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Theorem prdsxms 23142

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 1132	. . . 4 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝑆 ∈ 𝑊)
3		simp2 1133	. . . 4 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝐼 ∈ Fin)
4		eqid 2823	. . . 4 ⊢ (dist‘𝑌) = (dist‘𝑌)
5		eqid 2823	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
6		simp3 1134	. . . 4 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝑅:𝐼⟶∞MetSp)
7	1, 2, 3, 4, 5, 6	prdsxmslem1 23140	. . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (dist‘𝑌) ∈ (∞Met‘(Base‘𝑌)))
8		ssid 3991	. . 3 ⊢ (Base‘𝑌) ⊆ (Base‘𝑌)
9		xmetres2 22973	. . 3 ⊢ (((dist‘𝑌) ∈ (∞Met‘(Base‘𝑌)) ∧ (Base‘𝑌) ⊆ (Base‘𝑌)) → ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) ∈ (∞Met‘(Base‘𝑌)))
10	7, 8, 9	sylancl 588	. 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) ∈ (∞Met‘(Base‘𝑌)))
11		eqid 2823	. . . 4 ⊢ (TopOpen‘𝑌) = (TopOpen‘𝑌)
12		eqid 2823	. . . 4 ⊢ (Base‘(𝑅‘𝑘)) = (Base‘(𝑅‘𝑘))
13		eqid 2823	. . . 4 ⊢ ((dist‘(𝑅‘𝑘)) ↾ ((Base‘(𝑅‘𝑘)) × (Base‘(𝑅‘𝑘)))) = ((dist‘(𝑅‘𝑘)) ↾ ((Base‘(𝑅‘𝑘)) × (Base‘(𝑅‘𝑘))))
14		eqid 2823	. . . 4 ⊢ (TopOpen‘(𝑅‘𝑘)) = (TopOpen‘(𝑅‘𝑘))
15		eqid 2823	. . . 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 23141	. . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (TopOpen‘𝑌) = (MetOpen‘(dist‘𝑌)))
17		xmetf 22941	. . . . 5 ⊢ ((dist‘𝑌) ∈ (∞Met‘(Base‘𝑌)) → (dist‘𝑌):((Base‘𝑌) × (Base‘𝑌))⟶ℝ^*)
18		ffn 6516	. . . . 5 ⊢ ((dist‘𝑌):((Base‘𝑌) × (Base‘𝑌))⟶ℝ^* → (dist‘𝑌) Fn ((Base‘𝑌) × (Base‘𝑌)))
19		fnresdm 6468	. . . . 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 6676	. . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (MetOpen‘((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌)))) = (MetOpen‘(dist‘𝑌)))
22	16, 21	eqtr4d 2861	. 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → (TopOpen‘𝑌) = (MetOpen‘((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌)))))
23		eqid 2823	. . 3 ⊢ ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌))) = ((dist‘𝑌) ↾ ((Base‘𝑌) × (Base‘𝑌)))
24	11, 5, 23	isxms2 23060	. 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 585	1 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝑌 ∈ ∞MetSp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2801 ∀wral 3140 ∃wrex 3141 ∖ cdif 3935 ⊆ wss 3938 ∪ cuni 4840 × cxp 5555 ↾ cres 5559 ∘ ccom 5561 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Xcixp 8463 Fincfn 8511 ℝ^*cxr 10676 Basecbs 16485 distcds 16576 TopOpenctopn 16697 X_scprds 16721 ∞Metcxmet 20532 MetOpencmopn 20537 ∞MetSpcxms 22929

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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fi 8877 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-icc 12748 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-topgen 16719 df-pt 16720 df-prds 16723 df-psmet 20539 df-xmet 20540 df-bl 20542 df-mopn 20543 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-xms 22932

This theorem is referenced by: prdsms 23143 pwsxms 23144 xpsxms 23146

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