![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7214 | . . . 4 β’ ((π Fn πΌ β§ πΌ β π) β π β V) | |
6 | 3, 4, 5 | syl2anc 583 | . . 3 β’ (π β π β V) |
7 | prdsbasmpt.b | . . 3 β’ π΅ = (Baseβπ) | |
8 | fndm 6646 | . . . 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 17419 | . 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 6884 | . . . . . . . 8 β’ (π = πΉ β (πβπ₯) = (πΉβπ₯)) | |
13 | fveq1 6884 | . . . . . . . 8 β’ (π = πΊ β (πβπ₯) = (πΊβπ₯)) | |
14 | 12, 13 | oveqan12d 7424 | . . . . . . 7 β’ ((π = πΉ β§ π = πΊ) β ((πβπ₯)(distβ(π βπ₯))(πβπ₯)) = ((πΉβπ₯)(distβ(π βπ₯))(πΊβπ₯))) |
15 | 14 | adantl 481 | . . . . . 6 β’ ((π β§ (π = πΉ β§ π = πΊ)) β ((πβπ₯)(distβ(π βπ₯))(πβπ₯)) = ((πΉβπ₯)(distβ(π βπ₯))(πΊβπ₯))) |
16 | 15 | mpteq2dv 5243 | . . . . 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 5931 | . . . 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 4157 | . . 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 9443 | . 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 13126 | . . . 4 β’ < Or β* | |
23 | 22 | supex 9460 | . . 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 7556 | 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 395 = wceq 1533 β wcel 2098 Vcvv 3468 βͺ cun 3941 {csn 4623 β¦ cmpt 5224 dom cdm 5669 ran crn 5670 Fn wfn 6532 βcfv 6537 (class class class)co 7405 supcsup 9437 0cc0 11112 β*cxr 11251 < clt 11252 Basecbs 17153 distcds 17215 Xscprds 17400 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-hom 17230 df-cco 17231 df-prds 17402 |
This theorem is referenced by: prdsdsval2 17439 xpsdsval 24242 |
Copyright terms: Public domain | W3C validator |