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Theorem prdsxms 24543

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 24541	. . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (dist‘𝑌) ∈ (∞Met‘(Base‘𝑌)))
8		ssid 4006	. . 3 ⊢ (Base‘𝑌) ⊆ (Base‘𝑌)
9		xmetres2 24371	. . 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 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 24542	. . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (TopOpen‘𝑌) = (MetOpen‘(dist‘𝑌)))
17		xmetf 24339	. . . . 5 ⊢ ((dist‘𝑌) ∈ (∞Met‘(Base‘𝑌)) → (dist‘𝑌):((Base‘𝑌) × (Base‘𝑌))⟶ℝ^*)
18		ffn 6736	. . . . 5 ⊢ ((dist‘𝑌):((Base‘𝑌) × (Base‘𝑌))⟶ℝ^* → (dist‘𝑌) Fn ((Base‘𝑌) × (Base‘𝑌)))
19		fnresdm 6687	. . . . 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 6910	. . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (MetOpen‘((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌)))) = (MetOpen‘(dist‘𝑌)))
22	16, 21	eqtr4d 2780	. 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 24458	. 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 395 ∧ w3a 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 ∖ cdif 3948 ⊆ wss 3951 ∪ cuni 4907 × cxp 5683 ↾ cres 5687 ∘ ccom 5689 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Xcixp 8937 Fincfn 8985 ℝ^*cxr 11294 Basecbs 17247 distcds 17306 TopOpenctopn 17466 X_scprds 17490 ∞Metcxmet 21349 MetOpencmopn 21354 ∞MetSpcxms 24327

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-icc 13394 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-topgen 17488 df-pt 17489 df-prds 17492 df-psmet 21356 df-xmet 21357 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-xms 24330

This theorem is referenced by: prdsms 24544 pwsxms 24545 xpsxms 24547

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