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 10679 | . . 3 ⊢ ℝ ⊆ ℝ* | |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → ℝ ⊆ ℝ*) |
4 | 1, 3 | fssd 6523 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3936 ⟶wf 6346 ℝcr 10530 ℝ*cxr 10668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 df-un 3941 df-in 3943 df-ss 3952 df-f 6354 df-xr 10673 |
This theorem is referenced by: limsupubuz 41986 limsupreuz 42010 limsupvaluz2 42011 supcnvlimsup 42013 limsupgtlem 42050 liminflimsupclim 42080 climliminflimsup2 42082 climliminflimsup3 42083 climliminflimsup4 42084 xlimliminflimsup 42135 preimaioomnf 42990 incsmf 43012 issmfle 43015 decsmf 43036 |
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