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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 2726 | . . . . . 6 ⊢ (toMetSp‘𝑀) = (toMetSp‘𝑀) | |
3 | 2 | tmsxms 24350 | . . . . 5 ⊢ (𝑀 ∈ (∞Met‘𝑋) → (toMetSp‘𝑀) ∈ ∞MetSp) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → (toMetSp‘𝑀) ∈ ∞MetSp) |
5 | xmstps 24314 | . . . 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 2726 | . . . . . 6 ⊢ (toMetSp‘𝑁) = (toMetSp‘𝑁) | |
9 | 8 | tmsxms 24350 | . . . . 5 ⊢ (𝑁 ∈ (∞Met‘𝑌) → (toMetSp‘𝑁) ∈ ∞MetSp) |
10 | 7, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (toMetSp‘𝑁) ∈ ∞MetSp) |
11 | xmstps 24314 | . . . 4 ⊢ ((toMetSp‘𝑁) ∈ ∞MetSp → (toMetSp‘𝑁) ∈ TopSp) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝜑 → (toMetSp‘𝑁) ∈ TopSp) |
13 | eqid 2726 | . . . 4 ⊢ ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) = ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) | |
14 | eqid 2726 | . . . 4 ⊢ (TopOpen‘(toMetSp‘𝑀)) = (TopOpen‘(toMetSp‘𝑀)) | |
15 | eqid 2726 | . . . 4 ⊢ (TopOpen‘(toMetSp‘𝑁)) = (TopOpen‘(toMetSp‘𝑁)) | |
16 | eqid 2726 | . . . 4 ⊢ (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) | |
17 | 13, 14, 15, 16 | xpstopn 23671 | . . 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 583 | . 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 24398 | . . . . . 6 ⊢ (((toMetSp‘𝑀) ∈ ∞MetSp ∧ (toMetSp‘𝑁) ∈ ∞MetSp) → ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp) |
21 | 4, 10, 20 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp) |
22 | eqid 2726 | . . . . . 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 5971 | . . . . . 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 24312 | . . . . 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 2726 | . . . . . . 7 ⊢ (Base‘(toMetSp‘𝑀)) = (Base‘(toMetSp‘𝑀)) | |
28 | eqid 2726 | . . . . . . 7 ⊢ (Base‘(toMetSp‘𝑁)) = (Base‘(toMetSp‘𝑁)) | |
29 | 13, 27, 28, 4, 10, 23 | xpsdsfn2 24239 | . . . . . 6 ⊢ (𝜑 → 𝑃 Fn ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) |
30 | fnresdm 6663 | . . . . . 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 6889 | . . . 4 ⊢ (𝜑 → (MetOpen‘(𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))))) = (MetOpen‘𝑃)) |
33 | 26, 32 | eqtr2d 2767 | . . 3 ⊢ (𝜑 → (MetOpen‘𝑃) = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))) |
34 | 19, 33 | eqtrid 2778 | . 2 ⊢ (𝜑 → 𝐿 = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))) |
35 | tmsxpsmopn.j | . . . . 5 ⊢ 𝐽 = (MetOpen‘𝑀) | |
36 | 2, 35 | tmstopn 24349 | . . . 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 24349 | . . . 4 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝐾 = (TopOpen‘(toMetSp‘𝑁))) |
40 | 7, 39 | syl 17 | . . 3 ⊢ (𝜑 → 𝐾 = (TopOpen‘(toMetSp‘𝑁))) |
41 | 37, 40 | oveq12d 7423 | . 2 ⊢ (𝜑 → (𝐽 ×t 𝐾) = ((TopOpen‘(toMetSp‘𝑀)) ×t (TopOpen‘(toMetSp‘𝑁)))) |
42 | 18, 34, 41 | 3eqtr4d 2776 | 1 ⊢ (𝜑 → 𝐿 = (𝐽 ×t 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 × cxp 5667 ↾ cres 5671 Fn wfn 6532 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 distcds 17215 TopOpenctopn 17376 ×s cxps 17461 ∞Metcxmet 21225 MetOpencmopn 21230 TopSpctps 22789 ×t ctx 23419 ∞MetSpcxms 24178 toMetSpctms 24180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-icc 13337 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-psmet 21232 df-xmet 21233 df-bl 21235 df-mopn 21236 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cn 23086 df-cnp 23087 df-tx 23421 df-hmeo 23614 df-xms 24181 df-tms 24183 |
This theorem is referenced by: txmetcnp 24411 |
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