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| Mirrors > Home > MPE Home > Th. List > ioorcl | Structured version Visualization version GIF version | ||
| Description: The function 𝐹 does not always return real numbers, but it does on intervals of finite volume. (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 | 
|---|---|
| ioorcl | ⊢ ((𝐴 ∈ ran (,) ∧ (vol*‘𝐴) ∈ ℝ) → (𝐹‘𝐴) ∈ ( ≤ ∩ (ℝ × ℝ))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ioorf.1 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, 〈0, 0〉, 〈inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )〉)) | |
| 2 | 1 | ioorf 25608 | . . . . 5 ⊢ 𝐹:ran (,)⟶( ≤ ∩ (ℝ* × ℝ*)) | 
| 3 | 2 | ffvelcdmi 7103 | . . . 4 ⊢ (𝐴 ∈ ran (,) → (𝐹‘𝐴) ∈ ( ≤ ∩ (ℝ* × ℝ*))) | 
| 4 | 3 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ ran (,) ∧ (vol*‘𝐴) ∈ ℝ) → (𝐹‘𝐴) ∈ ( ≤ ∩ (ℝ* × ℝ*))) | 
| 5 | 4 | elin1d 4204 | . 2 ⊢ ((𝐴 ∈ ran (,) ∧ (vol*‘𝐴) ∈ ℝ) → (𝐹‘𝐴) ∈ ≤ ) | 
| 6 | 1 | ioorval 25609 | . . . . . 6 ⊢ (𝐴 ∈ ran (,) → (𝐹‘𝐴) = if(𝐴 = ∅, 〈0, 0〉, 〈inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )〉)) | 
| 7 | 6 | adantr 480 | . . . . 5 ⊢ ((𝐴 ∈ ran (,) ∧ (vol*‘𝐴) ∈ ℝ) → (𝐹‘𝐴) = if(𝐴 = ∅, 〈0, 0〉, 〈inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )〉)) | 
| 8 | iftrue 4531 | . . . . 5 ⊢ (𝐴 = ∅ → if(𝐴 = ∅, 〈0, 0〉, 〈inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )〉) = 〈0, 0〉) | |
| 9 | 7, 8 | sylan9eq 2797 | . . . 4 ⊢ (((𝐴 ∈ ran (,) ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 = ∅) → (𝐹‘𝐴) = 〈0, 0〉) | 
| 10 | 0re 11263 | . . . . 5 ⊢ 0 ∈ ℝ | |
| 11 | opelxpi 5722 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 0 ∈ ℝ) → 〈0, 0〉 ∈ (ℝ × ℝ)) | |
| 12 | 10, 10, 11 | mp2an 692 | . . . 4 ⊢ 〈0, 0〉 ∈ (ℝ × ℝ) | 
| 13 | 9, 12 | eqeltrdi 2849 | . . 3 ⊢ (((𝐴 ∈ ran (,) ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 = ∅) → (𝐹‘𝐴) ∈ (ℝ × ℝ)) | 
| 14 | ioof 13487 | . . . . . 6 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
| 15 | ffn 6736 | . . . . . 6 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
| 16 | ovelrn 7609 | . . . . . 6 ⊢ ((,) Fn (ℝ* × ℝ*) → (𝐴 ∈ ran (,) ↔ ∃𝑎 ∈ ℝ* ∃𝑏 ∈ ℝ* 𝐴 = (𝑎(,)𝑏))) | |
| 17 | 14, 15, 16 | mp2b 10 | . . . . 5 ⊢ (𝐴 ∈ ran (,) ↔ ∃𝑎 ∈ ℝ* ∃𝑏 ∈ ℝ* 𝐴 = (𝑎(,)𝑏)) | 
| 18 | 1 | ioorinv2 25610 | . . . . . . . . . 10 ⊢ ((𝑎(,)𝑏) ≠ ∅ → (𝐹‘(𝑎(,)𝑏)) = 〈𝑎, 𝑏〉) | 
| 19 | 18 | adantl 481 | . . . . . . . . 9 ⊢ (((vol*‘(𝑎(,)𝑏)) ∈ ℝ ∧ (𝑎(,)𝑏) ≠ ∅) → (𝐹‘(𝑎(,)𝑏)) = 〈𝑎, 𝑏〉) | 
| 20 | ioorcl2 25607 | . . . . . . . . . . 11 ⊢ (((𝑎(,)𝑏) ≠ ∅ ∧ (vol*‘(𝑎(,)𝑏)) ∈ ℝ) → (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ)) | |
| 21 | 20 | ancoms 458 | . . . . . . . . . 10 ⊢ (((vol*‘(𝑎(,)𝑏)) ∈ ℝ ∧ (𝑎(,)𝑏) ≠ ∅) → (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ)) | 
| 22 | opelxpi 5722 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → 〈𝑎, 𝑏〉 ∈ (ℝ × ℝ)) | |
| 23 | 21, 22 | syl 17 | . . . . . . . . 9 ⊢ (((vol*‘(𝑎(,)𝑏)) ∈ ℝ ∧ (𝑎(,)𝑏) ≠ ∅) → 〈𝑎, 𝑏〉 ∈ (ℝ × ℝ)) | 
| 24 | 19, 23 | eqeltrd 2841 | . . . . . . . 8 ⊢ (((vol*‘(𝑎(,)𝑏)) ∈ ℝ ∧ (𝑎(,)𝑏) ≠ ∅) → (𝐹‘(𝑎(,)𝑏)) ∈ (ℝ × ℝ)) | 
| 25 | fveq2 6906 | . . . . . . . . . . 11 ⊢ (𝐴 = (𝑎(,)𝑏) → (vol*‘𝐴) = (vol*‘(𝑎(,)𝑏))) | |
| 26 | 25 | eleq1d 2826 | . . . . . . . . . 10 ⊢ (𝐴 = (𝑎(,)𝑏) → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘(𝑎(,)𝑏)) ∈ ℝ)) | 
| 27 | neeq1 3003 | . . . . . . . . . 10 ⊢ (𝐴 = (𝑎(,)𝑏) → (𝐴 ≠ ∅ ↔ (𝑎(,)𝑏) ≠ ∅)) | |
| 28 | 26, 27 | anbi12d 632 | . . . . . . . . 9 ⊢ (𝐴 = (𝑎(,)𝑏) → (((vol*‘𝐴) ∈ ℝ ∧ 𝐴 ≠ ∅) ↔ ((vol*‘(𝑎(,)𝑏)) ∈ ℝ ∧ (𝑎(,)𝑏) ≠ ∅))) | 
| 29 | fveq2 6906 | . . . . . . . . . 10 ⊢ (𝐴 = (𝑎(,)𝑏) → (𝐹‘𝐴) = (𝐹‘(𝑎(,)𝑏))) | |
| 30 | 29 | eleq1d 2826 | . . . . . . . . 9 ⊢ (𝐴 = (𝑎(,)𝑏) → ((𝐹‘𝐴) ∈ (ℝ × ℝ) ↔ (𝐹‘(𝑎(,)𝑏)) ∈ (ℝ × ℝ))) | 
| 31 | 28, 30 | imbi12d 344 | . . . . . . . 8 ⊢ (𝐴 = (𝑎(,)𝑏) → ((((vol*‘𝐴) ∈ ℝ ∧ 𝐴 ≠ ∅) → (𝐹‘𝐴) ∈ (ℝ × ℝ)) ↔ (((vol*‘(𝑎(,)𝑏)) ∈ ℝ ∧ (𝑎(,)𝑏) ≠ ∅) → (𝐹‘(𝑎(,)𝑏)) ∈ (ℝ × ℝ)))) | 
| 32 | 24, 31 | mpbiri 258 | . . . . . . 7 ⊢ (𝐴 = (𝑎(,)𝑏) → (((vol*‘𝐴) ∈ ℝ ∧ 𝐴 ≠ ∅) → (𝐹‘𝐴) ∈ (ℝ × ℝ))) | 
| 33 | 32 | a1i 11 | . . . . . 6 ⊢ ((𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ*) → (𝐴 = (𝑎(,)𝑏) → (((vol*‘𝐴) ∈ ℝ ∧ 𝐴 ≠ ∅) → (𝐹‘𝐴) ∈ (ℝ × ℝ)))) | 
| 34 | 33 | rexlimivv 3201 | . . . . 5 ⊢ (∃𝑎 ∈ ℝ* ∃𝑏 ∈ ℝ* 𝐴 = (𝑎(,)𝑏) → (((vol*‘𝐴) ∈ ℝ ∧ 𝐴 ≠ ∅) → (𝐹‘𝐴) ∈ (ℝ × ℝ))) | 
| 35 | 17, 34 | sylbi 217 | . . . 4 ⊢ (𝐴 ∈ ran (,) → (((vol*‘𝐴) ∈ ℝ ∧ 𝐴 ≠ ∅) → (𝐹‘𝐴) ∈ (ℝ × ℝ))) | 
| 36 | 35 | impl 455 | . . 3 ⊢ (((𝐴 ∈ ran (,) ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ≠ ∅) → (𝐹‘𝐴) ∈ (ℝ × ℝ)) | 
| 37 | 13, 36 | pm2.61dane 3029 | . 2 ⊢ ((𝐴 ∈ ran (,) ∧ (vol*‘𝐴) ∈ ℝ) → (𝐹‘𝐴) ∈ (ℝ × ℝ)) | 
| 38 | 5, 37 | elind 4200 | 1 ⊢ ((𝐴 ∈ ran (,) ∧ (vol*‘𝐴) ∈ ℝ) → (𝐹‘𝐴) ∈ ( ≤ ∩ (ℝ × ℝ))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 ∩ cin 3950 ∅c0 4333 ifcif 4525 𝒫 cpw 4600 〈cop 4632 ↦ cmpt 5225 × cxp 5683 ran crn 5686 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 supcsup 9480 infcinf 9481 ℝcr 11154 0cc0 11155 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 (,)cioo 13387 vol*covol 25497 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-rest 17467 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 df-cmp 23395 df-ovol 25499 df-vol 25500 | 
| This theorem is referenced by: uniioombllem2 25618 | 
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