![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cnmpt2ds | Structured version Visualization version GIF version |
Description: Continuity of the metric function; analogue of cnmpt22f 21898 which cannot be used directly because 𝐷 is not necessarily a function. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
cnmpt1ds.d | ⊢ 𝐷 = (dist‘𝐺) |
cnmpt1ds.j | ⊢ 𝐽 = (TopOpen‘𝐺) |
cnmpt1ds.r | ⊢ 𝑅 = (topGen‘ran (,)) |
cnmpt1ds.g | ⊢ (𝜑 → 𝐺 ∈ MetSp) |
cnmpt1ds.k | ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋)) |
cnmpt2ds.l | ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌)) |
cnmpt2ds.a | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)) |
cnmpt2ds.b | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)) |
Ref | Expression |
---|---|
cnmpt2ds | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴𝐷𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmpt1ds.k | . . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋)) | |
2 | cnmpt2ds.l | . . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌)) | |
3 | txtopon 21814 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌))) | |
4 | 1, 2, 3 | syl2anc 579 | . . . . . . . . 9 ⊢ (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌))) |
5 | cnmpt1ds.g | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ MetSp) | |
6 | mstps 22679 | . . . . . . . . . . 11 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ TopSp) | |
7 | 5, 6 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ TopSp) |
8 | eqid 2778 | . . . . . . . . . . 11 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
9 | cnmpt1ds.j | . . . . . . . . . . 11 ⊢ 𝐽 = (TopOpen‘𝐺) | |
10 | 8, 9 | istps 21157 | . . . . . . . . . 10 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝐺))) |
11 | 7, 10 | sylib 210 | . . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘𝐺))) |
12 | cnmpt2ds.a | . . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)) | |
13 | cnf2 21472 | . . . . . . . . 9 ⊢ (((𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶(Base‘𝐺)) | |
14 | 4, 11, 12, 13 | syl3anc 1439 | . . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶(Base‘𝐺)) |
15 | eqid 2778 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) | |
16 | 15 | fmpt2 7519 | . . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ (Base‘𝐺) ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶(Base‘𝐺)) |
17 | 14, 16 | sylibr 226 | . . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ (Base‘𝐺)) |
18 | 17 | r19.21bi 3114 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 𝐴 ∈ (Base‘𝐺)) |
19 | 18 | r19.21bi 3114 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ (Base‘𝐺)) |
20 | cnmpt2ds.b | . . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)) | |
21 | cnf2 21472 | . . . . . . . . 9 ⊢ (((𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶(Base‘𝐺)) | |
22 | 4, 11, 20, 21 | syl3anc 1439 | . . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶(Base‘𝐺)) |
23 | eqid 2778 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) | |
24 | 23 | fmpt2 7519 | . . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ (Base‘𝐺) ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶(Base‘𝐺)) |
25 | 22, 24 | sylibr 226 | . . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ (Base‘𝐺)) |
26 | 25 | r19.21bi 3114 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 𝐵 ∈ (Base‘𝐺)) |
27 | 26 | r19.21bi 3114 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ (Base‘𝐺)) |
28 | 19, 27 | ovresd 7080 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝐴(𝐷 ↾ ((Base‘𝐺) × (Base‘𝐺)))𝐵) = (𝐴𝐷𝐵)) |
29 | 28 | 3impa 1097 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (𝐴(𝐷 ↾ ((Base‘𝐺) × (Base‘𝐺)))𝐵) = (𝐴𝐷𝐵)) |
30 | 29 | mpt2eq3dva 6998 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴(𝐷 ↾ ((Base‘𝐺) × (Base‘𝐺)))𝐵)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴𝐷𝐵))) |
31 | cnmpt1ds.d | . . . . 5 ⊢ 𝐷 = (dist‘𝐺) | |
32 | cnmpt1ds.r | . . . . 5 ⊢ 𝑅 = (topGen‘ran (,)) | |
33 | 8, 31, 9, 32 | msdcn 23063 | . . . 4 ⊢ (𝐺 ∈ MetSp → (𝐷 ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ ((𝐽 ×t 𝐽) Cn 𝑅)) |
34 | 5, 33 | syl 17 | . . 3 ⊢ (𝜑 → (𝐷 ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ ((𝐽 ×t 𝐽) Cn 𝑅)) |
35 | 1, 2, 12, 20, 34 | cnmpt22f 21898 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴(𝐷 ↾ ((Base‘𝐺) × (Base‘𝐺)))𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝑅)) |
36 | 30, 35 | eqeltrrd 2860 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴𝐷𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 × cxp 5355 ran crn 5358 ↾ cres 5359 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 ↦ cmpt2 6926 (,)cioo 12492 Basecbs 16266 distcds 16358 TopOpenctopn 16479 topGenctg 16495 TopOnctopon 21133 TopSpctps 21155 Cn ccn 21447 ×t ctx 21783 MetSpcms 22542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-ec 8030 df-map 8144 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-fi 8607 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-dec 11851 df-uz 11998 df-q 12101 df-rp 12143 df-xneg 12262 df-xadd 12263 df-xmul 12264 df-ioo 12496 df-ioc 12497 df-ico 12498 df-icc 12499 df-fz 12649 df-fzo 12790 df-seq 13125 df-exp 13184 df-hash 13442 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-sca 16365 df-vsca 16366 df-ip 16367 df-tset 16368 df-ple 16369 df-ds 16371 df-hom 16373 df-cco 16374 df-rest 16480 df-topn 16481 df-0g 16499 df-gsum 16500 df-topgen 16501 df-pt 16502 df-prds 16505 df-ordt 16558 df-xrs 16559 df-qtop 16564 df-imas 16565 df-xps 16567 df-mre 16643 df-mrc 16644 df-acs 16646 df-ps 17597 df-tsr 17598 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-submnd 17733 df-mulg 17939 df-cntz 18144 df-cmn 18592 df-psmet 20145 df-xmet 20146 df-met 20147 df-bl 20148 df-mopn 20149 df-top 21117 df-topon 21134 df-topsp 21156 df-bases 21169 df-cn 21450 df-cnp 21451 df-tx 21785 df-hmeo 21978 df-xms 22544 df-ms 22545 df-tms 22546 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |