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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 23064 | . . . . 5 ⊢ (𝑥 ∈ MetSp → 𝑥 ∈ ∞MetSp) | |
2 | 1 | ssriv 3971 | . . . 4 ⊢ MetSp ⊆ ∞MetSp |
3 | fss 6527 | . . . 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 23140 | . . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝑌 ∈ ∞MetSp) |
7 | 4, 6 | syl3an3 1161 | . 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → 𝑌 ∈ ∞MetSp) |
8 | simp1 1132 | . . . 4 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → 𝑆 ∈ 𝑊) | |
9 | simp2 1133 | . . . 4 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → 𝐼 ∈ Fin) | |
10 | eqid 2821 | . . . 4 ⊢ (dist‘𝑌) = (dist‘𝑌) | |
11 | eqid 2821 | . . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
12 | simp3 1134 | . . . 4 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → 𝑅:𝐼⟶MetSp) | |
13 | 5, 8, 9, 10, 11, 12 | prdsmslem1 23137 | . . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → (dist‘𝑌) ∈ (Met‘(Base‘𝑌))) |
14 | ssid 3989 | . . 3 ⊢ (Base‘𝑌) ⊆ (Base‘𝑌) | |
15 | metres2 22973 | . . 3 ⊢ (((dist‘𝑌) ∈ (Met‘(Base‘𝑌)) ∧ (Base‘𝑌) ⊆ (Base‘𝑌)) → ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) ∈ (Met‘(Base‘𝑌))) | |
16 | 13, 14, 15 | sylancl 588 | . 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) ∈ (Met‘(Base‘𝑌))) |
17 | eqid 2821 | . . 3 ⊢ (TopOpen‘𝑌) = (TopOpen‘𝑌) | |
18 | eqid 2821 | . . 3 ⊢ ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) = ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) | |
19 | 17, 11, 18 | isms 23059 | . 2 ⊢ (𝑌 ∈ MetSp ↔ (𝑌 ∈ ∞MetSp ∧ ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) ∈ (Met‘(Base‘𝑌)))) |
20 | 7, 16, 19 | sylanbrc 585 | 1 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → 𝑌 ∈ MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 × cxp 5553 ↾ cres 5557 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 Basecbs 16483 distcds 16574 TopOpenctopn 16695 Xscprds 16719 Metcmet 20531 ∞MetSpcxms 22927 MetSpcms 22928 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fi 8875 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-icc 12746 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-topgen 16717 df-pt 16718 df-prds 16721 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-xms 22930 df-ms 22931 |
This theorem is referenced by: pwsms 23143 xpsms 23145 |
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