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Mirrors > Home > MPE Home > Th. List > ovolfsf | Structured version Visualization version GIF version |
Description: Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.) |
Ref | Expression |
---|---|
ovolfs.1 | ⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹) |
Ref | Expression |
---|---|
ovolfsf | ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶(0[,)+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absf 13925 | . . . . . 6 ⊢ abs:ℂ⟶ℝ | |
2 | subf 10162 | . . . . . 6 ⊢ − :(ℂ × ℂ)⟶ℂ | |
3 | fco 5971 | . . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ) | |
4 | 1, 2, 3 | mp2an 704 | . . . . 5 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ |
5 | inss2 3796 | . . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ) | |
6 | ax-resscn 9872 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
7 | xpss12 5148 | . . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ) ⊆ (ℂ × ℂ)) | |
8 | 6, 6, 7 | mp2an 704 | . . . . . . 7 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ) |
9 | 5, 8 | sstri 3577 | . . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℂ × ℂ) |
10 | fss 5969 | . . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℂ × ℂ)) → 𝐹:ℕ⟶(ℂ × ℂ)) | |
11 | 9, 10 | mpan2 703 | . . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℂ × ℂ)) |
12 | fco 5971 | . . . . 5 ⊢ (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ 𝐹:ℕ⟶(ℂ × ℂ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶ℝ) | |
13 | 4, 11, 12 | sylancr 694 | . . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶ℝ) |
14 | ovolfs.1 | . . . . 5 ⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹) | |
15 | 14 | feq1i 5949 | . . . 4 ⊢ (𝐺:ℕ⟶ℝ ↔ ((abs ∘ − ) ∘ 𝐹):ℕ⟶ℝ) |
16 | 13, 15 | sylibr 223 | . . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶ℝ) |
17 | ffn 5958 | . . 3 ⊢ (𝐺:ℕ⟶ℝ → 𝐺 Fn ℕ) | |
18 | 16, 17 | syl 17 | . 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺 Fn ℕ) |
19 | 16 | ffvelrnda 6267 | . . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ ℝ) |
20 | ovolfcl 23042 | . . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑥)) ≤ (2nd ‘(𝐹‘𝑥)))) | |
21 | subge0 10420 | . . . . . . . 8 ⊢ (((2nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑥)) ∈ ℝ) → (0 ≤ ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥))) ↔ (1st ‘(𝐹‘𝑥)) ≤ (2nd ‘(𝐹‘𝑥)))) | |
22 | 21 | ancoms 468 | . . . . . . 7 ⊢ (((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ) → (0 ≤ ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥))) ↔ (1st ‘(𝐹‘𝑥)) ≤ (2nd ‘(𝐹‘𝑥)))) |
23 | 22 | biimp3ar 1425 | . . . . . 6 ⊢ (((1st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑥)) ≤ (2nd ‘(𝐹‘𝑥))) → 0 ≤ ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥)))) |
24 | 20, 23 | syl 17 | . . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → 0 ≤ ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥)))) |
25 | 14 | ovolfsval 23046 | . . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) = ((2nd ‘(𝐹‘𝑥)) − (1st ‘(𝐹‘𝑥)))) |
26 | 24, 25 | breqtrrd 4611 | . . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → 0 ≤ (𝐺‘𝑥)) |
27 | elrege0 12149 | . . . 4 ⊢ ((𝐺‘𝑥) ∈ (0[,)+∞) ↔ ((𝐺‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐺‘𝑥))) | |
28 | 19, 26, 27 | sylanbrc 695 | . . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ (0[,)+∞)) |
29 | 28 | ralrimiva 2949 | . 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑥 ∈ ℕ (𝐺‘𝑥) ∈ (0[,)+∞)) |
30 | ffnfv 6295 | . 2 ⊢ (𝐺:ℕ⟶(0[,)+∞) ↔ (𝐺 Fn ℕ ∧ ∀𝑥 ∈ ℕ (𝐺‘𝑥) ∈ (0[,)+∞))) | |
31 | 18, 29, 30 | sylanbrc 695 | 1 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶(0[,)+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∩ cin 3539 ⊆ wss 3540 class class class wbr 4583 × cxp 5036 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 ℂcc 9813 ℝcr 9814 0cc0 9815 +∞cpnf 9950 ≤ cle 9954 − cmin 10145 ℕcn 10897 [,)cico 12048 abscabs 13822 |
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-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-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
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-rmo 2904 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-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 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-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 |
This theorem is referenced by: ovolsf 23048 ovollb2lem 23063 ovolunlem1a 23071 ovoliunlem1 23077 ovolshftlem1 23084 ovolicc2lem4 23095 ioombl1lem4 23136 ovolfs2 23145 uniioombllem2 23157 uniioombllem6 23162 |
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