ((𝐴(,)𝐵) ≠ ∅ → 〈inf((𝐴(,)𝐵), ℝ*, < ), sup((𝐴(,)𝐵), ℝ*, < )〉 = 〈𝐴, 𝐵〉) |
23 | 5, 22 | eqtrd 2656 | . 2 ⊢ ((𝐴(,)𝐵) ≠ ∅ → if((𝐴(,)𝐵) = ∅, 〈0, 0〉, 〈inf((𝐴(,)𝐵), ℝ*, < ), sup((𝐴(,)𝐵), ℝ*, < )〉) = 〈𝐴, 𝐵〉) |
24 | 4, 23 | syl5eq 2668 | 1 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐹‘(𝐴(,)𝐵)) = 〈𝐴, 𝐵〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 ∅c0 3915 ifcif 4086 〈cop 4183 class class class wbr 4653 ↦ cmpt 4729 ran crn 5115 ‘cfv 5888 (class class class)co 6650 supcsup 8346 infcinf 8347 0cc0 9936 ℝ*cxr 10073 < clt 10074 (,)cioo 12175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-ioo 12179 |
This theorem is referenced by: ioorinv 23344 ioorcl 23345 |
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