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Mirrors > Home > MPE Home > Th. List > Mathboxes > fge0npnf | Structured version Visualization version GIF version |
Description: If 𝐹 maps to nonnegative reals, then +∞ is not in its range. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
fge0npnf.1 | ⊢ (𝜑 → 𝐹:𝑋⟶(0[,)+∞)) |
Ref | Expression |
---|---|
fge0npnf | ⊢ (𝜑 → ¬ +∞ ∈ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fge0npnf.1 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,)+∞)) | |
2 | 1 | frnd 6514 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ (0[,)+∞)) |
3 | 2 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → ran 𝐹 ⊆ (0[,)+∞)) |
4 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹) | |
5 | 3, 4 | sseldd 3965 | . 2 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ (0[,)+∞)) |
6 | 0xr 10676 | . . . 4 ⊢ 0 ∈ ℝ* | |
7 | icoub 41678 | . . . 4 ⊢ (0 ∈ ℝ* → ¬ +∞ ∈ (0[,)+∞)) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ ¬ +∞ ∈ (0[,)+∞) |
9 | 8 | a1i 11 | . 2 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ (0[,)+∞)) |
10 | 5, 9 | pm2.65da 813 | 1 ⊢ (𝜑 → ¬ +∞ ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∈ wcel 2105 ⊆ wss 3933 ran crn 5549 ⟶wf 6344 (class class class)co 7145 0cc0 10525 +∞cpnf 10660 ℝ*cxr 10662 [,)cico 12728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-addrcl 10586 ax-rnegex 10596 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-ico 12732 |
This theorem is referenced by: sge0reval 42531 sge0fsum 42546 |
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