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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 13493 | . . 3 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | fss 6753 | . . 3 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐹:ℝ⟶ℝ) | |
3 | 1, 2 | mpan2 691 | . 2 ⊢ (𝐹:ℝ⟶(0[,)+∞) → 𝐹:ℝ⟶ℝ) |
4 | ffvelcdm 7101 | . . . . 5 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) | |
5 | elrege0 13491 | . . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥))) | |
6 | 5 | baib 535 | . . . . 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 3174 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → (∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,)+∞) ↔ ∀𝑥 ∈ ℝ 0 ≤ (𝐹‘𝑥))) |
9 | ffn 6737 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ) | |
10 | ffnfv 7139 | . . . . 5 ⊢ (𝐹:ℝ⟶(0[,)+∞) ↔ (𝐹 Fn ℝ ∧ ∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,)+∞))) | |
11 | 10 | baib 535 | . . . 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 11251 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
14 | fnconstg 6797 | . . . . . . 7 ⊢ (0 ∈ ℂ → (ℂ × {0}) Fn ℂ) | |
15 | 13, 14 | ax-mp 5 | . . . . . 6 ⊢ (ℂ × {0}) Fn ℂ |
16 | df-0p 25719 | . . . . . . 7 ⊢ 0𝑝 = (ℂ × {0}) | |
17 | 16 | fneq1i 6666 | . . . . . 6 ⊢ (0𝑝 Fn ℂ ↔ (ℂ × {0}) Fn ℂ) |
18 | 15, 17 | mpbir 231 | . . . . 5 ⊢ 0𝑝 Fn ℂ |
19 | 18 | a1i 11 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → 0𝑝 Fn ℂ) |
20 | cnex 11234 | . . . . 5 ⊢ ℂ ∈ V | |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → ℂ ∈ V) |
22 | reex 11244 | . . . . 5 ⊢ ℝ ∈ V | |
23 | 22 | a1i 11 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → ℝ ∈ V) |
24 | ax-resscn 11210 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
25 | sseqin2 4231 | . . . . 5 ⊢ (ℝ ⊆ ℂ ↔ (ℂ ∩ ℝ) = ℝ) | |
26 | 24, 25 | mpbi 230 | . . . 4 ⊢ (ℂ ∩ ℝ) = ℝ |
27 | 0pval 25720 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (0𝑝‘𝑥) = 0) | |
28 | 27 | adantl 481 | . . . 4 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℂ) → (0𝑝‘𝑥) = 0) |
29 | eqidd 2736 | . . . 4 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
30 | 19, 9, 21, 23, 26, 28, 29 | ofrfval 7707 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → (0𝑝 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ ℝ 0 ≤ (𝐹‘𝑥))) |
31 | 8, 12, 30 | 3bitr4d 311 | . 2 ⊢ (𝐹:ℝ⟶ℝ → (𝐹:ℝ⟶(0[,)+∞) ↔ 0𝑝 ∘r ≤ 𝐹)) |
32 | 3, 31 | biadanii 822 | 1 ⊢ (𝐹:ℝ⟶(0[,)+∞) ↔ (𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 {csn 4631 class class class wbr 5148 × cxp 5687 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∘r cofr 7696 ℂcc 11151 ℝcr 11152 0cc0 11153 +∞cpnf 11290 ≤ cle 11294 [,)cico 13386 0𝑝c0p 25718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-i2m1 11221 ax-rnegex 11224 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-ofr 7698 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-ico 13390 df-0p 25719 |
This theorem is referenced by: itg2i1fseq 25805 itg2addlem 25808 ftc1anclem8 37687 |
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