Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > metres | Structured version Visualization version GIF version | ||
| Description: A restriction of a metric is a metric. (Contributed by NM, 26-Aug-2007.) (Revised by Mario Carneiro, 14-Aug-2015.) |
| Ref | Expression |
|---|---|
| metres | β’ (π· β (Metβπ) β (π· βΎ (π Γ π )) β (Metβ(π β© π ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | metf 24156 | . . 3 β’ (π· β (Metβπ) β π·:(π Γ π)βΆβ) | |
| 2 | fdm 6726 | . . 3 β’ (π·:(π Γ π)βΆβ β dom π· = (π Γ π)) | |
| 3 | metreslem 24188 | . . 3 β’ (dom π· = (π Γ π) β (π· βΎ (π Γ π )) = (π· βΎ ((π β© π ) Γ (π β© π )))) | |
| 4 | 1, 2, 3 | 3syl 18 | . 2 β’ (π· β (Metβπ) β (π· βΎ (π Γ π )) = (π· βΎ ((π β© π ) Γ (π β© π )))) |
| 5 | inss1 4228 | . . 3 β’ (π β© π ) β π | |
| 6 | metres2 24189 | . . 3 β’ ((π· β (Metβπ) β§ (π β© π ) β π) β (π· βΎ ((π β© π ) Γ (π β© π ))) β (Metβ(π β© π ))) | |
| 7 | 5, 6 | mpan2 688 | . 2 β’ (π· β (Metβπ) β (π· βΎ ((π β© π ) Γ (π β© π ))) β (Metβ(π β© π ))) |
| 8 | 4, 7 | eqeltrd 2832 | 1 β’ (π· β (Metβπ) β (π· βΎ (π Γ π )) β (Metβ(π β© π ))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1540 β wcel 2105 β© cin 3947 β wss 3948 Γ cxp 5674 dom cdm 5676 βΎ cres 5678 βΆwf 6539 βcfv 6543 βcr 11115 Metcmet 21219 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-mulcl 11178 ax-i2m1 11184 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-xadd 13100 df-xmet 21226 df-met 21227 |
| This theorem is referenced by: ressms 24355 |
| Copyright terms: Public domain | W3C validator |