Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fuzxrpmcn | Structured version Visualization version GIF version |
Description: A function mapping from an upper set of integers to the extended reals is a partial map on the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
fuzxrpmcn.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
fuzxrpmcn.2 | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
Ref | Expression |
---|---|
fuzxrpmcn | ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 10618 | . . 3 ⊢ ℂ ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ℂ ∈ V) |
3 | xrex 12387 | . . 3 ⊢ ℝ* ∈ V | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → ℝ* ∈ V) |
5 | fuzxrpmcn.1 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | 5 | uzsscn2 41774 | . . 3 ⊢ 𝑍 ⊆ ℂ |
7 | 6 | a1i 11 | . 2 ⊢ (𝜑 → 𝑍 ⊆ ℂ) |
8 | fuzxrpmcn.2 | . 2 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
9 | 2, 4, 7, 8 | fpmd 41558 | 1 ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑pm cpm 8407 ℂcc 10535 ℝ*cxr 10674 ℤ≥cuz 12244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-pm 8409 df-xr 10679 df-neg 10873 df-z 11983 df-uz 12245 |
This theorem is referenced by: xlimconst2 42136 xlimclim2lem 42140 climxlim2 42147 xlimliminflimsup 42163 |
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