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Mirrors > Home > ILE Home > Th. List > metcnpd | GIF version |
Description: Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous at point 𝑃. (Contributed by Jim Kingdon, 14-Jun-2023.) |
Ref | Expression |
---|---|
metcnpd.j | ⊢ (𝜑 → 𝐽 = (MetOpen‘𝐶)) |
metcnpd.k | ⊢ (𝜑 → 𝐾 = (MetOpen‘𝐷)) |
metcnpd.c | ⊢ (𝜑 → 𝐶 ∈ (∞Met‘𝑋)) |
metcnpd.d | ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑌)) |
metcnpd.p | ⊢ (𝜑 → 𝑃 ∈ 𝑋) |
Ref | Expression |
---|---|
metcnpd | ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metcnpd.j | . . . . 5 ⊢ (𝜑 → 𝐽 = (MetOpen‘𝐶)) | |
2 | metcnpd.k | . . . . 5 ⊢ (𝜑 → 𝐾 = (MetOpen‘𝐷)) | |
3 | 1, 2 | oveq12d 5792 | . . . 4 ⊢ (𝜑 → (𝐽 CnP 𝐾) = ((MetOpen‘𝐶) CnP (MetOpen‘𝐷))) |
4 | 3 | fveq1d 5423 | . . 3 ⊢ (𝜑 → ((𝐽 CnP 𝐾)‘𝑃) = (((MetOpen‘𝐶) CnP (MetOpen‘𝐷))‘𝑃)) |
5 | 4 | eleq2d 2209 | . 2 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ (((MetOpen‘𝐶) CnP (MetOpen‘𝐷))‘𝑃))) |
6 | metcnpd.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ (∞Met‘𝑋)) | |
7 | metcnpd.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑌)) | |
8 | metcnpd.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ 𝑋) | |
9 | eqid 2139 | . . . 4 ⊢ (MetOpen‘𝐶) = (MetOpen‘𝐶) | |
10 | eqid 2139 | . . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
11 | 9, 10 | metcnp 12686 | . . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ (((MetOpen‘𝐶) CnP (MetOpen‘𝐷))‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦)))) |
12 | 6, 7, 8, 11 | syl3anc 1216 | . 2 ⊢ (𝜑 → (𝐹 ∈ (((MetOpen‘𝐶) CnP (MetOpen‘𝐷))‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦)))) |
13 | 5, 12 | bitrd 187 | 1 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 class class class wbr 3929 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 < clt 7805 ℝ+crp 9446 ∞Metcxmet 12154 MetOpencmopn 12159 CnP ccnp 12360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7716 ax-resscn 7717 ax-1cn 7718 ax-1re 7719 ax-icn 7720 ax-addcl 7721 ax-addrcl 7722 ax-mulcl 7723 ax-mulrcl 7724 ax-addcom 7725 ax-mulcom 7726 ax-addass 7727 ax-mulass 7728 ax-distr 7729 ax-i2m1 7730 ax-0lt1 7731 ax-1rid 7732 ax-0id 7733 ax-rnegex 7734 ax-precex 7735 ax-cnre 7736 ax-pre-ltirr 7737 ax-pre-ltwlin 7738 ax-pre-lttrn 7739 ax-pre-apti 7740 ax-pre-ltadd 7741 ax-pre-mulgt0 7742 ax-pre-mulext 7743 ax-arch 7744 ax-caucvg 7745 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-map 6544 df-sup 6871 df-inf 6872 df-pnf 7807 df-mnf 7808 df-xr 7809 df-ltxr 7810 df-le 7811 df-sub 7940 df-neg 7941 df-reap 8342 df-ap 8349 df-div 8438 df-inn 8726 df-2 8784 df-3 8785 df-4 8786 df-n0 8983 df-z 9060 df-uz 9332 df-q 9417 df-rp 9447 df-xneg 9564 df-xadd 9565 df-seqfrec 10224 df-exp 10298 df-cj 10619 df-re 10620 df-im 10621 df-rsqrt 10775 df-abs 10776 df-topgen 12146 df-psmet 12161 df-xmet 12162 df-bl 12164 df-mopn 12165 df-top 12170 df-topon 12183 df-bases 12215 df-cnp 12363 |
This theorem is referenced by: cnplimcim 12810 limccnpcntop 12818 |
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