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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 12808 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
3 | 2, 1 | sseldi 3913 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
4 | pnfxr 10684 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → +∞ ∈ ℝ*) |
6 | ge0lere.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
7 | 6 | rexrd 10680 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
8 | ge0lere.l | . . . 4 ⊢ (𝜑 → 𝐵 ≤ 𝐴) | |
9 | 6 | ltpnfd 12504 | . . . 4 ⊢ (𝜑 → 𝐴 < +∞) |
10 | 3, 7, 5, 8, 9 | xrlelttrd 12541 | . . 3 ⊢ (𝜑 → 𝐵 < +∞) |
11 | 3, 5, 10 | xrltned 41989 | . 2 ⊢ (𝜑 → 𝐵 ≠ +∞) |
12 | ge0xrre 42168 | . 2 ⊢ ((𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≠ +∞) → 𝐵 ∈ ℝ) | |
13 | 1, 11, 12 | syl2anc 587 | 1 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 +∞cpnf 10661 ℝ*cxr 10663 ≤ cle 10665 [,]cicc 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-addrcl 10587 ax-rnegex 10597 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-ico 12732 df-icc 12733 |
This theorem is referenced by: hspmbllem2 43266 hspmbllem3 43267 |
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