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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 10334 | . . 3 ⊢ ℂ ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ℂ ∈ V) |
3 | xrex 12110 | . . 3 ⊢ ℝ* ∈ V | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → ℝ* ∈ V) |
5 | fuzxrpmcn.1 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | 5 | uzsscn2 40503 | . . 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 40284 | 1 ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 Vcvv 3415 ⊆ wss 3799 ⟶wf 6120 ‘cfv 6124 (class class class)co 6906 ↑pm cpm 8124 ℂcc 10251 ℝ*cxr 10391 ℤ≥cuz 11969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-fv 6132 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-pm 8126 df-xr 10396 df-neg 10589 df-z 11706 df-uz 11970 |
This theorem is referenced by: xlimconst2 40857 xlimclim2lem 40861 climxlim2 40868 |
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