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 10842 | . . 3 ⊢ ℝ ⊆ ℝ* | |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → ℝ ⊆ ℝ*) |
4 | 1, 3 | fssd 6541 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3853 ⟶wf 6354 ℝcr 10693 ℝ*cxr 10831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-un 3858 df-in 3860 df-ss 3870 df-f 6362 df-xr 10836 |
This theorem is referenced by: limsupubuz 42872 limsupreuz 42896 limsupvaluz2 42897 supcnvlimsup 42899 limsupgtlem 42936 liminflimsupclim 42966 climliminflimsup2 42968 climliminflimsup3 42969 climliminflimsup4 42970 xlimliminflimsup 43021 preimaioomnf 43871 incsmf 43893 issmfle 43896 decsmf 43917 |
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