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Mirrors > Home > MPE Home > Th. List > prdsdsval | Structured version Visualization version GIF version |
Description: Value of the metric in a structure product. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
prdsbasmpt.y | β’ π = (πXsπ ) |
prdsbasmpt.b | β’ π΅ = (Baseβπ) |
prdsbasmpt.s | β’ (π β π β π) |
prdsbasmpt.i | β’ (π β πΌ β π) |
prdsbasmpt.r | β’ (π β π Fn πΌ) |
prdsplusgval.f | β’ (π β πΉ β π΅) |
prdsplusgval.g | β’ (π β πΊ β π΅) |
prdsdsval.d | β’ π· = (distβπ) |
Ref | Expression |
---|---|
prdsdsval | β’ (π β (πΉπ·πΊ) = sup((ran (π₯ β πΌ β¦ ((πΉβπ₯)(distβ(π βπ₯))(πΊβπ₯))) βͺ {0}), β*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsbasmpt.y | . . 3 β’ π = (πXsπ ) | |
2 | prdsbasmpt.s | . . 3 β’ (π β π β π) | |
3 | prdsbasmpt.r | . . . 4 β’ (π β π Fn πΌ) | |
4 | prdsbasmpt.i | . . . 4 β’ (π β πΌ β π) | |
5 | fnex 7225 | . . . 4 β’ ((π Fn πΌ β§ πΌ β π) β π β V) | |
6 | 3, 4, 5 | syl2anc 582 | . . 3 β’ (π β π β V) |
7 | prdsbasmpt.b | . . 3 β’ π΅ = (Baseβπ) | |
8 | fndm 6652 | . . . 4 β’ (π Fn πΌ β dom π = πΌ) | |
9 | 3, 8 | syl 17 | . . 3 β’ (π β dom π = πΌ) |
10 | prdsdsval.d | . . 3 β’ π· = (distβπ) | |
11 | 1, 2, 6, 7, 9, 10 | prdsds 17445 | . 2 β’ (π β π· = (π β π΅, π β π΅ β¦ sup((ran (π₯ β πΌ β¦ ((πβπ₯)(distβ(π βπ₯))(πβπ₯))) βͺ {0}), β*, < ))) |
12 | fveq1 6891 | . . . . . . . 8 β’ (π = πΉ β (πβπ₯) = (πΉβπ₯)) | |
13 | fveq1 6891 | . . . . . . . 8 β’ (π = πΊ β (πβπ₯) = (πΊβπ₯)) | |
14 | 12, 13 | oveqan12d 7435 | . . . . . . 7 β’ ((π = πΉ β§ π = πΊ) β ((πβπ₯)(distβ(π βπ₯))(πβπ₯)) = ((πΉβπ₯)(distβ(π βπ₯))(πΊβπ₯))) |
15 | 14 | adantl 480 | . . . . . 6 β’ ((π β§ (π = πΉ β§ π = πΊ)) β ((πβπ₯)(distβ(π βπ₯))(πβπ₯)) = ((πΉβπ₯)(distβ(π βπ₯))(πΊβπ₯))) |
16 | 15 | mpteq2dv 5245 | . . . . 5 β’ ((π β§ (π = πΉ β§ π = πΊ)) β (π₯ β πΌ β¦ ((πβπ₯)(distβ(π βπ₯))(πβπ₯))) = (π₯ β πΌ β¦ ((πΉβπ₯)(distβ(π βπ₯))(πΊβπ₯)))) |
17 | 16 | rneqd 5934 | . . . 4 β’ ((π β§ (π = πΉ β§ π = πΊ)) β ran (π₯ β πΌ β¦ ((πβπ₯)(distβ(π βπ₯))(πβπ₯))) = ran (π₯ β πΌ β¦ ((πΉβπ₯)(distβ(π βπ₯))(πΊβπ₯)))) |
18 | 17 | uneq1d 4155 | . . 3 β’ ((π β§ (π = πΉ β§ π = πΊ)) β (ran (π₯ β πΌ β¦ ((πβπ₯)(distβ(π βπ₯))(πβπ₯))) βͺ {0}) = (ran (π₯ β πΌ β¦ ((πΉβπ₯)(distβ(π βπ₯))(πΊβπ₯))) βͺ {0})) |
19 | 18 | supeq1d 9469 | . 2 β’ ((π β§ (π = πΉ β§ π = πΊ)) β sup((ran (π₯ β πΌ β¦ ((πβπ₯)(distβ(π βπ₯))(πβπ₯))) βͺ {0}), β*, < ) = sup((ran (π₯ β πΌ β¦ ((πΉβπ₯)(distβ(π βπ₯))(πΊβπ₯))) βͺ {0}), β*, < )) |
20 | prdsplusgval.f | . 2 β’ (π β πΉ β π΅) | |
21 | prdsplusgval.g | . 2 β’ (π β πΊ β π΅) | |
22 | xrltso 13152 | . . . 4 β’ < Or β* | |
23 | 22 | supex 9486 | . . 3 β’ sup((ran (π₯ β πΌ β¦ ((πΉβπ₯)(distβ(π βπ₯))(πΊβπ₯))) βͺ {0}), β*, < ) β V |
24 | 23 | a1i 11 | . 2 β’ (π β sup((ran (π₯ β πΌ β¦ ((πΉβπ₯)(distβ(π βπ₯))(πΊβπ₯))) βͺ {0}), β*, < ) β V) |
25 | 11, 19, 20, 21, 24 | ovmpod 7570 | 1 β’ (π β (πΉπ·πΊ) = sup((ran (π₯ β πΌ β¦ ((πΉβπ₯)(distβ(π βπ₯))(πΊβπ₯))) βͺ {0}), β*, < )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3463 βͺ cun 3937 {csn 4624 β¦ cmpt 5226 dom cdm 5672 ran crn 5673 Fn wfn 6538 βcfv 6543 (class class class)co 7416 supcsup 9463 0cc0 11138 β*cxr 11277 < clt 11278 Basecbs 17179 distcds 17241 Xscprds 17426 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-hom 17256 df-cco 17257 df-prds 17428 |
This theorem is referenced by: prdsdsval2 17465 xpsdsval 24305 |
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