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Mirrors > Home > MPE Home > Th. List > prdsmslem1 | Structured version Visualization version GIF version |
Description: Lemma for prdsms 24460. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
prdsxms.y | β’ π = (πXsπ ) |
prdsxms.s | β’ (π β π β π) |
prdsxms.i | β’ (π β πΌ β Fin) |
prdsxms.d | β’ π· = (distβπ) |
prdsxms.b | β’ π΅ = (Baseβπ) |
prdsms.r | β’ (π β π :πΌβΆMetSp) |
Ref | Expression |
---|---|
prdsmslem1 | β’ (π β π· β (Metβπ΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . 3 β’ (πXs(π β πΌ β¦ (π βπ))) = (πXs(π β πΌ β¦ (π βπ))) | |
2 | eqid 2728 | . . 3 β’ (Baseβ(πXs(π β πΌ β¦ (π βπ)))) = (Baseβ(πXs(π β πΌ β¦ (π βπ)))) | |
3 | eqid 2728 | . . 3 β’ (Baseβ(π βπ)) = (Baseβ(π βπ)) | |
4 | eqid 2728 | . . 3 β’ ((distβ(π βπ)) βΎ ((Baseβ(π βπ)) Γ (Baseβ(π βπ)))) = ((distβ(π βπ)) βΎ ((Baseβ(π βπ)) Γ (Baseβ(π βπ)))) | |
5 | eqid 2728 | . . 3 β’ (distβ(πXs(π β πΌ β¦ (π βπ)))) = (distβ(πXs(π β πΌ β¦ (π βπ)))) | |
6 | prdsxms.s | . . 3 β’ (π β π β π) | |
7 | prdsxms.i | . . 3 β’ (π β πΌ β Fin) | |
8 | prdsms.r | . . . 4 β’ (π β π :πΌβΆMetSp) | |
9 | 8 | ffvelcdmda 7099 | . . 3 β’ ((π β§ π β πΌ) β (π βπ) β MetSp) |
10 | 3, 4 | msmet 24383 | . . . 4 β’ ((π βπ) β MetSp β ((distβ(π βπ)) βΎ ((Baseβ(π βπ)) Γ (Baseβ(π βπ)))) β (Metβ(Baseβ(π βπ)))) |
11 | 9, 10 | syl 17 | . . 3 β’ ((π β§ π β πΌ) β ((distβ(π βπ)) βΎ ((Baseβ(π βπ)) Γ (Baseβ(π βπ)))) β (Metβ(Baseβ(π βπ)))) |
12 | 1, 2, 3, 4, 5, 6, 7, 9, 11 | prdsmet 24296 | . 2 β’ (π β (distβ(πXs(π β πΌ β¦ (π βπ)))) β (Metβ(Baseβ(πXs(π β πΌ β¦ (π βπ)))))) |
13 | prdsxms.d | . . 3 β’ π· = (distβπ) | |
14 | prdsxms.y | . . . . 5 β’ π = (πXsπ ) | |
15 | 8 | feqmptd 6972 | . . . . . 6 β’ (π β π = (π β πΌ β¦ (π βπ))) |
16 | 15 | oveq2d 7442 | . . . . 5 β’ (π β (πXsπ ) = (πXs(π β πΌ β¦ (π βπ)))) |
17 | 14, 16 | eqtrid 2780 | . . . 4 β’ (π β π = (πXs(π β πΌ β¦ (π βπ)))) |
18 | 17 | fveq2d 6906 | . . 3 β’ (π β (distβπ) = (distβ(πXs(π β πΌ β¦ (π βπ))))) |
19 | 13, 18 | eqtrid 2780 | . 2 β’ (π β π· = (distβ(πXs(π β πΌ β¦ (π βπ))))) |
20 | prdsxms.b | . . . 4 β’ π΅ = (Baseβπ) | |
21 | 17 | fveq2d 6906 | . . . 4 β’ (π β (Baseβπ) = (Baseβ(πXs(π β πΌ β¦ (π βπ))))) |
22 | 20, 21 | eqtrid 2780 | . . 3 β’ (π β π΅ = (Baseβ(πXs(π β πΌ β¦ (π βπ))))) |
23 | 22 | fveq2d 6906 | . 2 β’ (π β (Metβπ΅) = (Metβ(Baseβ(πXs(π β πΌ β¦ (π βπ)))))) |
24 | 12, 19, 23 | 3eltr4d 2844 | 1 β’ (π β π· β (Metβπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β¦ cmpt 5235 Γ cxp 5680 βΎ cres 5684 βΆwf 6549 βcfv 6553 (class class class)co 7426 Fincfn 8970 Basecbs 17187 distcds 17249 Xscprds 17434 Metcmet 21272 MetSpcms 24244 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-icc 13371 df-fz 13525 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-hom 17264 df-cco 17265 df-topgen 17432 df-prds 17436 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-xms 24246 df-ms 24247 |
This theorem is referenced by: prdsms 24460 |
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