| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > frexr | Structured version Visualization version GIF version | ||
| Description: A function taking real values, is a function taking extended real values. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
| Ref | Expression |
|---|---|
| frexr.1 | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
| Ref | Expression |
|---|---|
| frexr | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | frexr.1 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
| 2 | ressxr 11305 | . . 3 ⊢ ℝ ⊆ ℝ* | |
| 3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → ℝ ⊆ ℝ*) |
| 4 | 1, 3 | fssd 6753 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ⊆ wss 3951 ⟶wf 6557 ℝcr 11154 ℝ*cxr 11294 |
| 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-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-f 6565 df-xr 11299 |
| This theorem is referenced by: limsupubuz 45728 limsupreuz 45752 limsupvaluz2 45753 supcnvlimsup 45755 limsupgtlem 45792 liminflimsupclim 45822 climliminflimsup2 45824 climliminflimsup3 45825 climliminflimsup4 45826 xlimliminflimsup 45877 preimaioomnf 46734 incsmf 46757 issmfle 46760 decsmf 46782 smfsupdmmbllem 46859 smfinfdmmbllem 46863 |
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