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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 2724 | . . . . 5 ⊢ ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) = ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) | |
2 | eqid 2724 | . . . . 5 ⊢ (Base‘(toMetSp‘𝑀)) = (Base‘(toMetSp‘𝑀)) | |
3 | eqid 2724 | . . . . 5 ⊢ (Base‘(toMetSp‘𝑁)) = (Base‘(toMetSp‘𝑁)) | |
4 | tmsxps.1 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋)) | |
5 | eqid 2724 | . . . . . . 7 ⊢ (toMetSp‘𝑀) = (toMetSp‘𝑀) | |
6 | 5 | tmsxms 24339 | . . . . . 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 2724 | . . . . . . 7 ⊢ (toMetSp‘𝑁) = (toMetSp‘𝑁) | |
10 | 9 | tmsxms 24339 | . . . . . 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 24228 | . . . 4 ⊢ (𝜑 → 𝑃 Fn ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) |
14 | fnresdm 6660 | . . . 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 24387 | . . . . 5 ⊢ (((toMetSp‘𝑀) ∈ ∞MetSp ∧ (toMetSp‘𝑁) ∈ ∞MetSp) → ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp) |
17 | 7, 11, 16 | syl2anc 583 | . . . 4 ⊢ (𝜑 → ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp) |
18 | eqid 2724 | . . . . 5 ⊢ (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) | |
19 | 18, 12 | xmsxmet2 24309 | . . . 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 2826 | . 2 ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) |
22 | 5 | tmsbas 24336 | . . . . . 6 ⊢ (𝑀 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘(toMetSp‘𝑀))) |
23 | 4, 22 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑋 = (Base‘(toMetSp‘𝑀))) |
24 | 9 | tmsbas 24336 | . . . . . 6 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝑌 = (Base‘(toMetSp‘𝑁))) |
25 | 8, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑌 = (Base‘(toMetSp‘𝑁))) |
26 | 23, 25 | xpeq12d 5698 | . . . 4 ⊢ (𝜑 → (𝑋 × 𝑌) = ((Base‘(toMetSp‘𝑀)) × (Base‘(toMetSp‘𝑁)))) |
27 | 1, 2, 3, 7, 11 | xpsbas 17523 | . . . 4 ⊢ (𝜑 → ((Base‘(toMetSp‘𝑀)) × (Base‘(toMetSp‘𝑁))) = (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))) |
28 | 26, 27 | eqtrd 2764 | . . 3 ⊢ (𝜑 → (𝑋 × 𝑌) = (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))) |
29 | 28 | fveq2d 6886 | . 2 ⊢ (𝜑 → (∞Met‘(𝑋 × 𝑌)) = (∞Met‘(Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) |
30 | 21, 29 | eleqtrrd 2828 | 1 ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 × cxp 5665 ↾ cres 5669 Fn wfn 6529 ‘cfv 6534 (class class class)co 7402 Basecbs 17149 distcds 17211 ×s cxps 17457 ∞Metcxmet 21219 ∞MetSpcxms 24167 toMetSpctms 24169 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12976 df-xneg 13093 df-xadd 13094 df-xmul 13095 df-icc 13332 df-fz 13486 df-fzo 13629 df-seq 13968 df-hash 14292 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-mulg 18992 df-cntz 19229 df-cmn 19698 df-psmet 21226 df-xmet 21227 df-bl 21229 df-mopn 21230 df-top 22740 df-topon 22757 df-topsp 22779 df-bases 22793 df-xms 24170 df-tms 24172 |
This theorem is referenced by: txmetcnp 24400 |
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