Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ge0lere | Structured version Visualization version GIF version |
Description: A nonnegative extended Real number smaller than or equal to a Real number, is a Real number. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
ge0lere.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ge0lere.b | ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) |
ge0lere.l | ⊢ (𝜑 → 𝐵 ≤ 𝐴) |
Ref | Expression |
---|---|
ge0lere | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ge0lere.b | . 2 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) | |
2 | iccssxr 13173 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
3 | 2, 1 | sselid 3924 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
4 | pnfxr 11040 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → +∞ ∈ ℝ*) |
6 | ge0lere.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
7 | 6 | rexrd 11036 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
8 | ge0lere.l | . . . 4 ⊢ (𝜑 → 𝐵 ≤ 𝐴) | |
9 | 6 | ltpnfd 12868 | . . . 4 ⊢ (𝜑 → 𝐴 < +∞) |
10 | 3, 7, 5, 8, 9 | xrlelttrd 12905 | . . 3 ⊢ (𝜑 → 𝐵 < +∞) |
11 | 3, 5, 10 | xrltned 42878 | . 2 ⊢ (𝜑 → 𝐵 ≠ +∞) |
12 | ge0xrre 43051 | . 2 ⊢ ((𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≠ +∞) → 𝐵 ∈ ℝ) | |
13 | 1, 11, 12 | syl2anc 584 | 1 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ≠ wne 2945 class class class wbr 5079 (class class class)co 7272 ℝcr 10881 0cc0 10882 +∞cpnf 11017 ℝ*cxr 11019 ≤ cle 11021 [,]cicc 13093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-addrcl 10943 ax-rnegex 10953 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-po 5504 df-so 5505 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7275 df-oprab 7276 df-mpo 7277 df-1st 7825 df-2nd 7826 df-er 8490 df-en 8726 df-dom 8727 df-sdom 8728 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-ico 13096 df-icc 13097 |
This theorem is referenced by: hspmbllem2 44147 hspmbllem3 44148 |
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