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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iccsuble | Structured version Visualization version GIF version |
Description: An upper bound to the distance of two elements in a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
iccsuble.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
iccsuble.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
iccsuble.3 | ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) |
iccsuble.4 | ⊢ (𝜑 → 𝐷 ∈ (𝐴[,]𝐵)) |
Ref | Expression |
---|---|
iccsuble | ⊢ (𝜑 → (𝐶 − 𝐷) ≤ (𝐵 − 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccsuble.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | iccsuble.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | iccsuble.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) | |
4 | eliccre 45425 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℝ) | |
5 | 1, 2, 3, 4 | syl3anc 1371 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
6 | iccsuble.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ (𝐴[,]𝐵)) | |
7 | eliccre 45425 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐷 ∈ (𝐴[,]𝐵)) → 𝐷 ∈ ℝ) | |
8 | 1, 2, 6, 7 | syl3anc 1371 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
9 | elicc2 13474 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
10 | 1, 2, 9 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
11 | 3, 10 | mpbid 232 | . . 3 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
12 | 11 | simp3d 1144 | . 2 ⊢ (𝜑 → 𝐶 ≤ 𝐵) |
13 | elicc2 13474 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐷 ∈ (𝐴[,]𝐵) ↔ (𝐷 ∈ ℝ ∧ 𝐴 ≤ 𝐷 ∧ 𝐷 ≤ 𝐵))) | |
14 | 1, 2, 13 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝐷 ∈ (𝐴[,]𝐵) ↔ (𝐷 ∈ ℝ ∧ 𝐴 ≤ 𝐷 ∧ 𝐷 ≤ 𝐵))) |
15 | 6, 14 | mpbid 232 | . . 3 ⊢ (𝜑 → (𝐷 ∈ ℝ ∧ 𝐴 ≤ 𝐷 ∧ 𝐷 ≤ 𝐵)) |
16 | 15 | simp2d 1143 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐷) |
17 | 5, 1, 2, 8, 12, 16 | le2subd 11912 | 1 ⊢ (𝜑 → (𝐶 − 𝐷) ≤ (𝐵 − 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 ∈ wcel 2108 class class class wbr 5166 (class class class)co 7450 ℝcr 11185 ≤ cle 11327 − cmin 11522 [,]cicc 13412 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-icc 13416 |
This theorem is referenced by: fourierdlem6 46036 fourierdlem42 46072 hoidmvlelem1 46518 |
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