Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 10688 | . . 3 ⊢ ℝ ⊆ ℝ* | |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → ℝ ⊆ ℝ*) |
4 | 1, 3 | fssd 6531 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3939 ⟶wf 6354 ℝcr 10539 ℝ*cxr 10677 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 df-un 3944 df-in 3946 df-ss 3955 df-f 6362 df-xr 10682 |
This theorem is referenced by: limsupubuz 42000 limsupreuz 42024 limsupvaluz2 42025 supcnvlimsup 42027 limsupgtlem 42064 liminflimsupclim 42094 climliminflimsup2 42096 climliminflimsup3 42097 climliminflimsup4 42098 xlimliminflimsup 42149 preimaioomnf 43004 incsmf 43026 issmfle 43029 decsmf 43050 |
Copyright terms: Public domain | W3C validator |