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Mirrors > Home > MPE Home > Th. List > tmsxps | 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)) |
Ref | Expression |
---|---|
tmsxps | ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . . 5 ⊢ ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) = ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) | |
2 | eqid 2821 | . . . . 5 ⊢ (Base‘(toMetSp‘𝑀)) = (Base‘(toMetSp‘𝑀)) | |
3 | eqid 2821 | . . . . 5 ⊢ (Base‘(toMetSp‘𝑁)) = (Base‘(toMetSp‘𝑁)) | |
4 | tmsxps.1 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋)) | |
5 | eqid 2821 | . . . . . . 7 ⊢ (toMetSp‘𝑀) = (toMetSp‘𝑀) | |
6 | 5 | tmsxms 23025 | . . . . . 6 ⊢ (𝑀 ∈ (∞Met‘𝑋) → (toMetSp‘𝑀) ∈ ∞MetSp) |
7 | 4, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → (toMetSp‘𝑀) ∈ ∞MetSp) |
8 | tmsxps.2 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌)) | |
9 | eqid 2821 | . . . . . . 7 ⊢ (toMetSp‘𝑁) = (toMetSp‘𝑁) | |
10 | 9 | tmsxms 23025 | . . . . . 6 ⊢ (𝑁 ∈ (∞Met‘𝑌) → (toMetSp‘𝑁) ∈ ∞MetSp) |
11 | 8, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → (toMetSp‘𝑁) ∈ ∞MetSp) |
12 | tmsxps.p | . . . . 5 ⊢ 𝑃 = (dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) | |
13 | 1, 2, 3, 7, 11, 12 | xpsdsfn2 22917 | . . . 4 ⊢ (𝜑 → 𝑃 Fn ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) |
14 | fnresdm 6460 | . . . 4 ⊢ (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) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ (𝜑 → (𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) = 𝑃) |
16 | 1 | xpsxms 23073 | . . . . 5 ⊢ (((toMetSp‘𝑀) ∈ ∞MetSp ∧ (toMetSp‘𝑁) ∈ ∞MetSp) → ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp) |
17 | 7, 11, 16 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp) |
18 | eqid 2821 | . . . . 5 ⊢ (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) | |
19 | 18, 12 | xmsxmet2 22998 | . . . 4 ⊢ (((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp → (𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) ∈ (∞Met‘(Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) |
20 | 17, 19 | syl 17 | . . 3 ⊢ (𝜑 → (𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) ∈ (∞Met‘(Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) |
21 | 15, 20 | eqeltrrd 2914 | . 2 ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) |
22 | 5 | tmsbas 23022 | . . . . . 6 ⊢ (𝑀 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘(toMetSp‘𝑀))) |
23 | 4, 22 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑋 = (Base‘(toMetSp‘𝑀))) |
24 | 9 | tmsbas 23022 | . . . . . 6 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝑌 = (Base‘(toMetSp‘𝑁))) |
25 | 8, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑌 = (Base‘(toMetSp‘𝑁))) |
26 | 23, 25 | xpeq12d 5580 | . . . 4 ⊢ (𝜑 → (𝑋 × 𝑌) = ((Base‘(toMetSp‘𝑀)) × (Base‘(toMetSp‘𝑁)))) |
27 | 1, 2, 3, 7, 11 | xpsbas 16835 | . . . 4 ⊢ (𝜑 → ((Base‘(toMetSp‘𝑀)) × (Base‘(toMetSp‘𝑁))) = (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))) |
28 | 26, 27 | eqtrd 2856 | . . 3 ⊢ (𝜑 → (𝑋 × 𝑌) = (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))) |
29 | 28 | fveq2d 6668 | . 2 ⊢ (𝜑 → (∞Met‘(𝑋 × 𝑌)) = (∞Met‘(Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) |
30 | 21, 29 | eleqtrrd 2916 | 1 ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌))) |
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 ×s cxps 16769 ∞Metcxmet 20460 ∞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-xms 22859 df-tms 22861 |
This theorem is referenced by: txmetcnp 23086 |
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