|   | Mathbox for Glauco Siliprandi | < Previous  
      Next > Nearby theorems | |
| 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 13470 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
| 3 | 2, 1 | sselid 3981 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | 
| 4 | pnfxr 11315 | . . . 4 ⊢ +∞ ∈ ℝ* | |
| 5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → +∞ ∈ ℝ*) | 
| 6 | ge0lere.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 7 | 6 | rexrd 11311 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | 
| 8 | ge0lere.l | . . . 4 ⊢ (𝜑 → 𝐵 ≤ 𝐴) | |
| 9 | 6 | ltpnfd 13163 | . . . 4 ⊢ (𝜑 → 𝐴 < +∞) | 
| 10 | 3, 7, 5, 8, 9 | xrlelttrd 13202 | . . 3 ⊢ (𝜑 → 𝐵 < +∞) | 
| 11 | 3, 5, 10 | xrltned 45368 | . 2 ⊢ (𝜑 → 𝐵 ≠ +∞) | 
| 12 | ge0xrre 45544 | . 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 2108 ≠ wne 2940 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 +∞cpnf 11292 ℝ*cxr 11294 ≤ cle 11296 [,]cicc 13390 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ico 13393 df-icc 13394 | 
| This theorem is referenced by: hspmbllem2 46642 hspmbllem3 46643 | 
| Copyright terms: Public domain | W3C validator |