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Mirrors > Home > MPE Home > Th. List > xrge0f | Structured version Visualization version GIF version |
Description: A real function is a nonnegative extended real function if all its values are greater than or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 28-Jul-2014.) |
Ref | Expression |
---|---|
xrge0f | ⊢ ((𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ 𝐹) → 𝐹:ℝ⟶(0[,]+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 6513 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ) | |
2 | 1 | adantr 483 | . 2 ⊢ ((𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ 𝐹) → 𝐹 Fn ℝ) |
3 | ax-resscn 10593 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝐹:ℝ⟶ℝ → ℝ ⊆ ℂ) |
5 | 4, 1 | 0pledm 24273 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → (0𝑝 ∘r ≤ 𝐹 ↔ (ℝ × {0}) ∘r ≤ 𝐹)) |
6 | 0re 10642 | . . . . . 6 ⊢ 0 ∈ ℝ | |
7 | fnconstg 6566 | . . . . . 6 ⊢ (0 ∈ ℝ → (ℝ × {0}) Fn ℝ) | |
8 | 6, 7 | mp1i 13 | . . . . 5 ⊢ (𝐹:ℝ⟶ℝ → (ℝ × {0}) Fn ℝ) |
9 | reex 10627 | . . . . . 6 ⊢ ℝ ∈ V | |
10 | 9 | a1i 11 | . . . . 5 ⊢ (𝐹:ℝ⟶ℝ → ℝ ∈ V) |
11 | inidm 4194 | . . . . 5 ⊢ (ℝ ∩ ℝ) = ℝ | |
12 | c0ex 10634 | . . . . . . 7 ⊢ 0 ∈ V | |
13 | 12 | fvconst2 6965 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → ((ℝ × {0})‘𝑥) = 0) |
14 | 13 | adantl 484 | . . . . 5 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → ((ℝ × {0})‘𝑥) = 0) |
15 | eqidd 2822 | . . . . 5 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
16 | 8, 1, 10, 10, 11, 14, 15 | ofrfval 7416 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → ((ℝ × {0}) ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ ℝ 0 ≤ (𝐹‘𝑥))) |
17 | ffvelrn 6848 | . . . . . . . 8 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) | |
18 | 17 | rexrd 10690 | . . . . . . 7 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ*) |
19 | 18 | biantrurd 535 | . . . . . 6 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (0 ≤ (𝐹‘𝑥) ↔ ((𝐹‘𝑥) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑥)))) |
20 | elxrge0 12844 | . . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (0[,]+∞) ↔ ((𝐹‘𝑥) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑥))) | |
21 | 19, 20 | syl6bbr 291 | . . . . 5 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (0 ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑥) ∈ (0[,]+∞))) |
22 | 21 | ralbidva 3196 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → (∀𝑥 ∈ ℝ 0 ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,]+∞))) |
23 | 5, 16, 22 | 3bitrd 307 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → (0𝑝 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,]+∞))) |
24 | 23 | biimpa 479 | . 2 ⊢ ((𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ 𝐹) → ∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,]+∞)) |
25 | ffnfv 6881 | . 2 ⊢ (𝐹:ℝ⟶(0[,]+∞) ↔ (𝐹 Fn ℝ ∧ ∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,]+∞))) | |
26 | 2, 24, 25 | sylanbrc 585 | 1 ⊢ ((𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ 𝐹) → 𝐹:ℝ⟶(0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ⊆ wss 3935 {csn 4566 class class class wbr 5065 × cxp 5552 Fn wfn 6349 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ∘r cofr 7407 ℂcc 10534 ℝcr 10535 0cc0 10536 +∞cpnf 10671 ℝ*cxr 10673 ≤ cle 10675 [,]cicc 12740 0𝑝c0p 24269 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-i2m1 10604 ax-rnegex 10607 ax-cnre 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-ofr 7409 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-icc 12744 df-0p 24270 |
This theorem is referenced by: itg2itg1 24336 |
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