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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 11187 | . . 3 ⊢ ℂ ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ℂ ∈ V) |
3 | xrex 12968 | . . 3 ⊢ ℝ* ∈ V | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → ℝ* ∈ V) |
5 | fuzxrpmcn.1 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | 5 | uzsscn2 44673 | . . 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 44453 | 1 ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ⊆ wss 3940 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 ↑pm cpm 8817 ℂcc 11104 ℝ*cxr 11244 ℤ≥cuz 12819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-pm 8819 df-xr 11249 df-neg 11444 df-z 12556 df-uz 12820 |
This theorem is referenced by: xlimconst2 45036 xlimclim2lem 45040 climxlim2 45047 xlimliminflimsup 45063 |
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