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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 7168 | . . . 4 β’ ((π Fn πΌ β§ πΌ β π) β π β V) | |
6 | 3, 4, 5 | syl2anc 585 | . . 3 β’ (π β π β V) |
7 | prdsbasmpt.b | . . 3 β’ π΅ = (Baseβπ) | |
8 | fndm 6606 | . . . 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 17351 | . 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 6842 | . . . . . . . 8 β’ (π = πΉ β (πβπ₯) = (πΉβπ₯)) | |
13 | fveq1 6842 | . . . . . . . 8 β’ (π = πΊ β (πβπ₯) = (πΊβπ₯)) | |
14 | 12, 13 | oveqan12d 7377 | . . . . . . 7 β’ ((π = πΉ β§ π = πΊ) β ((πβπ₯)(distβ(π βπ₯))(πβπ₯)) = ((πΉβπ₯)(distβ(π βπ₯))(πΊβπ₯))) |
15 | 14 | adantl 483 | . . . . . 6 β’ ((π β§ (π = πΉ β§ π = πΊ)) β ((πβπ₯)(distβ(π βπ₯))(πβπ₯)) = ((πΉβπ₯)(distβ(π βπ₯))(πΊβπ₯))) |
16 | 15 | mpteq2dv 5208 | . . . . 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 5894 | . . . 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 4123 | . . 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 9387 | . 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 13066 | . . . 4 β’ < Or β* | |
23 | 22 | supex 9404 | . . 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 7508 | 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 397 = wceq 1542 β wcel 2107 Vcvv 3444 βͺ cun 3909 {csn 4587 β¦ cmpt 5189 dom cdm 5634 ran crn 5635 Fn wfn 6492 βcfv 6497 (class class class)co 7358 supcsup 9381 0cc0 11056 β*cxr 11193 < clt 11194 Basecbs 17088 distcds 17147 Xscprds 17332 |
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-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 3446 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 8651 df-map 8770 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-hom 17162 df-cco 17163 df-prds 17334 |
This theorem is referenced by: prdsdsval2 17371 xpsdsval 23750 |
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