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 10620 | . . 3 ⊢ ℂ ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ℂ ∈ V) |
3 | xrex 12389 | . . 3 ⊢ ℝ* ∈ V | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → ℝ* ∈ V) |
5 | fuzxrpmcn.1 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | 5 | uzsscn2 41761 | . . 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 41545 | 1 ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑pm cpm 8409 ℂcc 10537 ℝ*cxr 10676 ℤ≥cuz 12246 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-pm 8411 df-xr 10681 df-neg 10875 df-z 11985 df-uz 12247 |
This theorem is referenced by: xlimconst2 42123 xlimclim2lem 42127 climxlim2 42134 xlimliminflimsup 42150 |
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