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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 2778 | . . . . . 6 ⊢ (toMetSp‘𝑀) = (toMetSp‘𝑀) | |
3 | 2 | tmsxms 22699 | . . . . 5 ⊢ (𝑀 ∈ (∞Met‘𝑋) → (toMetSp‘𝑀) ∈ ∞MetSp) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → (toMetSp‘𝑀) ∈ ∞MetSp) |
5 | xmstps 22666 | . . . 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 2778 | . . . . . 6 ⊢ (toMetSp‘𝑁) = (toMetSp‘𝑁) | |
9 | 8 | tmsxms 22699 | . . . . 5 ⊢ (𝑁 ∈ (∞Met‘𝑌) → (toMetSp‘𝑁) ∈ ∞MetSp) |
10 | 7, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (toMetSp‘𝑁) ∈ ∞MetSp) |
11 | xmstps 22666 | . . . 4 ⊢ ((toMetSp‘𝑁) ∈ ∞MetSp → (toMetSp‘𝑁) ∈ TopSp) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝜑 → (toMetSp‘𝑁) ∈ TopSp) |
13 | eqid 2778 | . . . 4 ⊢ ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) = ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) | |
14 | eqid 2778 | . . . 4 ⊢ (TopOpen‘(toMetSp‘𝑀)) = (TopOpen‘(toMetSp‘𝑀)) | |
15 | eqid 2778 | . . . 4 ⊢ (TopOpen‘(toMetSp‘𝑁)) = (TopOpen‘(toMetSp‘𝑁)) | |
16 | eqid 2778 | . . . 4 ⊢ (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) | |
17 | 13, 14, 15, 16 | xpstopn 22024 | . . 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 579 | . 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 22747 | . . . . . 6 ⊢ (((toMetSp‘𝑀) ∈ ∞MetSp ∧ (toMetSp‘𝑁) ∈ ∞MetSp) → ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp) |
21 | 4, 10, 20 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp) |
22 | eqid 2778 | . . . . . 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 5638 | . . . . . 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 22664 | . . . . 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 2778 | . . . . . . 7 ⊢ (Base‘(toMetSp‘𝑀)) = (Base‘(toMetSp‘𝑀)) | |
28 | eqid 2778 | . . . . . . 7 ⊢ (Base‘(toMetSp‘𝑁)) = (Base‘(toMetSp‘𝑁)) | |
29 | 13, 27, 28, 4, 10, 23 | xpsdsfn2 22591 | . . . . . 6 ⊢ (𝜑 → 𝑃 Fn ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) |
30 | fnresdm 6246 | . . . . . 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 6450 | . . . 4 ⊢ (𝜑 → (MetOpen‘(𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))))) = (MetOpen‘𝑃)) |
33 | 26, 32 | eqtr2d 2815 | . . 3 ⊢ (𝜑 → (MetOpen‘𝑃) = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))) |
34 | 19, 33 | syl5eq 2826 | . 2 ⊢ (𝜑 → 𝐿 = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))) |
35 | tmsxpsmopn.j | . . . . 5 ⊢ 𝐽 = (MetOpen‘𝑀) | |
36 | 2, 35 | tmstopn 22698 | . . . 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 22698 | . . . 4 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝐾 = (TopOpen‘(toMetSp‘𝑁))) |
40 | 7, 39 | syl 17 | . . 3 ⊢ (𝜑 → 𝐾 = (TopOpen‘(toMetSp‘𝑁))) |
41 | 37, 40 | oveq12d 6940 | . 2 ⊢ (𝜑 → (𝐽 ×t 𝐾) = ((TopOpen‘(toMetSp‘𝑀)) ×t (TopOpen‘(toMetSp‘𝑁)))) |
42 | 18, 34, 41 | 3eqtr4d 2824 | 1 ⊢ (𝜑 → 𝐿 = (𝐽 ×t 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 × cxp 5353 ↾ cres 5357 Fn wfn 6130 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 distcds 16347 TopOpenctopn 16468 ×s cxps 16552 ∞Metcxmet 20127 MetOpencmopn 20132 TopSpctps 21144 ×t ctx 21772 ∞MetSpcxms 22530 toMetSpctms 22532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-icc 12494 df-fz 12644 df-fzo 12785 df-seq 13120 df-hash 13436 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-bl 20137 df-mopn 20138 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cn 21439 df-cnp 21440 df-tx 21774 df-hmeo 21967 df-xms 22533 df-tms 22535 |
This theorem is referenced by: txmetcnp 22760 |
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