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Mirrors > Home > MPE Home > Th. List > prdsdsval3 | Structured version Visualization version GIF version |
Description: Value of the metric in a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.) |
Ref | Expression |
---|---|
prdsbasmpt2.y | β’ π = (πXs(π₯ β πΌ β¦ π )) |
prdsbasmpt2.b | β’ π΅ = (Baseβπ) |
prdsbasmpt2.s | β’ (π β π β π) |
prdsbasmpt2.i | β’ (π β πΌ β π) |
prdsbasmpt2.r | β’ (π β βπ₯ β πΌ π β π) |
prdsdsval2.f | β’ (π β πΉ β π΅) |
prdsdsval2.g | β’ (π β πΊ β π΅) |
prdsdsval3.k | β’ πΎ = (Baseβπ ) |
prdsdsval3.e | β’ πΈ = ((distβπ ) βΎ (πΎ Γ πΎ)) |
prdsdsval3.d | β’ π· = (distβπ) |
Ref | Expression |
---|---|
prdsdsval3 | β’ (π β (πΉπ·πΊ) = sup((ran (π₯ β πΌ β¦ ((πΉβπ₯)πΈ(πΊβπ₯))) βͺ {0}), β*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsbasmpt2.y | . . 3 β’ π = (πXs(π₯ β πΌ β¦ π )) | |
2 | prdsbasmpt2.b | . . 3 β’ π΅ = (Baseβπ) | |
3 | prdsbasmpt2.s | . . 3 β’ (π β π β π) | |
4 | prdsbasmpt2.i | . . 3 β’ (π β πΌ β π) | |
5 | prdsbasmpt2.r | . . 3 β’ (π β βπ₯ β πΌ π β π) | |
6 | prdsdsval2.f | . . 3 β’ (π β πΉ β π΅) | |
7 | prdsdsval2.g | . . 3 β’ (π β πΊ β π΅) | |
8 | eqid 2737 | . . 3 β’ (distβπ ) = (distβπ ) | |
9 | prdsdsval3.d | . . 3 β’ π· = (distβπ) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | prdsdsval2 17367 | . 2 β’ (π β (πΉπ·πΊ) = sup((ran (π₯ β πΌ β¦ ((πΉβπ₯)(distβπ )(πΊβπ₯))) βͺ {0}), β*, < )) |
11 | eqidd 2738 | . . . . . 6 β’ (π β πΌ = πΌ) | |
12 | prdsdsval3.k | . . . . . . . 8 β’ πΎ = (Baseβπ ) | |
13 | 1, 2, 3, 4, 5, 12, 6 | prdsbascl 17366 | . . . . . . 7 β’ (π β βπ₯ β πΌ (πΉβπ₯) β πΎ) |
14 | 1, 2, 3, 4, 5, 12, 7 | prdsbascl 17366 | . . . . . . 7 β’ (π β βπ₯ β πΌ (πΊβπ₯) β πΎ) |
15 | prdsdsval3.e | . . . . . . . . . . 11 β’ πΈ = ((distβπ ) βΎ (πΎ Γ πΎ)) | |
16 | 15 | oveqi 7371 | . . . . . . . . . 10 β’ ((πΉβπ₯)πΈ(πΊβπ₯)) = ((πΉβπ₯)((distβπ ) βΎ (πΎ Γ πΎ))(πΊβπ₯)) |
17 | ovres 7521 | . . . . . . . . . 10 β’ (((πΉβπ₯) β πΎ β§ (πΊβπ₯) β πΎ) β ((πΉβπ₯)((distβπ ) βΎ (πΎ Γ πΎ))(πΊβπ₯)) = ((πΉβπ₯)(distβπ )(πΊβπ₯))) | |
18 | 16, 17 | eqtrid 2789 | . . . . . . . . 9 β’ (((πΉβπ₯) β πΎ β§ (πΊβπ₯) β πΎ) β ((πΉβπ₯)πΈ(πΊβπ₯)) = ((πΉβπ₯)(distβπ )(πΊβπ₯))) |
19 | 18 | ex 414 | . . . . . . . 8 β’ ((πΉβπ₯) β πΎ β ((πΊβπ₯) β πΎ β ((πΉβπ₯)πΈ(πΊβπ₯)) = ((πΉβπ₯)(distβπ )(πΊβπ₯)))) |
20 | 19 | ral2imi 3089 | . . . . . . 7 β’ (βπ₯ β πΌ (πΉβπ₯) β πΎ β (βπ₯ β πΌ (πΊβπ₯) β πΎ β βπ₯ β πΌ ((πΉβπ₯)πΈ(πΊβπ₯)) = ((πΉβπ₯)(distβπ )(πΊβπ₯)))) |
21 | 13, 14, 20 | sylc 65 | . . . . . 6 β’ (π β βπ₯ β πΌ ((πΉβπ₯)πΈ(πΊβπ₯)) = ((πΉβπ₯)(distβπ )(πΊβπ₯))) |
22 | mpteq12 5198 | . . . . . 6 β’ ((πΌ = πΌ β§ βπ₯ β πΌ ((πΉβπ₯)πΈ(πΊβπ₯)) = ((πΉβπ₯)(distβπ )(πΊβπ₯))) β (π₯ β πΌ β¦ ((πΉβπ₯)πΈ(πΊβπ₯))) = (π₯ β πΌ β¦ ((πΉβπ₯)(distβπ )(πΊβπ₯)))) | |
23 | 11, 21, 22 | syl2anc 585 | . . . . 5 β’ (π β (π₯ β πΌ β¦ ((πΉβπ₯)πΈ(πΊβπ₯))) = (π₯ β πΌ β¦ ((πΉβπ₯)(distβπ )(πΊβπ₯)))) |
24 | 23 | rneqd 5894 | . . . 4 β’ (π β ran (π₯ β πΌ β¦ ((πΉβπ₯)πΈ(πΊβπ₯))) = ran (π₯ β πΌ β¦ ((πΉβπ₯)(distβπ )(πΊβπ₯)))) |
25 | 24 | uneq1d 4123 | . . 3 β’ (π β (ran (π₯ β πΌ β¦ ((πΉβπ₯)πΈ(πΊβπ₯))) βͺ {0}) = (ran (π₯ β πΌ β¦ ((πΉβπ₯)(distβπ )(πΊβπ₯))) βͺ {0})) |
26 | 25 | supeq1d 9383 | . 2 β’ (π β sup((ran (π₯ β πΌ β¦ ((πΉβπ₯)πΈ(πΊβπ₯))) βͺ {0}), β*, < ) = sup((ran (π₯ β πΌ β¦ ((πΉβπ₯)(distβπ )(πΊβπ₯))) βͺ {0}), β*, < )) |
27 | 10, 26 | eqtr4d 2780 | 1 β’ (π β (πΉπ·πΊ) = sup((ran (π₯ β πΌ β¦ ((πΉβπ₯)πΈ(πΊβπ₯))) βͺ {0}), β*, < )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3065 βͺ cun 3909 {csn 4587 β¦ cmpt 5189 Γ cxp 5632 ran crn 5635 βΎ cres 5636 βcfv 6497 (class class class)co 7358 supcsup 9377 0cc0 11052 β*cxr 11189 < clt 11190 Basecbs 17084 distcds 17143 Xscprds 17328 |
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-map 8768 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9379 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-hom 17158 df-cco 17159 df-prds 17330 |
This theorem is referenced by: prdsxmetlem 23724 prdsmet 23726 prdsbl 23850 prdsbnd 36255 rrnequiv 36297 |
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