Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 12146 | . . 3 ⊢ (𝐴(,)𝐵) ∈ ran (,) | |
2 | ioorf.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, 〈0, 0〉, 〈inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )〉)) | |
3 | 2 | ioorval 23148 | . . 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 4048 | . . 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 3890 | . . . . . . 7 ⊢ ((𝐴(,)𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴(,)𝐵)) | |
7 | eliooxr 12103 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) | |
8 | 7 | exlimiv 1845 | . . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
9 | 6, 8 | sylbi 206 | . . . . . 6 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
10 | 9 | simpld 474 | . . . . 5 ⊢ ((𝐴(,)𝐵) ≠ ∅ → 𝐴 ∈ ℝ*) |
11 | 9 | simprd 478 | . . . . 5 ⊢ ((𝐴(,)𝐵) ≠ ∅ → 𝐵 ∈ ℝ*) |
12 | id 22 | . . . . 5 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴(,)𝐵) ≠ ∅) | |
13 | df-ioo 12050 | . . . . . 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 11858 | . . . . . 6 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵)) | |
16 | idd 24 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 < 𝑤 → 𝐴 < 𝑤)) | |
17 | xrltle 11858 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 < 𝑤 → 𝐴 ≤ 𝑤)) | |
18 | 13, 14, 15, 16, 17 | ixxlb 12068 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝐴(,)𝐵) ≠ ∅) → inf((𝐴(,)𝐵), ℝ*, < ) = 𝐴) |
19 | 10, 11, 12, 18 | syl3anc 1318 | . . . 4 ⊢ ((𝐴(,)𝐵) ≠ ∅ → inf((𝐴(,)𝐵), ℝ*, < ) = 𝐴) |
20 | 13, 14, 15, 16, 17 | ixxub 12067 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝐴(,)𝐵) ≠ ∅) → sup((𝐴(,)𝐵), ℝ*, < ) = 𝐵) |
21 | 10, 11, 12, 20 | syl3anc 1318 | . . . 4 ⊢ ((𝐴(,)𝐵) ≠ ∅ → sup((𝐴(,)𝐵), ℝ*, < ) = 𝐵) |
22 | 19, 21 | opeq12d 4348 | . . 3 ⊢ ((𝐴(,)𝐵) ≠ ∅ → 〈inf((𝐴(,)𝐵), ℝ*, < ), sup((𝐴(,)𝐵), ℝ*, < )〉 = 〈𝐴, 𝐵〉) |
23 | 5, 22 | eqtrd 2644 | . 2 ⊢ ((𝐴(,)𝐵) ≠ ∅ → if((𝐴(,)𝐵) = ∅, 〈0, 0〉, 〈inf((𝐴(,)𝐵), ℝ*, < ), sup((𝐴(,)𝐵), ℝ*, < )〉) = 〈𝐴, 𝐵〉) |
24 | 4, 23 | syl5eq 2656 | 1 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐹‘(𝐴(,)𝐵)) = 〈𝐴, 𝐵〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 ifcif 4036 〈cop 4131 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ‘cfv 5804 (class class class)co 6549 supcsup 8229 infcinf 8230 0cc0 9815 ℝ*cxr 9952 < clt 9953 (,)cioo 12046 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-ioo 12050 |
This theorem is referenced by: ioorinv 23150 ioorcl 23151 |
Copyright terms: Public domain | W3C validator |