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 41883 | . . . . 5 ⊢ (𝑥 ∈ ran ([,) ↾ (ℝ × ℝ)) ↔ ∃𝑦 ∈ ℝ ∃𝑧 ∈ ℝ 𝑥 = (𝑦[,)𝑧)) | |
2 | 1 | biimpi 218 | . . . 4 ⊢ (𝑥 ∈ ran ([,) ↾ (ℝ × ℝ)) → ∃𝑦 ∈ ℝ ∃𝑧 ∈ ℝ 𝑥 = (𝑦[,)𝑧)) |
3 | simpr 487 | . . . . . . . 8 ⊢ (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ 𝑥 = (𝑦[,)𝑧)) → 𝑥 = (𝑦[,)𝑧)) | |
4 | simpl 485 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℝ) | |
5 | rexr 10680 | . . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℝ*) | |
6 | 5 | adantl 484 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ*) |
7 | icombl 24160 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ*) → (𝑦[,)𝑧) ∈ dom vol) | |
8 | 4, 6, 7 | syl2anc 586 | . . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦[,)𝑧) ∈ dom vol) |
9 | 8 | adantr 483 | . . . . . . . 8 ⊢ (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ 𝑥 = (𝑦[,)𝑧)) → (𝑦[,)𝑧) ∈ dom vol) |
10 | 3, 9 | eqeltrd 2912 | . . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ 𝑥 = (𝑦[,)𝑧)) → 𝑥 ∈ dom vol) |
11 | 10 | rexlimdva2 3286 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → (∃𝑧 ∈ ℝ 𝑥 = (𝑦[,)𝑧) → 𝑥 ∈ dom vol)) |
12 | 11 | rexlimiv 3279 | . . . . 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 3147 | . 2 ⊢ ∀𝑥 ∈ ran ([,) ↾ (ℝ × ℝ))𝑥 ∈ dom vol |
16 | dfss3 3949 | . 2 ⊢ (ran ([,) ↾ (ℝ × ℝ)) ⊆ dom vol ↔ ∀𝑥 ∈ ran ([,) ↾ (ℝ × ℝ))𝑥 ∈ dom vol) | |
17 | 15, 16 | mpbir 233 | 1 ⊢ ran ([,) ↾ (ℝ × ℝ)) ⊆ dom vol |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3137 ∃wrex 3138 ⊆ wss 3929 × cxp 5546 dom cdm 5548 ran crn 5549 ↾ cres 5550 (class class class)co 7149 ℝcr 10529 ℝ*cxr 10667 [,)cico 12734 volcvol 24059 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-er 8282 df-map 8401 df-pm 8402 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-inf 8900 df-oi 8967 df-dju 9323 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xadd 12502 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12890 df-fzo 13031 df-fl 13159 df-seq 13367 df-exp 13427 df-hash 13688 df-cj 14453 df-re 14454 df-im 14455 df-sqrt 14589 df-abs 14590 df-clim 14840 df-rlim 14841 df-sum 15038 df-xmet 20533 df-met 20534 df-ovol 24060 df-vol 24061 |
This theorem is referenced by: (None) |
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