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Mirrors > Home > MPE Home > Th. List > imasdsfn | Structured version Visualization version GIF version |
Description: The distance function is a function on the base set. (Contributed by Mario Carneiro, 20-Aug-2015.) (Proof shortened 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) |
Ref | Expression |
---|---|
imasdsfn | β’ (π β π· Fn (π΅ Γ π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . 3 β’ (π₯ β π΅, π¦ β π΅ β¦ inf(βͺ π β β ran (π β {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)))))} β¦ (β*π Ξ£g (πΈ β π))), β*, < )) = (π₯ β π΅, π¦ β π΅ β¦ inf(βͺ π β β ran (π β {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)))))} β¦ (β*π Ξ£g (πΈ β π))), β*, < )) | |
2 | xrltso 13061 | . . . 4 β’ < Or β* | |
3 | 2 | infex 9430 | . . 3 β’ inf(βͺ π β β ran (π β {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)))))} β¦ (β*π Ξ£g (πΈ β π))), β*, < ) β V |
4 | 1, 3 | fnmpoi 8003 | . 2 β’ (π₯ β π΅, π¦ β π΅ β¦ inf(βͺ π β β ran (π β {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)))))} β¦ (β*π Ξ£g (πΈ β π))), β*, < )) Fn (π΅ Γ π΅) |
5 | imasbas.u | . . . 4 β’ (π β π = (πΉ βs π )) | |
6 | imasbas.v | . . . 4 β’ (π β π = (Baseβπ )) | |
7 | imasbas.f | . . . 4 β’ (π β πΉ:πβontoβπ΅) | |
8 | imasbas.r | . . . 4 β’ (π β π β π) | |
9 | imasds.e | . . . 4 β’ πΈ = (distβπ ) | |
10 | imasds.d | . . . 4 β’ π· = (distβπ) | |
11 | 5, 6, 7, 8, 9, 10 | imasds 17396 | . . 3 β’ (π β π· = (π₯ β π΅, π¦ β π΅ β¦ inf(βͺ π β β ran (π β {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)))))} β¦ (β*π Ξ£g (πΈ β π))), β*, < ))) |
12 | 11 | fneq1d 6596 | . 2 β’ (π β (π· Fn (π΅ Γ π΅) β (π₯ β π΅, π¦ β π΅ β¦ inf(βͺ π β β ran (π β {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)))))} β¦ (β*π Ξ£g (πΈ β π))), β*, < )) Fn (π΅ Γ π΅))) |
13 | 4, 12 | mpbiri 258 | 1 β’ (π β π· Fn (π΅ Γ π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3065 {crab 3408 βͺ ciun 4955 β¦ cmpt 5189 Γ cxp 5632 ran crn 5635 β ccom 5638 Fn wfn 6492 βontoβwfo 6495 βcfv 6497 (class class class)co 7358 β cmpo 7360 1st c1st 7920 2nd c2nd 7921 βm cmap 8766 infcinf 9378 1c1 11053 + caddc 11055 β*cxr 11189 < clt 11190 β cmin 11386 βcn 12154 ...cfz 13425 Basecbs 17084 distcds 17143 Ξ£g cgsu 17323 β*π cxrs 17383 βs cimas 17387 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9379 df-inf 9380 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-dec 12620 df-uz 12765 df-fz 13426 df-struct 17020 df-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-mulr 17148 df-sca 17150 df-vsca 17151 df-ip 17152 df-tset 17153 df-ple 17154 df-ds 17156 df-imas 17391 |
This theorem is referenced by: imasf1oxmet 23731 imasf1omet 23732 xpsdsfn 23733 |
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