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Mirrors > Home > MPE Home > Th. List > psmetxrge0 | Structured version Visualization version GIF version |
Description: The distance function of a pseudometric space is a function into the nonnegative extended real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.) |
Ref | Expression |
---|---|
psmetxrge0 | β’ (π· β (PsMetβπ) β π·:(π Γ π)βΆ(0[,]+β)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psmetf 23581 | . . 3 β’ (π· β (PsMetβπ) β π·:(π Γ π)βΆβ*) | |
2 | 1 | ffnd 6665 | . 2 β’ (π· β (PsMetβπ) β π· Fn (π Γ π)) |
3 | 1 | ffvelcdmda 7030 | . . . . 5 β’ ((π· β (PsMetβπ) β§ π β (π Γ π)) β (π·βπ) β β*) |
4 | elxp6 7946 | . . . . . . . 8 β’ (π β (π Γ π) β (π = β¨(1st βπ), (2nd βπ)β© β§ ((1st βπ) β π β§ (2nd βπ) β π))) | |
5 | 4 | simprbi 498 | . . . . . . 7 β’ (π β (π Γ π) β ((1st βπ) β π β§ (2nd βπ) β π)) |
6 | psmetge0 23587 | . . . . . . . 8 β’ ((π· β (PsMetβπ) β§ (1st βπ) β π β§ (2nd βπ) β π) β 0 β€ ((1st βπ)π·(2nd βπ))) | |
7 | 6 | 3expb 1121 | . . . . . . 7 β’ ((π· β (PsMetβπ) β§ ((1st βπ) β π β§ (2nd βπ) β π)) β 0 β€ ((1st βπ)π·(2nd βπ))) |
8 | 5, 7 | sylan2 594 | . . . . . 6 β’ ((π· β (PsMetβπ) β§ π β (π Γ π)) β 0 β€ ((1st βπ)π·(2nd βπ))) |
9 | 1st2nd2 7951 | . . . . . . . . 9 β’ (π β (π Γ π) β π = β¨(1st βπ), (2nd βπ)β©) | |
10 | 9 | fveq2d 6842 | . . . . . . . 8 β’ (π β (π Γ π) β (π·βπ) = (π·ββ¨(1st βπ), (2nd βπ)β©)) |
11 | df-ov 7353 | . . . . . . . 8 β’ ((1st βπ)π·(2nd βπ)) = (π·ββ¨(1st βπ), (2nd βπ)β©) | |
12 | 10, 11 | eqtr4di 2796 | . . . . . . 7 β’ (π β (π Γ π) β (π·βπ) = ((1st βπ)π·(2nd βπ))) |
13 | 12 | adantl 483 | . . . . . 6 β’ ((π· β (PsMetβπ) β§ π β (π Γ π)) β (π·βπ) = ((1st βπ)π·(2nd βπ))) |
14 | 8, 13 | breqtrrd 5132 | . . . . 5 β’ ((π· β (PsMetβπ) β§ π β (π Γ π)) β 0 β€ (π·βπ)) |
15 | elxrge0 13303 | . . . . 5 β’ ((π·βπ) β (0[,]+β) β ((π·βπ) β β* β§ 0 β€ (π·βπ))) | |
16 | 3, 14, 15 | sylanbrc 584 | . . . 4 β’ ((π· β (PsMetβπ) β§ π β (π Γ π)) β (π·βπ) β (0[,]+β)) |
17 | 16 | ralrimiva 3142 | . . 3 β’ (π· β (PsMetβπ) β βπ β (π Γ π)(π·βπ) β (0[,]+β)) |
18 | fnfvrnss 7063 | . . 3 β’ ((π· Fn (π Γ π) β§ βπ β (π Γ π)(π·βπ) β (0[,]+β)) β ran π· β (0[,]+β)) | |
19 | 2, 17, 18 | syl2anc 585 | . 2 β’ (π· β (PsMetβπ) β ran π· β (0[,]+β)) |
20 | df-f 6496 | . 2 β’ (π·:(π Γ π)βΆ(0[,]+β) β (π· Fn (π Γ π) β§ ran π· β (0[,]+β))) | |
21 | 2, 19, 20 | sylanbrc 584 | 1 β’ (π· β (PsMetβπ) β π·:(π Γ π)βΆ(0[,]+β)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3063 β wss 3909 β¨cop 4591 class class class wbr 5104 Γ cxp 5629 ran crn 5632 Fn wfn 6487 βΆwf 6488 βcfv 6492 (class class class)co 7350 1st c1st 7910 2nd c2nd 7911 0cc0 10985 +βcpnf 11120 β*cxr 11122 β€ cle 11124 [,]cicc 13196 PsMetcpsmet 20703 |
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 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-1st 7912 df-2nd 7913 df-er 8582 df-map 8701 df-en 8818 df-dom 8819 df-sdom 8820 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-2 12150 df-rp 12845 df-xneg 12962 df-xadd 12963 df-xmul 12964 df-icc 13200 df-psmet 20711 |
This theorem is referenced by: sitmcl 32712 |
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