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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 2740 | . . . . . 6 ⊢ (toMetSp‘𝑀) = (toMetSp‘𝑀) | |
3 | 2 | tmsxms 23638 | . . . . 5 ⊢ (𝑀 ∈ (∞Met‘𝑋) → (toMetSp‘𝑀) ∈ ∞MetSp) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → (toMetSp‘𝑀) ∈ ∞MetSp) |
5 | xmstps 23602 | . . . 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 2740 | . . . . . 6 ⊢ (toMetSp‘𝑁) = (toMetSp‘𝑁) | |
9 | 8 | tmsxms 23638 | . . . . 5 ⊢ (𝑁 ∈ (∞Met‘𝑌) → (toMetSp‘𝑁) ∈ ∞MetSp) |
10 | 7, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (toMetSp‘𝑁) ∈ ∞MetSp) |
11 | xmstps 23602 | . . . 4 ⊢ ((toMetSp‘𝑁) ∈ ∞MetSp → (toMetSp‘𝑁) ∈ TopSp) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝜑 → (toMetSp‘𝑁) ∈ TopSp) |
13 | eqid 2740 | . . . 4 ⊢ ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) = ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) | |
14 | eqid 2740 | . . . 4 ⊢ (TopOpen‘(toMetSp‘𝑀)) = (TopOpen‘(toMetSp‘𝑀)) | |
15 | eqid 2740 | . . . 4 ⊢ (TopOpen‘(toMetSp‘𝑁)) = (TopOpen‘(toMetSp‘𝑁)) | |
16 | eqid 2740 | . . . 4 ⊢ (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) | |
17 | 13, 14, 15, 16 | xpstopn 22959 | . . 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 23686 | . . . . . 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 2740 | . . . . . 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 5885 | . . . . . 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 23600 | . . . . 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 2740 | . . . . . . 7 ⊢ (Base‘(toMetSp‘𝑀)) = (Base‘(toMetSp‘𝑀)) | |
28 | eqid 2740 | . . . . . . 7 ⊢ (Base‘(toMetSp‘𝑁)) = (Base‘(toMetSp‘𝑁)) | |
29 | 13, 27, 28, 4, 10, 23 | xpsdsfn2 23527 | . . . . . 6 ⊢ (𝜑 → 𝑃 Fn ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) |
30 | fnresdm 6548 | . . . . . 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 6773 | . . . 4 ⊢ (𝜑 → (MetOpen‘(𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))))) = (MetOpen‘𝑃)) |
33 | 26, 32 | eqtr2d 2781 | . . 3 ⊢ (𝜑 → (MetOpen‘𝑃) = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))) |
34 | 19, 33 | eqtrid 2792 | . 2 ⊢ (𝜑 → 𝐿 = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))) |
35 | tmsxpsmopn.j | . . . . 5 ⊢ 𝐽 = (MetOpen‘𝑀) | |
36 | 2, 35 | tmstopn 23637 | . . . 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 23637 | . . . 4 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝐾 = (TopOpen‘(toMetSp‘𝑁))) |
40 | 7, 39 | syl 17 | . . 3 ⊢ (𝜑 → 𝐾 = (TopOpen‘(toMetSp‘𝑁))) |
41 | 37, 40 | oveq12d 7287 | . 2 ⊢ (𝜑 → (𝐽 ×t 𝐾) = ((TopOpen‘(toMetSp‘𝑀)) ×t (TopOpen‘(toMetSp‘𝑁)))) |
42 | 18, 34, 41 | 3eqtr4d 2790 | 1 ⊢ (𝜑 → 𝐿 = (𝐽 ×t 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 × cxp 5587 ↾ cres 5591 Fn wfn 6426 ‘cfv 6431 (class class class)co 7269 Basecbs 16908 distcds 16967 TopOpenctopn 17128 ×s cxps 17213 ∞Metcxmet 20578 MetOpencmopn 20583 TopSpctps 22077 ×t ctx 22707 ∞MetSpcxms 23466 toMetSpctms 23468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-pre-sup 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-om 7705 df-1st 7822 df-2nd 7823 df-supp 7967 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-2o 8287 df-er 8479 df-map 8598 df-ixp 8667 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-fsupp 9105 df-fi 9146 df-sup 9177 df-inf 9178 df-oi 9245 df-card 9696 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12435 df-uz 12580 df-q 12686 df-rp 12728 df-xneg 12845 df-xadd 12846 df-xmul 12847 df-icc 13083 df-fz 13237 df-fzo 13380 df-seq 13718 df-hash 14041 df-struct 16844 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-ress 16938 df-plusg 16971 df-mulr 16972 df-sca 16974 df-vsca 16975 df-ip 16976 df-tset 16977 df-ple 16978 df-ds 16980 df-hom 16982 df-cco 16983 df-rest 17129 df-topn 17130 df-0g 17148 df-gsum 17149 df-topgen 17150 df-pt 17151 df-prds 17154 df-xrs 17209 df-qtop 17214 df-imas 17215 df-xps 17217 df-mre 17291 df-mrc 17292 df-acs 17294 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-submnd 18427 df-mulg 18697 df-cntz 18919 df-cmn 19384 df-psmet 20585 df-xmet 20586 df-bl 20588 df-mopn 20589 df-top 22039 df-topon 22056 df-topsp 22078 df-bases 22092 df-cn 22374 df-cnp 22375 df-tx 22709 df-hmeo 22902 df-xms 23469 df-tms 23471 |
This theorem is referenced by: txmetcnp 23699 |
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