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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 13396 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
7 | 4, 5, 6 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
8 | 1, 2, 3, 7 | mpbir3and 1341 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1086 ∈ wcel 2105 class class class wbr 5148 (class class class)co 7412 ℝcr 11115 ≤ cle 11256 [,]cicc 13334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-pre-lttri 11190 ax-pre-lttrn 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-icc 13338 |
This theorem is referenced by: iccshift 44689 iooiinicc 44713 sqrlearg 44724 limciccioolb 44795 cncfiooicclem1 45067 iblspltprt 45147 itgspltprt 45153 itgiccshift 45154 itgperiod 45155 itgsbtaddcnst 45156 fourierdlem15 45296 fourierdlem17 45298 fourierdlem40 45321 fourierdlem50 45330 fourierdlem51 45331 fourierdlem62 45342 fourierdlem63 45343 fourierdlem64 45344 fourierdlem65 45345 fourierdlem73 45353 fourierdlem74 45354 fourierdlem75 45355 fourierdlem76 45356 fourierdlem78 45358 fourierdlem81 45361 fourierdlem82 45362 fourierdlem92 45372 fourierdlem93 45373 fourierdlem101 45381 fourierdlem103 45383 fourierdlem104 45384 fourierdlem107 45387 fourierdlem111 45391 rrxsnicc 45474 salgencntex 45517 hoidmv1lelem2 45766 hoidmvlelem1 45769 hoidmvlelem2 45770 iinhoiicclem 45847 smfmullem1 45965 |
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