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Theorem prdsxms 23686

Description: The indexed product structure is an extended metric space when the index set is finite. (Although the extended metric is still valid when the index set is infinite, it no longer agrees with the product topology, which is not metrizable in any case.) (Contributed by Mario Carneiro, 28-Aug-2015.)

**Hypothesis**
Ref	Expression
prdsxms.y	⊢ 𝑌 = (𝑆X_s𝑅)

**Assertion**
Ref	Expression
prdsxms	⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝑌 ∈ ∞MetSp)

Proof of Theorem prdsxms Dummy variables 𝑔 𝑘 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdsxms.y	. . . 4 ⊢ 𝑌 = (𝑆X_s𝑅)
2		simp1 1135	. . . 4 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝑆 ∈ 𝑊)
3		simp2 1136	. . . 4 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝐼 ∈ Fin)
4		eqid 2738	. . . 4 ⊢ (dist‘𝑌) = (dist‘𝑌)
5		eqid 2738	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
6		simp3 1137	. . . 4 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝑅:𝐼⟶∞MetSp)
7	1, 2, 3, 4, 5, 6	prdsxmslem1 23684	. . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (dist‘𝑌) ∈ (∞Met‘(Base‘𝑌)))
8		ssid 3943	. . 3 ⊢ (Base‘𝑌) ⊆ (Base‘𝑌)
9		xmetres2 23514	. . 3 ⊢ (((dist‘𝑌) ∈ (∞Met‘(Base‘𝑌)) ∧ (Base‘𝑌) ⊆ (Base‘𝑌)) → ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) ∈ (∞Met‘(Base‘𝑌)))
10	7, 8, 9	sylancl 586	. 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) ∈ (∞Met‘(Base‘𝑌)))
11		eqid 2738	. . . 4 ⊢ (TopOpen‘𝑌) = (TopOpen‘𝑌)
12		eqid 2738	. . . 4 ⊢ (Base‘(𝑅‘𝑘)) = (Base‘(𝑅‘𝑘))
13		eqid 2738	. . . 4 ⊢ ((dist‘(𝑅‘𝑘)) ↾ ((Base‘(𝑅‘𝑘)) × (Base‘(𝑅‘𝑘)))) = ((dist‘(𝑅‘𝑘)) ↾ ((Base‘(𝑅‘𝑘)) × (Base‘(𝑅‘𝑘))))
14		eqid 2738	. . . 4 ⊢ (TopOpen‘(𝑅‘𝑘)) = (TopOpen‘(𝑅‘𝑘))
15		eqid 2738	. . . 4 ⊢ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))} = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))}
16	1, 2, 3, 4, 5, 6, 11, 12, 13, 14, 15	prdsxmslem2 23685	. . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (TopOpen‘𝑌) = (MetOpen‘(dist‘𝑌)))
17		xmetf 23482	. . . . 5 ⊢ ((dist‘𝑌) ∈ (∞Met‘(Base‘𝑌)) → (dist‘𝑌):((Base‘𝑌) × (Base‘𝑌))⟶ℝ^*)
18		ffn 6600	. . . . 5 ⊢ ((dist‘𝑌):((Base‘𝑌) × (Base‘𝑌))⟶ℝ^* → (dist‘𝑌) Fn ((Base‘𝑌) × (Base‘𝑌)))
19		fnresdm 6551	. . . . 5 ⊢ ((dist‘𝑌) Fn ((Base‘𝑌) × (Base‘𝑌)) → ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) = (dist‘𝑌))
20	7, 17, 18, 19	4syl 19	. . . 4 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) = (dist‘𝑌))
21	20	fveq2d 6778	. . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (MetOpen‘((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌)))) = (MetOpen‘(dist‘𝑌)))
22	16, 21	eqtr4d 2781	. 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (TopOpen‘𝑌) = (MetOpen‘((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌)))))
23		eqid 2738	. . 3 ⊢ ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) = ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌)))
24	11, 5, 23	isxms2 23601	. 2 ⊢ (𝑌 ∈ ∞MetSp ↔ (((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) ∈ (∞Met‘(Base‘𝑌)) ∧ (TopOpen‘𝑌) = (MetOpen‘((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))))))
25	10, 22, 24	sylanbrc 583	1 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝑌 ∈ ∞MetSp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∃wex 1782 ∈ wcel 2106 {cab 2715 ∀wral 3064 ∃wrex 3065 ∖ cdif 3884 ⊆ wss 3887 ∪ cuni 4839 × cxp 5587 ↾ cres 5591 ∘ ccom 5593 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 Xcixp 8685 Fincfn 8733 ℝ^*cxr 11008 Basecbs 16912 distcds 16971 TopOpenctopn 17132 X_scprds 17156 ∞Metcxmet 20582 MetOpencmopn 20587 ∞MetSpcxms 23470

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fi 9170 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-icc 13086 df-fz 13240 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-topgen 17154 df-pt 17155 df-prds 17158 df-psmet 20589 df-xmet 20590 df-bl 20592 df-mopn 20593 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-xms 23473

This theorem is referenced by: prdsms 23687 pwsxms 23688 xpsxms 23690

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