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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eliccd | Structured version Visualization version GIF version |
Description: Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
eliccd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
eliccd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
eliccd.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
eliccd.4 | ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
eliccd.5 | ⊢ (𝜑 → 𝐶 ≤ 𝐵) |
Ref | Expression |
---|---|
eliccd | ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliccd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
2 | eliccd.4 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐶) | |
3 | eliccd.5 | . 2 ⊢ (𝜑 → 𝐶 ≤ 𝐵) | |
4 | eliccd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
5 | eliccd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
6 | elicc2 12526 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
7 | 4, 5, 6 | syl2anc 581 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
8 | 1, 2, 3, 7 | mpbir3and 1448 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1113 ∈ wcel 2166 class class class wbr 4873 (class class class)co 6905 ℝcr 10251 ≤ cle 10392 [,]cicc 12466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-pre-lttri 10326 ax-pre-lttrn 10327 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-icc 12470 |
This theorem is referenced by: iccshift 40540 iooiinicc 40564 sqrlearg 40575 limciccioolb 40648 cncfiooicclem1 40901 iblspltprt 40983 itgspltprt 40989 itgiccshift 40990 itgperiod 40991 itgsbtaddcnst 40992 fourierdlem15 41133 fourierdlem17 41135 fourierdlem40 41158 fourierdlem50 41167 fourierdlem51 41168 fourierdlem62 41179 fourierdlem63 41180 fourierdlem64 41181 fourierdlem65 41182 fourierdlem73 41190 fourierdlem74 41191 fourierdlem75 41192 fourierdlem76 41193 fourierdlem78 41195 fourierdlem81 41198 fourierdlem82 41199 fourierdlem92 41209 fourierdlem93 41210 fourierdlem101 41218 fourierdlem103 41220 fourierdlem104 41221 fourierdlem107 41224 fourierdlem111 41228 rrxsnicc 41311 salgencntex 41352 hoidmv1lelem2 41600 hoidmvlelem1 41603 hoidmvlelem2 41604 iinhoiicclem 41681 smfmullem1 41792 |
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