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Mirrors > Home > MPE Home > Th. List > xmetdcn2 | Structured version Visualization version GIF version |
Description: The metric function of an extended metric space is always continuous in the topology generated by it. In this variation of xmetdcn 22449 we use the metric topology instead of the order topology on ℝ*, which makes the theorem a bit stronger. Since +∞ is an isolated point in the metric topology, this is saying that for any points 𝐴, 𝐵 which are an infinite distance apart, there is a product neighborhood around 〈𝐴, 𝐵〉 such that 𝑑(𝑎, 𝑏) = +∞ for any 𝑎 near 𝐴 and 𝑏 near 𝐵, i.e. the distance function is locally constant +∞. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 4-Sep-2015.) |
Ref | Expression |
---|---|
xmetdcn2.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
xmetdcn2.2 | ⊢ 𝐶 = (dist‘ℝ*𝑠) |
xmetdcn2.3 | ⊢ 𝐾 = (MetOpen‘𝐶) |
Ref | Expression |
---|---|
xmetdcn2 | ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmetf 21944 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
2 | rphalfcl 11734 | . . . . . 6 ⊢ (𝑟 ∈ ℝ+ → (𝑟 / 2) ∈ ℝ+) | |
3 | 2 | adantl 481 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑟 ∈ ℝ+) → (𝑟 / 2) ∈ ℝ+) |
4 | xmetdcn2.1 | . . . . . . . 8 ⊢ 𝐽 = (MetOpen‘𝐷) | |
5 | xmetdcn2.2 | . . . . . . . 8 ⊢ 𝐶 = (dist‘ℝ*𝑠) | |
6 | xmetdcn2.3 | . . . . . . . 8 ⊢ 𝐾 = (MetOpen‘𝐶) | |
7 | simp-4l 802 | . . . . . . . 8 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ ((𝑥𝐷𝑧) < (𝑟 / 2) ∧ (𝑦𝐷𝑤) < (𝑟 / 2))) → 𝐷 ∈ (∞Met‘𝑋)) | |
8 | simplrl 796 | . . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑟 ∈ ℝ+) → 𝑥 ∈ 𝑋) | |
9 | 8 | ad2antrr 758 | . . . . . . . 8 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ ((𝑥𝐷𝑧) < (𝑟 / 2) ∧ (𝑦𝐷𝑤) < (𝑟 / 2))) → 𝑥 ∈ 𝑋) |
10 | simplrr 797 | . . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑟 ∈ ℝ+) → 𝑦 ∈ 𝑋) | |
11 | 10 | ad2antrr 758 | . . . . . . . 8 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ ((𝑥𝐷𝑧) < (𝑟 / 2) ∧ (𝑦𝐷𝑤) < (𝑟 / 2))) → 𝑦 ∈ 𝑋) |
12 | simpllr 795 | . . . . . . . 8 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ ((𝑥𝐷𝑧) < (𝑟 / 2) ∧ (𝑦𝐷𝑤) < (𝑟 / 2))) → 𝑟 ∈ ℝ+) | |
13 | simplrl 796 | . . . . . . . 8 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ ((𝑥𝐷𝑧) < (𝑟 / 2) ∧ (𝑦𝐷𝑤) < (𝑟 / 2))) → 𝑧 ∈ 𝑋) | |
14 | simplrr 797 | . . . . . . . 8 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ ((𝑥𝐷𝑧) < (𝑟 / 2) ∧ (𝑦𝐷𝑤) < (𝑟 / 2))) → 𝑤 ∈ 𝑋) | |
15 | simprl 790 | . . . . . . . 8 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ ((𝑥𝐷𝑧) < (𝑟 / 2) ∧ (𝑦𝐷𝑤) < (𝑟 / 2))) → (𝑥𝐷𝑧) < (𝑟 / 2)) | |
16 | simprr 792 | . . . . . . . 8 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ ((𝑥𝐷𝑧) < (𝑟 / 2) ∧ (𝑦𝐷𝑤) < (𝑟 / 2))) → (𝑦𝐷𝑤) < (𝑟 / 2)) | |
17 | 4, 5, 6, 7, 9, 11, 12, 13, 14, 15, 16 | metdcnlem 22447 | . . . . . . 7 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ ((𝑥𝐷𝑧) < (𝑟 / 2) ∧ (𝑦𝐷𝑤) < (𝑟 / 2))) → ((𝑥𝐷𝑦)𝐶(𝑧𝐷𝑤)) < 𝑟) |
18 | 17 | ex 449 | . . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (((𝑥𝐷𝑧) < (𝑟 / 2) ∧ (𝑦𝐷𝑤) < (𝑟 / 2)) → ((𝑥𝐷𝑦)𝐶(𝑧𝐷𝑤)) < 𝑟)) |
19 | 18 | ralrimivva 2954 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑟 ∈ ℝ+) → ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (((𝑥𝐷𝑧) < (𝑟 / 2) ∧ (𝑦𝐷𝑤) < (𝑟 / 2)) → ((𝑥𝐷𝑦)𝐶(𝑧𝐷𝑤)) < 𝑟)) |
20 | breq2 4587 | . . . . . . . . 9 ⊢ (𝑠 = (𝑟 / 2) → ((𝑥𝐷𝑧) < 𝑠 ↔ (𝑥𝐷𝑧) < (𝑟 / 2))) | |
21 | breq2 4587 | . . . . . . . . 9 ⊢ (𝑠 = (𝑟 / 2) → ((𝑦𝐷𝑤) < 𝑠 ↔ (𝑦𝐷𝑤) < (𝑟 / 2))) | |
22 | 20, 21 | anbi12d 743 | . . . . . . . 8 ⊢ (𝑠 = (𝑟 / 2) → (((𝑥𝐷𝑧) < 𝑠 ∧ (𝑦𝐷𝑤) < 𝑠) ↔ ((𝑥𝐷𝑧) < (𝑟 / 2) ∧ (𝑦𝐷𝑤) < (𝑟 / 2)))) |
23 | 22 | imbi1d 330 | . . . . . . 7 ⊢ (𝑠 = (𝑟 / 2) → ((((𝑥𝐷𝑧) < 𝑠 ∧ (𝑦𝐷𝑤) < 𝑠) → ((𝑥𝐷𝑦)𝐶(𝑧𝐷𝑤)) < 𝑟) ↔ (((𝑥𝐷𝑧) < (𝑟 / 2) ∧ (𝑦𝐷𝑤) < (𝑟 / 2)) → ((𝑥𝐷𝑦)𝐶(𝑧𝐷𝑤)) < 𝑟))) |
24 | 23 | 2ralbidv 2972 | . . . . . 6 ⊢ (𝑠 = (𝑟 / 2) → (∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (((𝑥𝐷𝑧) < 𝑠 ∧ (𝑦𝐷𝑤) < 𝑠) → ((𝑥𝐷𝑦)𝐶(𝑧𝐷𝑤)) < 𝑟) ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (((𝑥𝐷𝑧) < (𝑟 / 2) ∧ (𝑦𝐷𝑤) < (𝑟 / 2)) → ((𝑥𝐷𝑦)𝐶(𝑧𝐷𝑤)) < 𝑟))) |
25 | 24 | rspcev 3282 | . . . . 5 ⊢ (((𝑟 / 2) ∈ ℝ+ ∧ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (((𝑥𝐷𝑧) < (𝑟 / 2) ∧ (𝑦𝐷𝑤) < (𝑟 / 2)) → ((𝑥𝐷𝑦)𝐶(𝑧𝐷𝑤)) < 𝑟)) → ∃𝑠 ∈ ℝ+ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (((𝑥𝐷𝑧) < 𝑠 ∧ (𝑦𝐷𝑤) < 𝑠) → ((𝑥𝐷𝑦)𝐶(𝑧𝐷𝑤)) < 𝑟)) |
26 | 3, 19, 25 | syl2anc 691 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑟 ∈ ℝ+) → ∃𝑠 ∈ ℝ+ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (((𝑥𝐷𝑧) < 𝑠 ∧ (𝑦𝐷𝑤) < 𝑠) → ((𝑥𝐷𝑦)𝐶(𝑧𝐷𝑤)) < 𝑟)) |
27 | 26 | ralrimiva 2949 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (((𝑥𝐷𝑧) < 𝑠 ∧ (𝑦𝐷𝑤) < 𝑠) → ((𝑥𝐷𝑦)𝐶(𝑧𝐷𝑤)) < 𝑟)) |
28 | 27 | ralrimivva 2954 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (((𝑥𝐷𝑧) < 𝑠 ∧ (𝑦𝐷𝑤) < 𝑠) → ((𝑥𝐷𝑦)𝐶(𝑧𝐷𝑤)) < 𝑟)) |
29 | id 22 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | |
30 | 5 | xrsxmet 22420 | . . . 4 ⊢ 𝐶 ∈ (∞Met‘ℝ*) |
31 | 30 | a1i 11 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐶 ∈ (∞Met‘ℝ*)) |
32 | 4, 4, 6 | txmetcn 22163 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋) ∧ 𝐶 ∈ (∞Met‘ℝ*)) → (𝐷 ∈ ((𝐽 ×t 𝐽) Cn 𝐾) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (((𝑥𝐷𝑧) < 𝑠 ∧ (𝑦𝐷𝑤) < 𝑠) → ((𝑥𝐷𝑦)𝐶(𝑧𝐷𝑤)) < 𝑟)))) |
33 | 29, 31, 32 | mpd3an23 1418 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ∈ ((𝐽 ×t 𝐽) Cn 𝐾) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (((𝑥𝐷𝑧) < 𝑠 ∧ (𝑦𝐷𝑤) < 𝑠) → ((𝑥𝐷𝑦)𝐶(𝑧𝐷𝑤)) < 𝑟)))) |
34 | 1, 28, 33 | mpbir2and 959 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 class class class wbr 4583 × cxp 5036 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℝ*cxr 9952 < clt 9953 / cdiv 10563 2c2 10947 ℝ+crp 11708 distcds 15777 ℝ*𝑠cxrs 15983 ∞Metcxmt 19552 MetOpencmopn 19557 Cn ccn 20838 ×t ctx 21173 |
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-inf2 8421 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-int 4411 df-iun 4457 df-iin 4458 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-se 4998 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-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 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-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cn 20841 df-cnp 20842 df-tx 21175 df-hmeo 21368 df-xms 21935 df-tms 21937 |
This theorem is referenced by: xmetdcn 22449 |
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