Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > icoresmbl | Structured version Visualization version GIF version |
Description: A closed-below, open-above real interval is measurable, when the bounds are real. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
icoresmbl | ⊢ ran ([,) ↾ (ℝ × ℝ)) ⊆ dom vol |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elicores 43053 | . . . . 5 ⊢ (𝑥 ∈ ran ([,) ↾ (ℝ × ℝ)) ↔ ∃𝑦 ∈ ℝ ∃𝑧 ∈ ℝ 𝑥 = (𝑦[,)𝑧)) | |
2 | 1 | biimpi 215 | . . . 4 ⊢ (𝑥 ∈ ran ([,) ↾ (ℝ × ℝ)) → ∃𝑦 ∈ ℝ ∃𝑧 ∈ ℝ 𝑥 = (𝑦[,)𝑧)) |
3 | simpr 485 | . . . . . . . 8 ⊢ (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ 𝑥 = (𝑦[,)𝑧)) → 𝑥 = (𝑦[,)𝑧)) | |
4 | simpl 483 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℝ) | |
5 | rexr 11032 | . . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℝ*) | |
6 | 5 | adantl 482 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ*) |
7 | icombl 24739 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ*) → (𝑦[,)𝑧) ∈ dom vol) | |
8 | 4, 6, 7 | syl2anc 584 | . . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦[,)𝑧) ∈ dom vol) |
9 | 8 | adantr 481 | . . . . . . . 8 ⊢ (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ 𝑥 = (𝑦[,)𝑧)) → (𝑦[,)𝑧) ∈ dom vol) |
10 | 3, 9 | eqeltrd 2841 | . . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ 𝑥 = (𝑦[,)𝑧)) → 𝑥 ∈ dom vol) |
11 | 10 | rexlimdva2 3218 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → (∃𝑧 ∈ ℝ 𝑥 = (𝑦[,)𝑧) → 𝑥 ∈ dom vol)) |
12 | 11 | rexlimiv 3211 | . . . . 5 ⊢ (∃𝑦 ∈ ℝ ∃𝑧 ∈ ℝ 𝑥 = (𝑦[,)𝑧) → 𝑥 ∈ dom vol) |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝑥 ∈ ran ([,) ↾ (ℝ × ℝ)) → (∃𝑦 ∈ ℝ ∃𝑧 ∈ ℝ 𝑥 = (𝑦[,)𝑧) → 𝑥 ∈ dom vol)) |
14 | 2, 13 | mpd 15 | . . 3 ⊢ (𝑥 ∈ ran ([,) ↾ (ℝ × ℝ)) → 𝑥 ∈ dom vol) |
15 | 14 | rgen 3076 | . 2 ⊢ ∀𝑥 ∈ ran ([,) ↾ (ℝ × ℝ))𝑥 ∈ dom vol |
16 | dfss3 3914 | . 2 ⊢ (ran ([,) ↾ (ℝ × ℝ)) ⊆ dom vol ↔ ∀𝑥 ∈ ran ([,) ↾ (ℝ × ℝ))𝑥 ∈ dom vol) | |
17 | 15, 16 | mpbir 230 | 1 ⊢ ran ([,) ↾ (ℝ × ℝ)) ⊆ dom vol |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∀wral 3066 ∃wrex 3067 ⊆ wss 3892 × cxp 5588 dom cdm 5590 ran crn 5591 ↾ cres 5592 (class class class)co 7272 ℝcr 10881 ℝ*cxr 11019 [,)cico 13092 volcvol 24638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-inf2 9387 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-pre-sup 10960 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-of 7528 df-om 7708 df-1st 7825 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-2o 8290 df-er 8490 df-map 8609 df-pm 8610 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-sup 9189 df-inf 9190 df-oi 9257 df-dju 9670 df-card 9708 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-div 11644 df-nn 11985 df-2 12047 df-3 12048 df-n0 12245 df-z 12331 df-uz 12594 df-q 12700 df-rp 12742 df-xadd 12860 df-ioo 13094 df-ico 13096 df-icc 13097 df-fz 13251 df-fzo 13394 df-fl 13523 df-seq 13733 df-exp 13794 df-hash 14056 df-cj 14821 df-re 14822 df-im 14823 df-sqrt 14957 df-abs 14958 df-clim 15208 df-rlim 15209 df-sum 15409 df-xmet 20601 df-met 20602 df-ovol 24639 df-vol 24640 |
This theorem is referenced by: (None) |
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