metcnpd - Intuitionistic Logic Explorer

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Theorem metcnpd 15621

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 6103	. . . 4 ⊢ (𝜑 → (𝐽 CnP 𝐾) = ((MetOpen‘𝐶) CnP (MetOpen‘𝐷)))
4	3	fveq1d 5697	. . 3 ⊢ (𝜑 → ((𝐽 CnP 𝐾)‘𝑃) = (((MetOpen‘𝐶) CnP (MetOpen‘𝐷))‘𝑃))
5	4	eleq2d 2308	. 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 2238	. . . 4 ⊢ (MetOpen‘𝐶) = (MetOpen‘𝐶)
10		eqid 2238	. . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
11	9, 10	metcnp 15613	. . 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 1278	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1402 ∈ wcel 2209 ∀wral 2528 ∃wrex 2529 class class class wbr 4130 ⟶wf 5373 ‘cfv 5377 (class class class)co 6085 < clt 8360 ℝ⁺crp 10054 ∞Metcxmet 14873 MetOpencmopn 14878 CnP ccnp 15287

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-map 6924 df-sup 7324 df-inf 7325 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 df-inn 9305 df-2 9363 df-3 9364 df-4 9365 df-n0 9564 df-z 9645 df-uz 9922 df-q 10020 df-rp 10055 df-xneg 10174 df-xadd 10175 df-seqfrec 10885 df-exp 10976 df-cj 11607 df-re 11608 df-im 11609 df-rsqrt 11764 df-abs 11765 df-topgen 13614 df-psmet 14880 df-xmet 14881 df-bl 14883 df-mopn 14884 df-top 15099 df-topon 15112 df-bases 15144 df-cnp 15290

This theorem is used by: cnplimcim 15768 limccnpcntop 15776

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