Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6516 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ) | |
2 | 1 | adantr 483 | . 2 ⊢ ((𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ 𝐹) → 𝐹 Fn ℝ) |
3 | ax-resscn 10596 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝐹:ℝ⟶ℝ → ℝ ⊆ ℂ) |
5 | 4, 1 | 0pledm 24276 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → (0𝑝 ∘r ≤ 𝐹 ↔ (ℝ × {0}) ∘r ≤ 𝐹)) |
6 | 0re 10645 | . . . . . 6 ⊢ 0 ∈ ℝ | |
7 | fnconstg 6569 | . . . . . 6 ⊢ (0 ∈ ℝ → (ℝ × {0}) Fn ℝ) | |
8 | 6, 7 | mp1i 13 | . . . . 5 ⊢ (𝐹:ℝ⟶ℝ → (ℝ × {0}) Fn ℝ) |
9 | reex 10630 | . . . . . 6 ⊢ ℝ ∈ V | |
10 | 9 | a1i 11 | . . . . 5 ⊢ (𝐹:ℝ⟶ℝ → ℝ ∈ V) |
11 | inidm 4197 | . . . . 5 ⊢ (ℝ ∩ ℝ) = ℝ | |
12 | c0ex 10637 | . . . . . . 7 ⊢ 0 ∈ V | |
13 | 12 | fvconst2 6968 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → ((ℝ × {0})‘𝑥) = 0) |
14 | 13 | adantl 484 | . . . . 5 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → ((ℝ × {0})‘𝑥) = 0) |
15 | eqidd 2824 | . . . . 5 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
16 | 8, 1, 10, 10, 11, 14, 15 | ofrfval 7419 | . . . 4 ⊢ (𝐹:ℝ⟶ℝ → ((ℝ × {0}) ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ ℝ 0 ≤ (𝐹‘𝑥))) |
17 | ffvelrn 6851 | . . . . . . . 8 ⊢ ((𝐹:ℝ⟶ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) | |
18 | 17 | rexrd 10693 | . . . . . . 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 12848 | . . . . . 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 3198 | . . . 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 6884 | . 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 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ⊆ wss 3938 {csn 4569 class class class wbr 5068 × cxp 5555 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∘r cofr 7410 ℂcc 10537 ℝcr 10538 0cc0 10539 +∞cpnf 10674 ℝ*cxr 10676 ≤ cle 10678 [,]cicc 12744 0𝑝c0p 24272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-i2m1 10607 ax-rnegex 10610 ax-cnre 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-ofr 7412 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-icc 12748 df-0p 24273 |
This theorem is referenced by: itg2itg1 24339 |
Copyright terms: Public domain | W3C validator |