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Mirrors > Home > MPE Home > Th. List > imasdsval2 | Structured version Visualization version GIF version |
Description: The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised by AV, 6-Oct-2020.) |
Ref | Expression |
---|---|
imasbas.u | β’ (π β π = (πΉ βs π )) |
imasbas.v | β’ (π β π = (Baseβπ )) |
imasbas.f | β’ (π β πΉ:πβontoβπ΅) |
imasbas.r | β’ (π β π β π) |
imasds.e | β’ πΈ = (distβπ ) |
imasds.d | β’ π· = (distβπ) |
imasdsval.x | β’ (π β π β π΅) |
imasdsval.y | β’ (π β π β π΅) |
imasdsval.s | β’ π = {β β ((π Γ π) βm (1...π)) β£ ((πΉβ(1st β(ββ1))) = π β§ (πΉβ(2nd β(ββπ))) = π β§ βπ β (1...(π β 1))(πΉβ(2nd β(ββπ))) = (πΉβ(1st β(ββ(π + 1)))))} |
imasds.u | β’ π = (πΈ βΎ (π Γ π)) |
Ref | Expression |
---|---|
imasdsval2 | β’ (π β (ππ·π) = inf(βͺ π β β ran (π β π β¦ (β*π Ξ£g (π β π))), β*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imasbas.u | . . 3 β’ (π β π = (πΉ βs π )) | |
2 | imasbas.v | . . 3 β’ (π β π = (Baseβπ )) | |
3 | imasbas.f | . . 3 β’ (π β πΉ:πβontoβπ΅) | |
4 | imasbas.r | . . 3 β’ (π β π β π) | |
5 | imasds.e | . . 3 β’ πΈ = (distβπ ) | |
6 | imasds.d | . . 3 β’ π· = (distβπ) | |
7 | imasdsval.x | . . 3 β’ (π β π β π΅) | |
8 | imasdsval.y | . . 3 β’ (π β π β π΅) | |
9 | imasdsval.s | . . 3 β’ π = {β β ((π Γ π) βm (1...π)) β£ ((πΉβ(1st β(ββ1))) = π β§ (πΉβ(2nd β(ββπ))) = π β§ βπ β (1...(π β 1))(πΉβ(2nd β(ββπ))) = (πΉβ(1st β(ββ(π + 1)))))} | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | imasdsval 17496 | . 2 β’ (π β (ππ·π) = inf(βͺ π β β ran (π β π β¦ (β*π Ξ£g (πΈ β π))), β*, < )) |
11 | imasds.u | . . . . . . . . . 10 β’ π = (πΈ βΎ (π Γ π)) | |
12 | 11 | coeq1i 5856 | . . . . . . . . 9 β’ (π β π) = ((πΈ βΎ (π Γ π)) β π) |
13 | 9 | ssrab3 4072 | . . . . . . . . . . 11 β’ π β ((π Γ π) βm (1...π)) |
14 | 13 | sseli 3968 | . . . . . . . . . 10 β’ (π β π β π β ((π Γ π) βm (1...π))) |
15 | elmapi 8866 | . . . . . . . . . 10 β’ (π β ((π Γ π) βm (1...π)) β π:(1...π)βΆ(π Γ π)) | |
16 | frn 6724 | . . . . . . . . . 10 β’ (π:(1...π)βΆ(π Γ π) β ran π β (π Γ π)) | |
17 | cores 6248 | . . . . . . . . . 10 β’ (ran π β (π Γ π) β ((πΈ βΎ (π Γ π)) β π) = (πΈ β π)) | |
18 | 14, 15, 16, 17 | 4syl 19 | . . . . . . . . 9 β’ (π β π β ((πΈ βΎ (π Γ π)) β π) = (πΈ β π)) |
19 | 12, 18 | eqtrid 2777 | . . . . . . . 8 β’ (π β π β (π β π) = (πΈ β π)) |
20 | 19 | oveq2d 7432 | . . . . . . 7 β’ (π β π β (β*π Ξ£g (π β π)) = (β*π Ξ£g (πΈ β π))) |
21 | 20 | mpteq2ia 5246 | . . . . . 6 β’ (π β π β¦ (β*π Ξ£g (π β π))) = (π β π β¦ (β*π Ξ£g (πΈ β π))) |
22 | 21 | rneqi 5933 | . . . . 5 β’ ran (π β π β¦ (β*π Ξ£g (π β π))) = ran (π β π β¦ (β*π Ξ£g (πΈ β π))) |
23 | 22 | a1i 11 | . . . 4 β’ (π β β β ran (π β π β¦ (β*π Ξ£g (π β π))) = ran (π β π β¦ (β*π Ξ£g (πΈ β π)))) |
24 | 23 | iuneq2i 5012 | . . 3 β’ βͺ π β β ran (π β π β¦ (β*π Ξ£g (π β π))) = βͺ π β β ran (π β π β¦ (β*π Ξ£g (πΈ β π))) |
25 | 24 | infeq1i 9501 | . 2 β’ inf(βͺ π β β ran (π β π β¦ (β*π Ξ£g (π β π))), β*, < ) = inf(βͺ π β β ran (π β π β¦ (β*π Ξ£g (πΈ β π))), β*, < ) |
26 | 10, 25 | eqtr4di 2783 | 1 β’ (π β (ππ·π) = inf(βͺ π β β ran (π β π β¦ (β*π Ξ£g (π β π))), β*, < )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3051 {crab 3419 β wss 3939 βͺ ciun 4991 β¦ cmpt 5226 Γ cxp 5670 ran crn 5673 βΎ cres 5674 β ccom 5676 βΆwf 6539 βontoβwfo 6541 βcfv 6543 (class class class)co 7416 1st c1st 7989 2nd c2nd 7990 βm cmap 8843 infcinf 9464 1c1 11139 + caddc 11141 β*cxr 11277 < clt 11278 β cmin 11474 βcn 12242 ...cfz 13516 Basecbs 17179 distcds 17241 Ξ£g cgsu 17421 β*π cxrs 17481 βs cimas 17485 |
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-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-inf 9466 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-imas 17489 |
This theorem is referenced by: imasdsf1olem 24297 |
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