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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 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 5958 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ) | |
2 | 1 | adantr 480 | . 2 ⊢ ((𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘𝑟 ≤ 𝐹) → 𝐹 Fn ℝ) |
3 | ax-resscn 9872 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝐹:ℝ⟶ℝ → ℝ ⊆ ℂ) |
5 | 4, 1 | 0pledm 23246 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → (0𝑝 ∘𝑟 ≤ 𝐹 ↔ (ℝ × {0}) ∘𝑟 ≤ 𝐹)) |
6 | 0re 9919 | . . . . . 6 ⊢ 0 ∈ ℝ | |
7 | fnconstg 6006 | . . . . . 6 ⊢ (0 ∈ ℝ → (ℝ × {0}) Fn ℝ) | |
8 | 6, 7 | mp1i 13 | . . . . 5 ⊢ (𝐹:ℝ⟶ℝ → (ℝ × {0}) Fn ℝ) |
9 | reex 9906 | . . . . . 6 ⊢ ℝ ∈ V | |
10 | 9 | a1i 11 | . . . . 5 ⊢ (𝐹:ℝ⟶ℝ → ℝ ∈ V) |
11 | inidm 3784 | . . . . 5 ⊢ (ℝ ∩ ℝ) = ℝ | |
12 | c0ex 9913 | . . . . . . 7 ⊢ 0 ∈ V | |
13 | 12 | fvconst2 6374 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → ((ℝ × {0})‘𝑥) = 0) |
14 | 13 | adantl 481 | . . . . 5 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → ((ℝ × {0})‘𝑥) = 0) |
15 | eqidd 2611 | . . . . 5 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
16 | 8, 1, 10, 10, 11, 14, 15 | ofrfval 6803 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → ((ℝ × {0}) ∘𝑟 ≤ 𝐹 ↔ ∀𝑥 ∈ ℝ 0 ≤ (𝐹‘𝑥))) |
17 | ffvelrn 6265 | . . . . . . . 8 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) | |
18 | 17 | rexrd 9968 | . . . . . . 7 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ*) |
19 | 18 | biantrurd 528 | . . . . . 6 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (0 ≤ (𝐹‘𝑥) ↔ ((𝐹‘𝑥) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑥)))) |
20 | elxrge0 12152 | . . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (0[,]+∞) ↔ ((𝐹‘𝑥) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑥))) | |
21 | 19, 20 | syl6bbr 277 | . . . . 5 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (0 ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑥) ∈ (0[,]+∞))) |
22 | 21 | ralbidva 2968 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → (∀𝑥 ∈ ℝ 0 ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,]+∞))) |
23 | 5, 16, 22 | 3bitrd 293 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → (0𝑝 ∘𝑟 ≤ 𝐹 ↔ ∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,]+∞))) |
24 | 23 | biimpa 500 | . 2 ⊢ ((𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘𝑟 ≤ 𝐹) → ∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,]+∞)) |
25 | ffnfv 6295 | . 2 ⊢ (𝐹:ℝ⟶(0[,]+∞) ↔ (𝐹 Fn ℝ ∧ ∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,]+∞))) | |
26 | 2, 24, 25 | sylanbrc 695 | 1 ⊢ ((𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘𝑟 ≤ 𝐹) → 𝐹:ℝ⟶(0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 {csn 4125 class class class wbr 4583 × cxp 5036 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘𝑟 cofr 6794 ℂcc 9813 ℝcr 9814 0cc0 9815 +∞cpnf 9950 ℝ*cxr 9952 ≤ cle 9954 [,]cicc 12049 0𝑝c0p 23242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-ofr 6796 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-icc 12053 df-0p 23243 |
This theorem is referenced by: itg2itg1 23309 |
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