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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iccgelbd | Structured version Visualization version GIF version |
Description: An element of a closed interval is more than or equal to its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
iccgelbd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
iccgelbd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
iccgelbd.3 | ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) |
Ref | Expression |
---|---|
iccgelbd | ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccgelbd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | iccgelbd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
3 | iccgelbd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) | |
4 | iccgelb 12542 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶) | |
5 | 1, 2, 3, 4 | syl3anc 1439 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 4886 (class class class)co 6922 ℝ*cxr 10410 ≤ cle 10412 [,]cicc 12490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-xr 10415 df-icc 12494 |
This theorem is referenced by: sqrlearg 40670 |
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