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Mirrors > Home > MPE Home > Th. List > prdsxmslem1 | Structured version Visualization version GIF version |
Description: Lemma for prdsms 23887. 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) |
prdsxms.r | β’ (π β π :πΌβΆβMetSp) |
Ref | Expression |
---|---|
prdsxmslem1 | β’ (π β π· β (βMetβπ΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 β’ (πXs(π β πΌ β¦ (π βπ))) = (πXs(π β πΌ β¦ (π βπ))) | |
2 | eqid 2736 | . . 3 β’ (Baseβ(πXs(π β πΌ β¦ (π βπ)))) = (Baseβ(πXs(π β πΌ β¦ (π βπ)))) | |
3 | eqid 2736 | . . 3 β’ (Baseβ(π βπ)) = (Baseβ(π βπ)) | |
4 | eqid 2736 | . . 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 2736 | . . 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 | prdsxms.r | . . . 4 β’ (π β π :πΌβΆβMetSp) | |
9 | 8 | ffvelcdmda 7035 | . . 3 β’ ((π β§ π β πΌ) β (π βπ) β βMetSp) |
10 | 3, 4 | xmsxmet 23809 | . . . 4 β’ ((π βπ) β βMetSp β ((distβ(π βπ)) βΎ ((Baseβ(π βπ)) Γ (Baseβ(π βπ)))) β (βMetβ(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 (βMetβ(Baseβ(π βπ)))) |
12 | 1, 2, 3, 4, 5, 6, 7, 9, 11 | prdsxmet 23722 | . 2 β’ (π β (distβ(πXs(π β πΌ β¦ (π βπ)))) β (βMetβ(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 6910 | . . . . . 6 β’ (π β π = (π β πΌ β¦ (π βπ))) |
16 | 15 | oveq2d 7373 | . . . . 5 β’ (π β (πXsπ ) = (πXs(π β πΌ β¦ (π βπ)))) |
17 | 14, 16 | eqtrid 2788 | . . . 4 β’ (π β π = (πXs(π β πΌ β¦ (π βπ)))) |
18 | 17 | fveq2d 6846 | . . 3 β’ (π β (distβπ) = (distβ(πXs(π β πΌ β¦ (π βπ))))) |
19 | 13, 18 | eqtrid 2788 | . 2 β’ (π β π· = (distβ(πXs(π β πΌ β¦ (π βπ))))) |
20 | prdsxms.b | . . . 4 β’ π΅ = (Baseβπ) | |
21 | 17 | fveq2d 6846 | . . . 4 β’ (π β (Baseβπ) = (Baseβ(πXs(π β πΌ β¦ (π βπ))))) |
22 | 20, 21 | eqtrid 2788 | . . 3 β’ (π β π΅ = (Baseβ(πXs(π β πΌ β¦ (π βπ))))) |
23 | 22 | fveq2d 6846 | . 2 β’ (π β (βMetβπ΅) = (βMetβ(Baseβ(πXs(π β πΌ β¦ (π βπ)))))) |
24 | 12, 19, 23 | 3eltr4d 2852 | 1 β’ (π β π· β (βMetβπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β¦ cmpt 5188 Γ cxp 5631 βΎ cres 5635 βΆwf 6492 βcfv 6496 (class class class)co 7357 Fincfn 8883 Basecbs 17083 distcds 17142 Xscprds 17327 βMetcxmet 20781 βMetSpcxms 23670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-icc 13271 df-fz 13425 df-struct 17019 df-slot 17054 df-ndx 17066 df-base 17084 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-hom 17157 df-cco 17158 df-topgen 17325 df-prds 17329 df-psmet 20788 df-xmet 20789 df-bl 20791 df-mopn 20792 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-xms 23673 |
This theorem is referenced by: prdsxmslem2 23885 prdsxms 23886 |
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