Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 41788 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℝ) | |
5 | 1, 2, 3, 4 | syl3anc 1367 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
6 | iccsuble.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ (𝐴[,]𝐵)) | |
7 | eliccre 41788 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐷 ∈ (𝐴[,]𝐵)) → 𝐷 ∈ ℝ) | |
8 | 1, 2, 6, 7 | syl3anc 1367 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
9 | elicc2 12804 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
10 | 1, 2, 9 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
11 | 3, 10 | mpbid 234 | . . 3 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
12 | 11 | simp3d 1140 | . 2 ⊢ (𝜑 → 𝐶 ≤ 𝐵) |
13 | elicc2 12804 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐷 ∈ (𝐴[,]𝐵) ↔ (𝐷 ∈ ℝ ∧ 𝐴 ≤ 𝐷 ∧ 𝐷 ≤ 𝐵))) | |
14 | 1, 2, 13 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐷 ∈ (𝐴[,]𝐵) ↔ (𝐷 ∈ ℝ ∧ 𝐴 ≤ 𝐷 ∧ 𝐷 ≤ 𝐵))) |
15 | 6, 14 | mpbid 234 | . . 3 ⊢ (𝜑 → (𝐷 ∈ ℝ ∧ 𝐴 ≤ 𝐷 ∧ 𝐷 ≤ 𝐵)) |
16 | 15 | simp2d 1139 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐷) |
17 | 5, 1, 2, 8, 12, 16 | le2subd 11262 | 1 ⊢ (𝜑 → (𝐶 − 𝐷) ≤ (𝐵 − 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 ≤ cle 10678 − cmin 10872 [,]cicc 12744 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-icc 12748 |
This theorem is referenced by: fourierdlem6 42405 fourierdlem42 42441 hoidmvlelem1 42884 |
Copyright terms: Public domain | W3C validator |