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| 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 13497 | . . 3 ⊢ (0[,)+∞) ⊆ ℝ | |
| 2 | fss 6751 | . . 3 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐹:ℝ⟶ℝ) | |
| 3 | 1, 2 | mpan2 691 | . 2 ⊢ (𝐹:ℝ⟶(0[,)+∞) → 𝐹:ℝ⟶ℝ) | 
| 4 | ffvelcdm 7100 | . . . . 5 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) | |
| 5 | elrege0 13495 | . . . . . 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 3175 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → (∀𝑥 ∈ ℝ (𝐹‘𝑥) ∈ (0[,)+∞) ↔ ∀𝑥 ∈ ℝ 0 ≤ (𝐹‘𝑥))) | 
| 9 | ffn 6735 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ) | |
| 10 | ffnfv 7138 | . . . . 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 11254 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
| 14 | fnconstg 6795 | . . . . . . 7 ⊢ (0 ∈ ℂ → (ℂ × {0}) Fn ℂ) | |
| 15 | 13, 14 | ax-mp 5 | . . . . . 6 ⊢ (ℂ × {0}) Fn ℂ | 
| 16 | df-0p 25706 | . . . . . . 7 ⊢ 0𝑝 = (ℂ × {0}) | |
| 17 | 16 | fneq1i 6664 | . . . . . 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 11237 | . . . . 5 ⊢ ℂ ∈ V | |
| 21 | 20 | a1i 11 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → ℂ ∈ V) | 
| 22 | reex 11247 | . . . . 5 ⊢ ℝ ∈ V | |
| 23 | 22 | a1i 11 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → ℝ ∈ V) | 
| 24 | ax-resscn 11213 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
| 25 | sseqin2 4222 | . . . . 5 ⊢ (ℝ ⊆ ℂ ↔ (ℂ ∩ ℝ) = ℝ) | |
| 26 | 24, 25 | mpbi 230 | . . . 4 ⊢ (ℂ ∩ ℝ) = ℝ | 
| 27 | 0pval 25707 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (0𝑝‘𝑥) = 0) | |
| 28 | 27 | adantl 481 | . . . 4 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℂ) → (0𝑝‘𝑥) = 0) | 
| 29 | eqidd 2737 | . . . 4 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
| 30 | 19, 9, 21, 23, 26, 28, 29 | ofrfval 7708 | . . 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 821 | 1 ⊢ (𝐹:ℝ⟶(0[,)+∞) ↔ (𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ 𝐹)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 Vcvv 3479 ∩ cin 3949 ⊆ wss 3950 {csn 4625 class class class wbr 5142 × cxp 5682 Fn wfn 6555 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ∘r cofr 7697 ℂcc 11154 ℝcr 11155 0cc0 11156 +∞cpnf 11293 ≤ cle 11297 [,)cico 13390 0𝑝c0p 25705 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-i2m1 11224 ax-rnegex 11227 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-ofr 7699 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-ico 13394 df-0p 25706 | 
| This theorem is referenced by: itg2i1fseq 25791 itg2addlem 25794 ftc1anclem8 37708 | 
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