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Mirrors > Home > MPE Home > Th. List > tmsxpsmopn | Structured version Visualization version GIF version |
Description: Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tmsxps.p | ⊢ 𝑃 = (dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) |
tmsxps.1 | ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋)) |
tmsxps.2 | ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌)) |
tmsxpsmopn.j | ⊢ 𝐽 = (MetOpen‘𝑀) |
tmsxpsmopn.k | ⊢ 𝐾 = (MetOpen‘𝑁) |
tmsxpsmopn.l | ⊢ 𝐿 = (MetOpen‘𝑃) |
Ref | Expression |
---|---|
tmsxpsmopn | ⊢ (𝜑 → 𝐿 = (𝐽 ×t 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tmsxps.1 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋)) | |
2 | eqid 2821 | . . . . . 6 ⊢ (toMetSp‘𝑀) = (toMetSp‘𝑀) | |
3 | 2 | tmsxms 23025 | . . . . 5 ⊢ (𝑀 ∈ (∞Met‘𝑋) → (toMetSp‘𝑀) ∈ ∞MetSp) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → (toMetSp‘𝑀) ∈ ∞MetSp) |
5 | xmstps 22992 | . . . 4 ⊢ ((toMetSp‘𝑀) ∈ ∞MetSp → (toMetSp‘𝑀) ∈ TopSp) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → (toMetSp‘𝑀) ∈ TopSp) |
7 | tmsxps.2 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌)) | |
8 | eqid 2821 | . . . . . 6 ⊢ (toMetSp‘𝑁) = (toMetSp‘𝑁) | |
9 | 8 | tmsxms 23025 | . . . . 5 ⊢ (𝑁 ∈ (∞Met‘𝑌) → (toMetSp‘𝑁) ∈ ∞MetSp) |
10 | 7, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (toMetSp‘𝑁) ∈ ∞MetSp) |
11 | xmstps 22992 | . . . 4 ⊢ ((toMetSp‘𝑁) ∈ ∞MetSp → (toMetSp‘𝑁) ∈ TopSp) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝜑 → (toMetSp‘𝑁) ∈ TopSp) |
13 | eqid 2821 | . . . 4 ⊢ ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) = ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) | |
14 | eqid 2821 | . . . 4 ⊢ (TopOpen‘(toMetSp‘𝑀)) = (TopOpen‘(toMetSp‘𝑀)) | |
15 | eqid 2821 | . . . 4 ⊢ (TopOpen‘(toMetSp‘𝑁)) = (TopOpen‘(toMetSp‘𝑁)) | |
16 | eqid 2821 | . . . 4 ⊢ (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) | |
17 | 13, 14, 15, 16 | xpstopn 22350 | . . 3 ⊢ (((toMetSp‘𝑀) ∈ TopSp ∧ (toMetSp‘𝑁) ∈ TopSp) → (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = ((TopOpen‘(toMetSp‘𝑀)) ×t (TopOpen‘(toMetSp‘𝑁)))) |
18 | 6, 12, 17 | syl2anc 584 | . 2 ⊢ (𝜑 → (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = ((TopOpen‘(toMetSp‘𝑀)) ×t (TopOpen‘(toMetSp‘𝑁)))) |
19 | tmsxpsmopn.l | . . 3 ⊢ 𝐿 = (MetOpen‘𝑃) | |
20 | 13 | xpsxms 23073 | . . . . . 6 ⊢ (((toMetSp‘𝑀) ∈ ∞MetSp ∧ (toMetSp‘𝑁) ∈ ∞MetSp) → ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp) |
21 | 4, 10, 20 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp) |
22 | eqid 2821 | . . . . . 6 ⊢ (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) | |
23 | tmsxps.p | . . . . . . 7 ⊢ 𝑃 = (dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) | |
24 | 23 | reseq1i 5843 | . . . . . 6 ⊢ (𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) = ((dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) |
25 | 16, 22, 24 | xmstopn 22990 | . . . . 5 ⊢ (((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp → (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = (MetOpen‘(𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))))) |
26 | 21, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = (MetOpen‘(𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))))) |
27 | eqid 2821 | . . . . . . 7 ⊢ (Base‘(toMetSp‘𝑀)) = (Base‘(toMetSp‘𝑀)) | |
28 | eqid 2821 | . . . . . . 7 ⊢ (Base‘(toMetSp‘𝑁)) = (Base‘(toMetSp‘𝑁)) | |
29 | 13, 27, 28, 4, 10, 23 | xpsdsfn2 22917 | . . . . . 6 ⊢ (𝜑 → 𝑃 Fn ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) |
30 | fnresdm 6460 | . . . . . 6 ⊢ (𝑃 Fn ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))) → (𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) = 𝑃) | |
31 | 29, 30 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) = 𝑃) |
32 | 31 | fveq2d 6668 | . . . 4 ⊢ (𝜑 → (MetOpen‘(𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))))) = (MetOpen‘𝑃)) |
33 | 26, 32 | eqtr2d 2857 | . . 3 ⊢ (𝜑 → (MetOpen‘𝑃) = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))) |
34 | 19, 33 | syl5eq 2868 | . 2 ⊢ (𝜑 → 𝐿 = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))) |
35 | tmsxpsmopn.j | . . . . 5 ⊢ 𝐽 = (MetOpen‘𝑀) | |
36 | 2, 35 | tmstopn 23024 | . . . 4 ⊢ (𝑀 ∈ (∞Met‘𝑋) → 𝐽 = (TopOpen‘(toMetSp‘𝑀))) |
37 | 1, 36 | syl 17 | . . 3 ⊢ (𝜑 → 𝐽 = (TopOpen‘(toMetSp‘𝑀))) |
38 | tmsxpsmopn.k | . . . . 5 ⊢ 𝐾 = (MetOpen‘𝑁) | |
39 | 8, 38 | tmstopn 23024 | . . . 4 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝐾 = (TopOpen‘(toMetSp‘𝑁))) |
40 | 7, 39 | syl 17 | . . 3 ⊢ (𝜑 → 𝐾 = (TopOpen‘(toMetSp‘𝑁))) |
41 | 37, 40 | oveq12d 7163 | . 2 ⊢ (𝜑 → (𝐽 ×t 𝐾) = ((TopOpen‘(toMetSp‘𝑀)) ×t (TopOpen‘(toMetSp‘𝑁)))) |
42 | 18, 34, 41 | 3eqtr4d 2866 | 1 ⊢ (𝜑 → 𝐿 = (𝐽 ×t 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 × cxp 5547 ↾ cres 5551 Fn wfn 6344 ‘cfv 6349 (class class class)co 7145 Basecbs 16473 distcds 16564 TopOpenctopn 16685 ×s cxps 16769 ∞Metcxmet 20460 MetOpencmopn 20465 TopSpctps 21470 ×t ctx 22098 ∞MetSpcxms 22856 toMetSpctms 22858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7569 df-1st 7680 df-2nd 7681 df-supp 7822 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-2o 8094 df-oadd 8097 df-er 8279 df-map 8398 df-ixp 8451 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-fsupp 8823 df-fi 8864 df-sup 8895 df-inf 8896 df-oi 8963 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-z 11971 df-dec 12088 df-uz 12233 df-q 12338 df-rp 12380 df-xneg 12497 df-xadd 12498 df-xmul 12499 df-icc 12735 df-fz 12883 df-fzo 13024 df-seq 13360 df-hash 13681 df-struct 16475 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-ress 16481 df-plusg 16568 df-mulr 16569 df-sca 16571 df-vsca 16572 df-ip 16573 df-tset 16574 df-ple 16575 df-ds 16577 df-hom 16579 df-cco 16580 df-rest 16686 df-topn 16687 df-0g 16705 df-gsum 16706 df-topgen 16707 df-pt 16708 df-prds 16711 df-xrs 16765 df-qtop 16770 df-imas 16771 df-xps 16773 df-mre 16847 df-mrc 16848 df-acs 16850 df-mgm 17842 df-sgrp 17891 df-mnd 17902 df-submnd 17947 df-mulg 18165 df-cntz 18387 df-cmn 18839 df-psmet 20467 df-xmet 20468 df-bl 20470 df-mopn 20471 df-top 21432 df-topon 21449 df-topsp 21471 df-bases 21484 df-cn 21765 df-cnp 21766 df-tx 22100 df-hmeo 22293 df-xms 22859 df-tms 22861 |
This theorem is referenced by: txmetcnp 23086 |
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