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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 6736 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ) | |
| 2 | 1 | adantr 480 | . 2 ⊢ ((𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ 𝐹) → 𝐹 Fn ℝ) |
| 3 | ax-resscn 11212 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
| 4 | 3 | a1i 11 | . . . . 5 ⊢ (𝐹:ℝ⟶ℝ → ℝ ⊆ ℂ) |
| 5 | 4, 1 | 0pledm 25708 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → (0𝑝 ∘r ≤ 𝐹 ↔ (ℝ × {0}) ∘r ≤ 𝐹)) |
| 6 | 0re 11263 | . . . . . 6 ⊢ 0 ∈ ℝ | |
| 7 | fnconstg 6796 | . . . . . 6 ⊢ (0 ∈ ℝ → (ℝ × {0}) Fn ℝ) | |
| 8 | 6, 7 | mp1i 13 | . . . . 5 ⊢ (𝐹:ℝ⟶ℝ → (ℝ × {0}) Fn ℝ) |
| 9 | reex 11246 | . . . . . 6 ⊢ ℝ ∈ V | |
| 10 | 9 | a1i 11 | . . . . 5 ⊢ (𝐹:ℝ⟶ℝ → ℝ ∈ V) |
| 11 | inidm 4227 | . . . . 5 ⊢ (ℝ ∩ ℝ) = ℝ | |
| 12 | c0ex 11255 | . . . . . . 7 ⊢ 0 ∈ V | |
| 13 | 12 | fvconst2 7224 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → ((ℝ × {0})‘𝑥) = 0) |
| 14 | 13 | adantl 481 | . . . . 5 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → ((ℝ × {0})‘𝑥) = 0) |
| 15 | eqidd 2738 | . . . . 5 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
| 16 | 8, 1, 10, 10, 11, 14, 15 | ofrfval 7707 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → ((ℝ × {0}) ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ ℝ 0 ≤ (𝐹‘𝑥))) |
| 17 | ffvelcdm 7101 | . . . . . . . 8 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) | |
| 18 | 17 | rexrd 11311 | . . . . . . 7 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ*) |
| 19 | 18 | biantrurd 532 | . . . . . 6 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (0 ≤ (𝐹‘𝑥) ↔ ((𝐹‘𝑥) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑥)))) |
| 20 | elxrge0 13497 | . . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (0[,]+∞) ↔ ((𝐹‘𝑥) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑥))) | |
| 21 | 19, 20 | bitr4di 289 | . . . . 5 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (0 ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑥) ∈ (0[,]+∞))) |
| 22 | 21 | ralbidva 3176 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → (∀𝑥 ∈ ℝ 0 ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,]+∞))) |
| 23 | 5, 16, 22 | 3bitrd 305 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → (0𝑝 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,]+∞))) |
| 24 | 23 | biimpa 476 | . 2 ⊢ ((𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ 𝐹) → ∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,]+∞)) |
| 25 | ffnfv 7139 | . 2 ⊢ (𝐹:ℝ⟶(0[,]+∞) ↔ (𝐹 Fn ℝ ∧ ∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,]+∞))) | |
| 26 | 2, 24, 25 | sylanbrc 583 | 1 ⊢ ((𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ 𝐹) → 𝐹:ℝ⟶(0[,]+∞)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 {csn 4626 class class class wbr 5143 × cxp 5683 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘r cofr 7696 ℂcc 11153 ℝcr 11154 0cc0 11155 +∞cpnf 11292 ℝ*cxr 11294 ≤ cle 11296 [,]cicc 13390 0𝑝c0p 25704 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-i2m1 11223 ax-rnegex 11226 ax-cnre 11228 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-ofr 7698 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-icc 13394 df-0p 25705 |
| This theorem is referenced by: itg2itg1 25771 |
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