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Mirrors > Home > MPE Home > Th. List > tmsxpsval2 | Structured version Visualization version GIF version |
Description: Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tmsxps.p | β’ π = (distβ((toMetSpβπ) Γs (toMetSpβπ))) |
tmsxps.1 | β’ (π β π β (βMetβπ)) |
tmsxps.2 | β’ (π β π β (βMetβπ)) |
tmsxpsval.a | β’ (π β π΄ β π) |
tmsxpsval.b | β’ (π β π΅ β π) |
tmsxpsval.c | β’ (π β πΆ β π) |
tmsxpsval.d | β’ (π β π· β π) |
Ref | Expression |
---|---|
tmsxpsval2 | β’ (π β (β¨π΄, π΅β©πβ¨πΆ, π·β©) = if((π΄ππΆ) β€ (π΅ππ·), (π΅ππ·), (π΄ππΆ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tmsxps.p | . . 3 β’ π = (distβ((toMetSpβπ) Γs (toMetSpβπ))) | |
2 | tmsxps.1 | . . 3 β’ (π β π β (βMetβπ)) | |
3 | tmsxps.2 | . . 3 β’ (π β π β (βMetβπ)) | |
4 | tmsxpsval.a | . . 3 β’ (π β π΄ β π) | |
5 | tmsxpsval.b | . . 3 β’ (π β π΅ β π) | |
6 | tmsxpsval.c | . . 3 β’ (π β πΆ β π) | |
7 | tmsxpsval.d | . . 3 β’ (π β π· β π) | |
8 | 1, 2, 3, 4, 5, 6, 7 | tmsxpsval 23917 | . 2 β’ (π β (β¨π΄, π΅β©πβ¨πΆ, π·β©) = sup({(π΄ππΆ), (π΅ππ·)}, β*, < )) |
9 | xrltso 13069 | . . 3 β’ < Or β* | |
10 | xmetcl 23707 | . . . 4 β’ ((π β (βMetβπ) β§ π΄ β π β§ πΆ β π) β (π΄ππΆ) β β*) | |
11 | 2, 4, 6, 10 | syl3anc 1372 | . . 3 β’ (π β (π΄ππΆ) β β*) |
12 | xmetcl 23707 | . . . 4 β’ ((π β (βMetβπ) β§ π΅ β π β§ π· β π) β (π΅ππ·) β β*) | |
13 | 3, 5, 7, 12 | syl3anc 1372 | . . 3 β’ (π β (π΅ππ·) β β*) |
14 | suppr 9415 | . . 3 β’ (( < Or β* β§ (π΄ππΆ) β β* β§ (π΅ππ·) β β*) β sup({(π΄ππΆ), (π΅ππ·)}, β*, < ) = if((π΅ππ·) < (π΄ππΆ), (π΄ππΆ), (π΅ππ·))) | |
15 | 9, 11, 13, 14 | mp3an2i 1467 | . 2 β’ (π β sup({(π΄ππΆ), (π΅ππ·)}, β*, < ) = if((π΅ππ·) < (π΄ππΆ), (π΄ππΆ), (π΅ππ·))) |
16 | xrltnle 11230 | . . . . 5 β’ (((π΅ππ·) β β* β§ (π΄ππΆ) β β*) β ((π΅ππ·) < (π΄ππΆ) β Β¬ (π΄ππΆ) β€ (π΅ππ·))) | |
17 | 13, 11, 16 | syl2anc 585 | . . . 4 β’ (π β ((π΅ππ·) < (π΄ππΆ) β Β¬ (π΄ππΆ) β€ (π΅ππ·))) |
18 | 17 | ifbid 4513 | . . 3 β’ (π β if((π΅ππ·) < (π΄ππΆ), (π΄ππΆ), (π΅ππ·)) = if(Β¬ (π΄ππΆ) β€ (π΅ππ·), (π΄ππΆ), (π΅ππ·))) |
19 | ifnot 4542 | . . 3 β’ if(Β¬ (π΄ππΆ) β€ (π΅ππ·), (π΄ππΆ), (π΅ππ·)) = if((π΄ππΆ) β€ (π΅ππ·), (π΅ππ·), (π΄ππΆ)) | |
20 | 18, 19 | eqtrdi 2789 | . 2 β’ (π β if((π΅ππ·) < (π΄ππΆ), (π΄ππΆ), (π΅ππ·)) = if((π΄ππΆ) β€ (π΅ππ·), (π΅ππ·), (π΄ππΆ))) |
21 | 8, 15, 20 | 3eqtrd 2777 | 1 β’ (π β (β¨π΄, π΅β©πβ¨πΆ, π·β©) = if((π΄ππΆ) β€ (π΅ππ·), (π΅ππ·), (π΄ππΆ))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 = wceq 1542 β wcel 2107 ifcif 4490 {cpr 4592 β¨cop 4596 class class class wbr 5109 Or wor 5548 βcfv 6500 (class class class)co 7361 supcsup 9384 β*cxr 11196 < clt 11197 β€ cle 11198 distcds 17150 Γs cxps 17396 βMetcxmet 20804 toMetSpctms 23695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-sup 9386 df-inf 9387 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-icc 13280 df-fz 13434 df-fzo 13577 df-seq 13916 df-hash 14240 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-hom 17165 df-cco 17166 df-rest 17312 df-topn 17313 df-0g 17331 df-gsum 17332 df-topgen 17333 df-prds 17337 df-xrs 17392 df-imas 17398 df-xps 17400 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-mulg 18881 df-cntz 19105 df-cmn 19572 df-psmet 20811 df-xmet 20812 df-bl 20814 df-mopn 20815 df-top 22266 df-topon 22283 df-topsp 22305 df-bases 22319 df-xms 23696 df-tms 23698 |
This theorem is referenced by: txmetcnp 23926 |
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