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Mirrors > Home > MPE Home > Th. List > 0plef | Structured version Visualization version GIF version |
Description: Two ways to say that the function 𝐹 on the reals is nonnegative. (Contributed by Mario Carneiro, 17-Aug-2014.) |
Ref | Expression |
---|---|
0plef | ⊢ (𝐹:ℝ⟶(0[,)+∞) ↔ (𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rge0ssre 12832 | . . 3 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | fss 6520 | . . 3 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐹:ℝ⟶ℝ) | |
3 | 1, 2 | mpan2 687 | . 2 ⊢ (𝐹:ℝ⟶(0[,)+∞) → 𝐹:ℝ⟶ℝ) |
4 | ffvelrn 6841 | . . . . 5 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) | |
5 | elrege0 12830 | . . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥))) | |
6 | 5 | baib 536 | . . . . 5 ⊢ ((𝐹‘𝑥) ∈ ℝ → ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ 0 ≤ (𝐹‘𝑥))) |
7 | 4, 6 | syl 17 | . . . 4 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ 0 ≤ (𝐹‘𝑥))) |
8 | 7 | ralbidva 3193 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → (∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,)+∞) ↔ ∀𝑥 ∈ ℝ 0 ≤ (𝐹‘𝑥))) |
9 | ffn 6507 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ) | |
10 | ffnfv 6874 | . . . . 5 ⊢ (𝐹:ℝ⟶(0[,)+∞) ↔ (𝐹 Fn ℝ ∧ ∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,)+∞))) | |
11 | 10 | baib 536 | . . . 4 ⊢ (𝐹 Fn ℝ → (𝐹:ℝ⟶(0[,)+∞) ↔ ∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,)+∞))) |
12 | 9, 11 | syl 17 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → (𝐹:ℝ⟶(0[,)+∞) ↔ ∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,)+∞))) |
13 | 0cn 10621 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
14 | fnconstg 6560 | . . . . . . 7 ⊢ (0 ∈ ℂ → (ℂ × {0}) Fn ℂ) | |
15 | 13, 14 | ax-mp 5 | . . . . . 6 ⊢ (ℂ × {0}) Fn ℂ |
16 | df-0p 24198 | . . . . . . 7 ⊢ 0𝑝 = (ℂ × {0}) | |
17 | 16 | fneq1i 6443 | . . . . . 6 ⊢ (0𝑝 Fn ℂ ↔ (ℂ × {0}) Fn ℂ) |
18 | 15, 17 | mpbir 232 | . . . . 5 ⊢ 0𝑝 Fn ℂ |
19 | 18 | a1i 11 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → 0𝑝 Fn ℂ) |
20 | cnex 10606 | . . . . 5 ⊢ ℂ ∈ V | |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → ℂ ∈ V) |
22 | reex 10616 | . . . . 5 ⊢ ℝ ∈ V | |
23 | 22 | a1i 11 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → ℝ ∈ V) |
24 | ax-resscn 10582 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
25 | sseqin2 4189 | . . . . 5 ⊢ (ℝ ⊆ ℂ ↔ (ℂ ∩ ℝ) = ℝ) | |
26 | 24, 25 | mpbi 231 | . . . 4 ⊢ (ℂ ∩ ℝ) = ℝ |
27 | 0pval 24199 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (0𝑝‘𝑥) = 0) | |
28 | 27 | adantl 482 | . . . 4 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℂ) → (0𝑝‘𝑥) = 0) |
29 | eqidd 2819 | . . . 4 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
30 | 19, 9, 21, 23, 26, 28, 29 | ofrfval 7406 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → (0𝑝 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ ℝ 0 ≤ (𝐹‘𝑥))) |
31 | 8, 12, 30 | 3bitr4d 312 | . 2 ⊢ (𝐹:ℝ⟶ℝ → (𝐹:ℝ⟶(0[,)+∞) ↔ 0𝑝 ∘r ≤ 𝐹)) |
32 | 3, 31 | biadanii 818 | 1 ⊢ (𝐹:ℝ⟶(0[,)+∞) ↔ (𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ∩ cin 3932 ⊆ wss 3933 {csn 4557 class class class wbr 5057 × cxp 5546 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ∘r cofr 7397 ℂcc 10523 ℝcr 10524 0cc0 10525 +∞cpnf 10660 ≤ cle 10664 [,)cico 12728 0𝑝c0p 24197 |
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-rep 5181 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-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-i2m1 10593 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-reu 3142 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-ofr 7399 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-le 10669 df-ico 12732 df-0p 24198 |
This theorem is referenced by: itg2i1fseq 24283 itg2addlem 24286 ftc1anclem8 34855 |
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