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 10688 | . 2 ⊢ ℝ ⊆ ℝ* | |
2 | uzxrd.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | uzxrd.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑍) | |
4 | 2, 3 | uzred 41723 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
5 | 1, 4 | sseldi 3968 | 1 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 ℝcr 10539 ℝ*cxr 10677 ℤ≥cuz 12246 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-cnex 10596 ax-resscn 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-xr 10682 df-neg 10876 df-z 11985 df-uz 12247 |
This theorem is referenced by: uzxr 41750 liminflelimsupuz 42072 |
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