Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > mettri2 | Structured version Visualization version GIF version | ||
| Description: Triangle inequality for the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.) |
| Ref | Expression |
|---|---|
| mettri2 | β’ ((π· β (Metβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄π·π΅) β€ ((πΆπ·π΄) + (πΆπ·π΅))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | metxmet 24190 | . . 3 β’ (π· β (Metβπ) β π· β (βMetβπ)) | |
| 2 | xmettri2 24196 | . . 3 β’ ((π· β (βMetβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄π·π΅) β€ ((πΆπ·π΄) +π (πΆπ·π΅))) | |
| 3 | 1, 2 | sylan 579 | . 2 β’ ((π· β (Metβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄π·π΅) β€ ((πΆπ·π΄) +π (πΆπ·π΅))) |
| 4 | metcl 24188 | . . . 4 β’ ((π· β (Metβπ) β§ πΆ β π β§ π΄ β π) β (πΆπ·π΄) β β) | |
| 5 | 4 | 3adant3r3 1181 | . . 3 β’ ((π· β (Metβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (πΆπ·π΄) β β) |
| 6 | metcl 24188 | . . . 4 β’ ((π· β (Metβπ) β§ πΆ β π β§ π΅ β π) β (πΆπ·π΅) β β) | |
| 7 | 6 | 3adant3r2 1180 | . . 3 β’ ((π· β (Metβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (πΆπ·π΅) β β) |
| 8 | 5, 7 | rexaddd 13216 | . 2 β’ ((π· β (Metβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β ((πΆπ·π΄) +π (πΆπ·π΅)) = ((πΆπ·π΄) + (πΆπ·π΅))) |
| 9 | 3, 8 | breqtrd 5167 | 1 β’ ((π· β (Metβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄π·π΅) β€ ((πΆπ·π΄) + (πΆπ·π΅))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 β wcel 2098 class class class wbr 5141 βcfv 6536 (class class class)co 7404 βcr 11108 + caddc 11112 β€ cle 11250 +π cxad 13093 βMetcxmet 21220 Metcmet 21221 |
| 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-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-mulcl 11171 ax-i2m1 11177 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 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-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-xadd 13096 df-xmet 21228 df-met 21229 |
| This theorem is referenced by: mettri 24208 mstri2 24323 metf1o 37135 isbnd3 37164 heibor1lem 37189 bfplem2 37203 |
| Copyright terms: Public domain | W3C validator |