metcnpd - Intuitionistic Logic Explorer

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Theorem metcnpd 15243

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 6035	. . . 4 ⊢ (𝜑 → (𝐽 CnP 𝐾) = ((MetOpen‘𝐶) CnP (MetOpen‘𝐷)))
4	3	fveq1d 5641	. . 3 ⊢ (𝜑 → ((𝐽 CnP 𝐾)‘𝑃) = (((MetOpen‘𝐶) CnP (MetOpen‘𝐷))‘𝑃))
5	4	eleq2d 2301	. 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 2231	. . . 4 ⊢ (MetOpen‘𝐶) = (MetOpen‘𝐶)
10		eqid 2231	. . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
11	9, 10	metcnp 15235	. . 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 1273	. 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 1397 ∈ wcel 2202 ∀wral 2510 ∃wrex 2511 class class class wbr 4088 ⟶wf 5322 ‘cfv 5326 (class class class)co 6017 < clt 8213 ℝ⁺crp 9887 ∞Metcxmet 14549 MetOpencmopn 14554 CnP ccnp 14909

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148 ax-pre-mulext 8149 ax-arch 8150 ax-caucvg 8151

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-recs 6470 df-frec 6556 df-map 6818 df-sup 7182 df-inf 7183 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-reap 8754 df-ap 8761 df-div 8852 df-inn 9143 df-2 9201 df-3 9202 df-4 9203 df-n0 9402 df-z 9479 df-uz 9755 df-q 9853 df-rp 9888 df-xneg 10006 df-xadd 10007 df-seqfrec 10709 df-exp 10800 df-cj 11402 df-re 11403 df-im 11404 df-rsqrt 11558 df-abs 11559 df-topgen 13342 df-psmet 14556 df-xmet 14557 df-bl 14559 df-mopn 14560 df-top 14721 df-topon 14734 df-bases 14766 df-cnp 14912

This theorem is referenced by: cnplimcim 15390 limccnpcntop 15398

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