Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > uzxrd | Structured version Visualization version GIF version |
Description: An upper integer is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
uzxrd.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
uzxrd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑍) |
Ref | Expression |
---|---|
uzxrd | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressxr 10763 | . 2 ⊢ ℝ ⊆ ℝ* | |
2 | uzxrd.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | uzxrd.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑍) | |
4 | 2, 3 | uzred 42521 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
5 | 1, 4 | sseldi 3875 | 1 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ‘cfv 6339 ℝcr 10614 ℝ*cxr 10752 ℤ≥cuz 12324 |
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 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-cnex 10671 ax-resscn 10672 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-fv 6347 df-ov 7173 df-xr 10757 df-neg 10951 df-z 12063 df-uz 12325 |
This theorem is referenced by: uzxr 42548 liminflelimsupuz 42868 |
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