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Theorem prdsxms 24021

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 2733	. . . 4 ⊢ (dist‘𝑌) = (dist‘𝑌)
5		eqid 2733	. . . 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 24019	. . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (dist‘𝑌) ∈ (∞Met‘(Base‘𝑌)))
8		ssid 4003	. . 3 ⊢ (Base‘𝑌) ⊆ (Base‘𝑌)
9		xmetres2 23849	. . 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 2733	. . . 4 ⊢ (TopOpen‘𝑌) = (TopOpen‘𝑌)
12		eqid 2733	. . . 4 ⊢ (Base‘(𝑅‘𝑘)) = (Base‘(𝑅‘𝑘))
13		eqid 2733	. . . 4 ⊢ ((dist‘(𝑅‘𝑘)) ↾ ((Base‘(𝑅‘𝑘)) × (Base‘(𝑅‘𝑘)))) = ((dist‘(𝑅‘𝑘)) ↾ ((Base‘(𝑅‘𝑘)) × (Base‘(𝑅‘𝑘))))
14		eqid 2733	. . . 4 ⊢ (TopOpen‘(𝑅‘𝑘)) = (TopOpen‘(𝑅‘𝑘))
15		eqid 2733	. . . 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 24020	. . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (TopOpen‘𝑌) = (MetOpen‘(dist‘𝑌)))
17		xmetf 23817	. . . . 5 ⊢ ((dist‘𝑌) ∈ (∞Met‘(Base‘𝑌)) → (dist‘𝑌):((Base‘𝑌) × (Base‘𝑌))⟶ℝ^*)
18		ffn 6714	. . . . 5 ⊢ ((dist‘𝑌):((Base‘𝑌) × (Base‘𝑌))⟶ℝ^* → (dist‘𝑌) Fn ((Base‘𝑌) × (Base‘𝑌)))
19		fnresdm 6666	. . . . 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 6892	. . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (MetOpen‘((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌)))) = (MetOpen‘(dist‘𝑌)))
22	16, 21	eqtr4d 2776	. 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (TopOpen‘𝑌) = (MetOpen‘((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌)))))
23		eqid 2733	. . 3 ⊢ ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) = ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌)))
24	11, 5, 23	isxms2 23936	. 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 397 ∧ w3a 1088 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2710 ∀wral 3062 ∃wrex 3071 ∖ cdif 3944 ⊆ wss 3947 ∪ cuni 4907 × cxp 5673 ↾ cres 5677 ∘ ccom 5679 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 Xcixp 8887 Fincfn 8935 ℝ^*cxr 11243 Basecbs 17140 distcds 17202 TopOpenctopn 17363 X_scprds 17387 ∞Metcxmet 20914 MetOpencmopn 20919 ∞MetSpcxms 23805

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-icc 13327 df-fz 13481 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-topgen 17385 df-pt 17386 df-prds 17389 df-psmet 20921 df-xmet 20922 df-bl 20924 df-mopn 20925 df-top 22378 df-topon 22395 df-topsp 22417 df-bases 22431 df-xms 23808

This theorem is referenced by: prdsms 24022 pwsxms 24023 xpsxms 24025

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