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Theorem prdsxms 24508

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 1137	. . . 4 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝑆 ∈ 𝑊)
3		simp2 1138	. . . 4 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝐼 ∈ Fin)
4		eqid 2737	. . . 4 ⊢ (dist‘𝑌) = (dist‘𝑌)
5		eqid 2737	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
6		simp3 1139	. . . 4 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝑅:𝐼⟶∞MetSp)
7	1, 2, 3, 4, 5, 6	prdsxmslem1 24506	. . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (dist‘𝑌) ∈ (∞Met‘(Base‘𝑌)))
8		ssid 3945	. . 3 ⊢ (Base‘𝑌) ⊆ (Base‘𝑌)
9		xmetres2 24339	. . 3 ⊢ (((dist‘𝑌) ∈ (∞Met‘(Base‘𝑌)) ∧ (Base‘𝑌) ⊆ (Base‘𝑌)) → ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) ∈ (∞Met‘(Base‘𝑌)))
10	7, 8, 9	sylancl 587	. 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) ∈ (∞Met‘(Base‘𝑌)))
11		eqid 2737	. . . 4 ⊢ (TopOpen‘𝑌) = (TopOpen‘𝑌)
12		eqid 2737	. . . 4 ⊢ (Base‘(𝑅‘𝑘)) = (Base‘(𝑅‘𝑘))
13		eqid 2737	. . . 4 ⊢ ((dist‘(𝑅‘𝑘)) ↾ ((Base‘(𝑅‘𝑘)) × (Base‘(𝑅‘𝑘)))) = ((dist‘(𝑅‘𝑘)) ↾ ((Base‘(𝑅‘𝑘)) × (Base‘(𝑅‘𝑘))))
14		eqid 2737	. . . 4 ⊢ (TopOpen‘(𝑅‘𝑘)) = (TopOpen‘(𝑅‘𝑘))
15		eqid 2737	. . . 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 24507	. . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (TopOpen‘𝑌) = (MetOpen‘(dist‘𝑌)))
17		xmetf 24307	. . . . 5 ⊢ ((dist‘𝑌) ∈ (∞Met‘(Base‘𝑌)) → (dist‘𝑌):((Base‘𝑌) × (Base‘𝑌))⟶ℝ^*)
18		ffn 6663	. . . . 5 ⊢ ((dist‘𝑌):((Base‘𝑌) × (Base‘𝑌))⟶ℝ^* → (dist‘𝑌) Fn ((Base‘𝑌) × (Base‘𝑌)))
19		fnresdm 6612	. . . . 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 6839	. . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (MetOpen‘((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌)))) = (MetOpen‘(dist‘𝑌)))
22	16, 21	eqtr4d 2775	. 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (TopOpen‘𝑌) = (MetOpen‘((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌)))))
23		eqid 2737	. . 3 ⊢ ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) = ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌)))
24	11, 5, 23	isxms2 24426	. 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 584	1 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝑌 ∈ ∞MetSp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∃wex 1781 ∈ wcel 2114 {cab 2715 ∀wral 3052 ∃wrex 3062 ∖ cdif 3887 ⊆ wss 3890 ∪ cuni 4851 × cxp 5623 ↾ cres 5627 ∘ ccom 5629 Fn wfn 6488 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 Xcixp 8839 Fincfn 8887 ℝ^*cxr 11172 Basecbs 17173 distcds 17223 TopOpenctopn 17378 X_scprds 17402 ∞Metcxmet 21332 MetOpencmopn 21337 ∞MetSpcxms 24295

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fi 9318 df-sup 9349 df-inf 9350 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-q 12893 df-rp 12937 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-icc 13299 df-fz 13456 df-struct 17111 df-slot 17146 df-ndx 17158 df-base 17174 df-plusg 17227 df-mulr 17228 df-sca 17230 df-vsca 17231 df-ip 17232 df-tset 17233 df-ple 17234 df-ds 17236 df-hom 17238 df-cco 17239 df-rest 17379 df-topn 17380 df-topgen 17400 df-pt 17401 df-prds 17404 df-psmet 21339 df-xmet 21340 df-bl 21342 df-mopn 21343 df-top 22872 df-topon 22889 df-topsp 22911 df-bases 22924 df-xms 24298

This theorem is referenced by: prdsms 24509 pwsxms 24510 xpsxms 24512

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