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Mirrors > Home > MPE Home > Th. List > prdsms | Structured version Visualization version GIF version |
Description: The indexed product structure is a metric space when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
prdsxms.y | ⊢ 𝑌 = (𝑆Xs𝑅) |
Ref | Expression |
---|---|
prdsms | ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → 𝑌 ∈ MetSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | msxms 23803 | . . . . 5 ⊢ (𝑥 ∈ MetSp → 𝑥 ∈ ∞MetSp) | |
2 | 1 | ssriv 3947 | . . . 4 ⊢ MetSp ⊆ ∞MetSp |
3 | fss 6683 | . . . 4 ⊢ ((𝑅:𝐼⟶MetSp ∧ MetSp ⊆ ∞MetSp) → 𝑅:𝐼⟶∞MetSp) | |
4 | 2, 3 | mpan2 689 | . . 3 ⊢ (𝑅:𝐼⟶MetSp → 𝑅:𝐼⟶∞MetSp) |
5 | prdsxms.y | . . . 4 ⊢ 𝑌 = (𝑆Xs𝑅) | |
6 | 5 | prdsxms 23882 | . . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝑌 ∈ ∞MetSp) |
7 | 4, 6 | syl3an3 1165 | . 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → 𝑌 ∈ ∞MetSp) |
8 | simp1 1136 | . . . 4 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → 𝑆 ∈ 𝑊) | |
9 | simp2 1137 | . . . 4 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → 𝐼 ∈ Fin) | |
10 | eqid 2736 | . . . 4 ⊢ (dist‘𝑌) = (dist‘𝑌) | |
11 | eqid 2736 | . . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
12 | simp3 1138 | . . . 4 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → 𝑅:𝐼⟶MetSp) | |
13 | 5, 8, 9, 10, 11, 12 | prdsmslem1 23879 | . . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → (dist‘𝑌) ∈ (Met‘(Base‘𝑌))) |
14 | ssid 3965 | . . 3 ⊢ (Base‘𝑌) ⊆ (Base‘𝑌) | |
15 | metres2 23712 | . . 3 ⊢ (((dist‘𝑌) ∈ (Met‘(Base‘𝑌)) ∧ (Base‘𝑌) ⊆ (Base‘𝑌)) → ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) ∈ (Met‘(Base‘𝑌))) | |
16 | 13, 14, 15 | sylancl 586 | . 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) ∈ (Met‘(Base‘𝑌))) |
17 | eqid 2736 | . . 3 ⊢ (TopOpen‘𝑌) = (TopOpen‘𝑌) | |
18 | eqid 2736 | . . 3 ⊢ ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) = ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) | |
19 | 17, 11, 18 | isms 23798 | . 2 ⊢ (𝑌 ∈ MetSp ↔ (𝑌 ∈ ∞MetSp ∧ ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) ∈ (Met‘(Base‘𝑌)))) |
20 | 7, 16, 19 | sylanbrc 583 | 1 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → 𝑌 ∈ MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ⊆ wss 3909 × cxp 5630 ↾ cres 5634 ⟶wf 6490 ‘cfv 6494 (class class class)co 7354 Fincfn 8880 Basecbs 17080 distcds 17139 TopOpenctopn 17300 Xscprds 17324 Metcmet 20778 ∞MetSpcxms 23666 MetSpcms 23667 |
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 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 ax-pre-sup 11126 |
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 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-1st 7918 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-1o 8409 df-er 8645 df-map 8764 df-ixp 8833 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 df-fi 9344 df-sup 9375 df-inf 9376 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-div 11810 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12411 df-z 12497 df-dec 12616 df-uz 12761 df-q 12871 df-rp 12913 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-icc 13268 df-fz 13422 df-struct 17016 df-slot 17051 df-ndx 17063 df-base 17081 df-plusg 17143 df-mulr 17144 df-sca 17146 df-vsca 17147 df-ip 17148 df-tset 17149 df-ple 17150 df-ds 17152 df-hom 17154 df-cco 17155 df-rest 17301 df-topn 17302 df-topgen 17322 df-pt 17323 df-prds 17326 df-psmet 20784 df-xmet 20785 df-met 20786 df-bl 20787 df-mopn 20788 df-top 22239 df-topon 22256 df-topsp 22278 df-bases 22292 df-xms 23669 df-ms 23670 |
This theorem is referenced by: pwsms 23885 xpsms 23887 |
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