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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 12151 | . . 3 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | fss 5969 | . . 3 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐹:ℝ⟶ℝ) | |
3 | 1, 2 | mpan2 703 | . 2 ⊢ (𝐹:ℝ⟶(0[,)+∞) → 𝐹:ℝ⟶ℝ) |
4 | ffvelrn 6265 | . . . . 5 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) | |
5 | elrege0 12149 | . . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥))) | |
6 | 5 | baib 942 | . . . . 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 2968 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → (∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,)+∞) ↔ ∀𝑥 ∈ ℝ 0 ≤ (𝐹‘𝑥))) |
9 | ffn 5958 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ) | |
10 | ffnfv 6295 | . . . . 5 ⊢ (𝐹:ℝ⟶(0[,)+∞) ↔ (𝐹 Fn ℝ ∧ ∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,)+∞))) | |
11 | 10 | baib 942 | . . . 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 9911 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
14 | fnconstg 6006 | . . . . . . 7 ⊢ (0 ∈ ℂ → (ℂ × {0}) Fn ℂ) | |
15 | 13, 14 | ax-mp 5 | . . . . . 6 ⊢ (ℂ × {0}) Fn ℂ |
16 | df-0p 23243 | . . . . . . 7 ⊢ 0𝑝 = (ℂ × {0}) | |
17 | 16 | fneq1i 5899 | . . . . . 6 ⊢ (0𝑝 Fn ℂ ↔ (ℂ × {0}) Fn ℂ) |
18 | 15, 17 | mpbir 220 | . . . . 5 ⊢ 0𝑝 Fn ℂ |
19 | 18 | a1i 11 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → 0𝑝 Fn ℂ) |
20 | cnex 9896 | . . . . 5 ⊢ ℂ ∈ V | |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → ℂ ∈ V) |
22 | reex 9906 | . . . . 5 ⊢ ℝ ∈ V | |
23 | 22 | a1i 11 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → ℝ ∈ V) |
24 | ax-resscn 9872 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
25 | sseqin2 3779 | . . . . 5 ⊢ (ℝ ⊆ ℂ ↔ (ℂ ∩ ℝ) = ℝ) | |
26 | 24, 25 | mpbi 219 | . . . 4 ⊢ (ℂ ∩ ℝ) = ℝ |
27 | 0pval 23244 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (0𝑝‘𝑥) = 0) | |
28 | 27 | adantl 481 | . . . 4 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℂ) → (0𝑝‘𝑥) = 0) |
29 | eqidd 2611 | . . . 4 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
30 | 19, 9, 21, 23, 26, 28, 29 | ofrfval 6803 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → (0𝑝 ∘𝑟 ≤ 𝐹 ↔ ∀𝑥 ∈ ℝ 0 ≤ (𝐹‘𝑥))) |
31 | 8, 12, 30 | 3bitr4d 299 | . 2 ⊢ (𝐹:ℝ⟶ℝ → (𝐹:ℝ⟶(0[,)+∞) ↔ 0𝑝 ∘𝑟 ≤ 𝐹)) |
32 | 3, 31 | biadan2 672 | 1 ⊢ (𝐹:ℝ⟶(0[,)+∞) ↔ (𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘𝑟 ≤ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∩ cin 3539 ⊆ 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 ≤ cle 9954 [,)cico 12048 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 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-po 4959 df-so 4960 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-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-ico 12052 df-0p 23243 |
This theorem is referenced by: itg2i1fseq 23328 itg2addlem 23331 ftc1anclem8 32662 |
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