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Mirrors > Home > MPE Home > Th. List > xrsdsre | Structured version Visualization version GIF version |
Description: The metric on the extended reals coincides with the usual metric on the reals. (Contributed by Mario Carneiro, 4-Sep-2015.) |
Ref | Expression |
---|---|
xrsxmet.1 | β’ π· = (distββ*π ) |
Ref | Expression |
---|---|
xrsdsre | β’ (π· βΎ (β Γ β)) = ((abs β β ) βΎ (β Γ β)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrsxmet.1 | . . . . 5 β’ π· = (distββ*π ) | |
2 | 1 | xrsdsreval 20865 | . . . 4 β’ ((π₯ β β β§ π¦ β β) β (π₯π·π¦) = (absβ(π₯ β π¦))) |
3 | ovres 7524 | . . . 4 β’ ((π₯ β β β§ π¦ β β) β (π₯(π· βΎ (β Γ β))π¦) = (π₯π·π¦)) | |
4 | eqid 2733 | . . . . 5 β’ ((abs β β ) βΎ (β Γ β)) = ((abs β β ) βΎ (β Γ β)) | |
5 | 4 | remetdval 24175 | . . . 4 β’ ((π₯ β β β§ π¦ β β) β (π₯((abs β β ) βΎ (β Γ β))π¦) = (absβ(π₯ β π¦))) |
6 | 2, 3, 5 | 3eqtr4d 2783 | . . 3 β’ ((π₯ β β β§ π¦ β β) β (π₯(π· βΎ (β Γ β))π¦) = (π₯((abs β β ) βΎ (β Γ β))π¦)) |
7 | 6 | rgen2 3191 | . 2 β’ βπ₯ β β βπ¦ β β (π₯(π· βΎ (β Γ β))π¦) = (π₯((abs β β ) βΎ (β Γ β))π¦) |
8 | 1 | xrsxmet 24195 | . . . . 5 β’ π· β (βMetββ*) |
9 | xmetf 23705 | . . . . 5 β’ (π· β (βMetββ*) β π·:(β* Γ β*)βΆβ*) | |
10 | ffn 6672 | . . . . 5 β’ (π·:(β* Γ β*)βΆβ* β π· Fn (β* Γ β*)) | |
11 | 8, 9, 10 | mp2b 10 | . . . 4 β’ π· Fn (β* Γ β*) |
12 | rexpssxrxp 11208 | . . . 4 β’ (β Γ β) β (β* Γ β*) | |
13 | fnssres 6628 | . . . 4 β’ ((π· Fn (β* Γ β*) β§ (β Γ β) β (β* Γ β*)) β (π· βΎ (β Γ β)) Fn (β Γ β)) | |
14 | 11, 12, 13 | mp2an 691 | . . 3 β’ (π· βΎ (β Γ β)) Fn (β Γ β) |
15 | cnmet 24158 | . . . . 5 β’ (abs β β ) β (Metββ) | |
16 | metf 23706 | . . . . 5 β’ ((abs β β ) β (Metββ) β (abs β β ):(β Γ β)βΆβ) | |
17 | ffn 6672 | . . . . 5 β’ ((abs β β ):(β Γ β)βΆβ β (abs β β ) Fn (β Γ β)) | |
18 | 15, 16, 17 | mp2b 10 | . . . 4 β’ (abs β β ) Fn (β Γ β) |
19 | ax-resscn 11116 | . . . . 5 β’ β β β | |
20 | xpss12 5652 | . . . . 5 β’ ((β β β β§ β β β) β (β Γ β) β (β Γ β)) | |
21 | 19, 19, 20 | mp2an 691 | . . . 4 β’ (β Γ β) β (β Γ β) |
22 | fnssres 6628 | . . . 4 β’ (((abs β β ) Fn (β Γ β) β§ (β Γ β) β (β Γ β)) β ((abs β β ) βΎ (β Γ β)) Fn (β Γ β)) | |
23 | 18, 21, 22 | mp2an 691 | . . 3 β’ ((abs β β ) βΎ (β Γ β)) Fn (β Γ β) |
24 | eqfnov2 7490 | . . 3 β’ (((π· βΎ (β Γ β)) Fn (β Γ β) β§ ((abs β β ) βΎ (β Γ β)) Fn (β Γ β)) β ((π· βΎ (β Γ β)) = ((abs β β ) βΎ (β Γ β)) β βπ₯ β β βπ¦ β β (π₯(π· βΎ (β Γ β))π¦) = (π₯((abs β β ) βΎ (β Γ β))π¦))) | |
25 | 14, 23, 24 | mp2an 691 | . 2 β’ ((π· βΎ (β Γ β)) = ((abs β β ) βΎ (β Γ β)) β βπ₯ β β βπ¦ β β (π₯(π· βΎ (β Γ β))π¦) = (π₯((abs β β ) βΎ (β Γ β))π¦)) |
26 | 7, 25 | mpbir 230 | 1 β’ (π· βΎ (β Γ β)) = ((abs β β ) βΎ (β Γ β)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 β wss 3914 Γ cxp 5635 βΎ cres 5639 β ccom 5641 Fn wfn 6495 βΆwf 6496 βcfv 6500 (class class class)co 7361 βcc 11057 βcr 11058 β*cxr 11196 β cmin 11393 abscabs 15128 distcds 17150 β*π cxrs 17390 βMetcxmet 20804 Metcmet 20805 |
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-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-iun 4960 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-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-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 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-rp 12924 df-xneg 13041 df-xadd 13042 df-icc 13280 df-fz 13434 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-struct 17027 df-slot 17062 df-ndx 17074 df-base 17092 df-plusg 17154 df-mulr 17155 df-tset 17160 df-ple 17161 df-ds 17163 df-xrs 17392 df-xmet 20812 df-met 20813 |
This theorem is referenced by: xrsmopn 24198 metdscn2 24243 |
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