metcnpd - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > metcnpd		GIF version

Theorem metcnpd 15188

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.)

**Hypotheses**
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)

**Assertion**
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))))

Distinct variable groups: 𝑤,𝐶,𝑦,𝑧 𝑤,𝐷,𝑦,𝑧 𝑤,𝐹,𝑦,𝑧 𝑤,𝑃,𝑦,𝑧 𝑤,𝑋,𝑦,𝑧 𝑤,𝑌,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑦,𝑧,𝑤) 𝐽(𝑦,𝑧,𝑤) 𝐾(𝑦,𝑧,𝑤)

**Proof of Theorem metcnpd**
Step	Hyp	Ref	Expression
1		metcnpd.j	. . . . 5 ⊢ (𝜑 → 𝐽 = (MetOpen‘𝐶))
2		metcnpd.k	. . . . 5 ⊢ (𝜑 → 𝐾 = (MetOpen‘𝐷))
3	1, 2	oveq12d 6018	. . . 4 ⊢ (𝜑 → (𝐽 CnP 𝐾) = ((MetOpen‘𝐶) CnP (MetOpen‘𝐷)))
4	3	fveq1d 5628	. . 3 ⊢ (𝜑 → ((𝐽 CnP 𝐾)‘𝑃) = (((MetOpen‘𝐶) CnP (MetOpen‘𝐷))‘𝑃))
5	4	eleq2d 2299	. 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 2229	. . . 4 ⊢ (MetOpen‘𝐶) = (MetOpen‘𝐶)
10		eqid 2229	. . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
11	9, 10	metcnp 15180	. . 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 1271	. 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 188	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 104 ↔ wb 105 = wceq 1395 ∈ wcel 2200 ∀wral 2508 ∃wrex 2509 class class class wbr 4082 ⟶wf 5313 ‘cfv 5317 (class class class)co 6000 < clt 8177 ℝ⁺crp 9845 ∞Metcxmet 14494 MetOpencmopn 14499 CnP ccnp 14854

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113 ax-arch 8114 ax-caucvg 8115

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4383 df-po 4386 df-iso 4387 df-iord 4456 df-on 4458 df-ilim 4459 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-isom 5326 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-recs 6449 df-frec 6535 df-map 6795 df-sup 7147 df-inf 7148 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 df-div 8816 df-inn 9107 df-2 9165 df-3 9166 df-4 9167 df-n0 9366 df-z 9443 df-uz 9719 df-q 9811 df-rp 9846 df-xneg 9964 df-xadd 9965 df-seqfrec 10665 df-exp 10756 df-cj 11348 df-re 11349 df-im 11350 df-rsqrt 11504 df-abs 11505 df-topgen 13288 df-psmet 14501 df-xmet 14502 df-bl 14504 df-mopn 14505 df-top 14666 df-topon 14679 df-bases 14711 df-cnp 14857

This theorem is referenced by: cnplimcim 15335 limccnpcntop 15343

W3C validator