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Theorem txmetcn 24487

Description: Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)

**Hypotheses**
Ref	Expression
metcn.2	⊢ 𝐽 = (MetOpen‘𝐶)
metcn.4	⊢ 𝐾 = (MetOpen‘𝐷)
txmetcnp.4	⊢ 𝐿 = (MetOpen‘𝐸)

**Assertion**
Ref	Expression
txmetcn	⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) → (𝐹 ∈ ((𝐽 ×_t 𝐾) Cn 𝐿) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ∀𝑧 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝑥𝐶𝑢) < 𝑤 ∧ (𝑦𝐷𝑣) < 𝑤) → ((𝑥𝐹𝑦)𝐸(𝑢𝐹𝑣)) < 𝑧))))

Distinct variable groups: 𝑣,𝑢,𝑤,𝑥,𝑦,𝑧,𝐹 𝑢,𝐽,𝑣,𝑤,𝑥,𝑦,𝑧 𝑢,𝐾,𝑣,𝑤,𝑥,𝑦,𝑧 𝑢,𝑋,𝑣,𝑤,𝑥,𝑦,𝑧 𝑢,𝑌,𝑣,𝑤,𝑥,𝑦,𝑧 𝑢,𝑍,𝑣,𝑤,𝑥,𝑦,𝑧 𝑢,𝐶,𝑣,𝑤,𝑥,𝑦,𝑧 𝑢,𝐷,𝑣,𝑤,𝑥,𝑦,𝑧 𝑢,𝐸,𝑣,𝑤,𝑥,𝑦,𝑧 𝑤,𝐿,𝑥,𝑦,𝑧 Allowed substitution hints: 𝐿(𝑣,𝑢)

Proof of Theorem txmetcn Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		metcn.2	. . . . . 6 ⊢ 𝐽 = (MetOpen‘𝐶)
2	1	mopntopon 24378	. . . . 5 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
3		metcn.4	. . . . . 6 ⊢ 𝐾 = (MetOpen‘𝐷)
4	3	mopntopon 24378	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌))
5		txtopon 23529	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×_t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6	2, 4, 5	syl2an 596	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) → (𝐽 ×_t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
7	6	3adant3 1132	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) → (𝐽 ×_t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
8		txmetcnp.4	. . . . 5 ⊢ 𝐿 = (MetOpen‘𝐸)
9	8	mopntopon 24378	. . . 4 ⊢ (𝐸 ∈ (∞Met‘𝑍) → 𝐿 ∈ (TopOn‘𝑍))
10	9	3ad2ant3 1135	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) → 𝐿 ∈ (TopOn‘𝑍))
11		cncnp 23218	. . 3 ⊢ (((𝐽 ×_t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘𝑍)) → (𝐹 ∈ ((𝐽 ×_t 𝐾) Cn 𝐿) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑡 ∈ (𝑋 × 𝑌)𝐹 ∈ (((𝐽 ×_t 𝐾) CnP 𝐿)‘𝑡))))
12	7, 10, 11	syl2anc 584	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) → (𝐹 ∈ ((𝐽 ×_t 𝐾) Cn 𝐿) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑡 ∈ (𝑋 × 𝑌)𝐹 ∈ (((𝐽 ×_t 𝐾) CnP 𝐿)‘𝑡))))
13		fveq2 6876	. . . . . 6 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (((𝐽 ×_t 𝐾) CnP 𝐿)‘𝑡) = (((𝐽 ×_t 𝐾) CnP 𝐿)‘⟨𝑥, 𝑦⟩))
14	13	eleq2d 2820	. . . . 5 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (𝐹 ∈ (((𝐽 ×_t 𝐾) CnP 𝐿)‘𝑡) ↔ 𝐹 ∈ (((𝐽 ×_t 𝐾) CnP 𝐿)‘⟨𝑥, 𝑦⟩)))
15	14	ralxp 5821	. . . 4 ⊢ (∀𝑡 ∈ (𝑋 × 𝑌)𝐹 ∈ (((𝐽 ×_t 𝐾) CnP 𝐿)‘𝑡) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐹 ∈ (((𝐽 ×_t 𝐾) CnP 𝐿)‘⟨𝑥, 𝑦⟩))
16		simplr 768	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ 𝐹:(𝑋 × 𝑌)⟶𝑍) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝐹:(𝑋 × 𝑌)⟶𝑍)
17	1, 3, 8	txmetcnp 24486	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝐹 ∈ (((𝐽 ×_t 𝐾) CnP 𝐿)‘⟨𝑥, 𝑦⟩) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑧 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝑥𝐶𝑢) < 𝑤 ∧ (𝑦𝐷𝑣) < 𝑤) → ((𝑥𝐹𝑦)𝐸(𝑢𝐹𝑣)) < 𝑧))))
18	17	adantlr 715	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ 𝐹:(𝑋 × 𝑌)⟶𝑍) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝐹 ∈ (((𝐽 ×_t 𝐾) CnP 𝐿)‘⟨𝑥, 𝑦⟩) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑧 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝑥𝐶𝑢) < 𝑤 ∧ (𝑦𝐷𝑣) < 𝑤) → ((𝑥𝐹𝑦)𝐸(𝑢𝐹𝑣)) < 𝑧))))
19	16, 18	mpbirand 707	. . . . 5 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ 𝐹:(𝑋 × 𝑌)⟶𝑍) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝐹 ∈ (((𝐽 ×_t 𝐾) CnP 𝐿)‘⟨𝑥, 𝑦⟩) ↔ ∀𝑧 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝑥𝐶𝑢) < 𝑤 ∧ (𝑦𝐷𝑣) < 𝑤) → ((𝑥𝐹𝑦)𝐸(𝑢𝐹𝑣)) < 𝑧)))
20	19	2ralbidva 3203	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ 𝐹:(𝑋 × 𝑌)⟶𝑍) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐹 ∈ (((𝐽 ×_t 𝐾) CnP 𝐿)‘⟨𝑥, 𝑦⟩) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ∀𝑧 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝑥𝐶𝑢) < 𝑤 ∧ (𝑦𝐷𝑣) < 𝑤) → ((𝑥𝐹𝑦)𝐸(𝑢𝐹𝑣)) < 𝑧)))
21	15, 20	bitrid 283	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ 𝐹:(𝑋 × 𝑌)⟶𝑍) → (∀𝑡 ∈ (𝑋 × 𝑌)𝐹 ∈ (((𝐽 ×_t 𝐾) CnP 𝐿)‘𝑡) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ∀𝑧 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝑥𝐶𝑢) < 𝑤 ∧ (𝑦𝐷𝑣) < 𝑤) → ((𝑥𝐹𝑦)𝐸(𝑢𝐹𝑣)) < 𝑧)))
22	21	pm5.32da 579	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) → ((𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑡 ∈ (𝑋 × 𝑌)𝐹 ∈ (((𝐽 ×_t 𝐾) CnP 𝐿)‘𝑡)) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ∀𝑧 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝑥𝐶𝑢) < 𝑤 ∧ (𝑦𝐷𝑣) < 𝑤) → ((𝑥𝐹𝑦)𝐸(𝑢𝐹𝑣)) < 𝑧))))
23	12, 22	bitrd 279	1 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) → (𝐹 ∈ ((𝐽 ×_t 𝐾) Cn 𝐿) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ∀𝑧 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝑥𝐶𝑢) < 𝑤 ∧ (𝑦𝐷𝑣) < 𝑤) → ((𝑥𝐹𝑦)𝐸(𝑢𝐹𝑣)) < 𝑧))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ∀wral 3051 ∃wrex 3060 ⟨cop 4607 class class class wbr 5119 × cxp 5652 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 < clt 11269 ℝ⁺crp 13008 ∞Metcxmet 21300 MetOpencmopn 21305 TopOnctopon 22848 Cn ccn 23162 CnP ccnp 23163 ×_t ctx 23498

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-map 8842 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-icc 13369 df-fz 13525 df-fzo 13672 df-seq 14020 df-hash 14349 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-hom 17295 df-cco 17296 df-rest 17436 df-topn 17437 df-0g 17455 df-gsum 17456 df-topgen 17457 df-pt 17458 df-prds 17461 df-xrs 17516 df-qtop 17521 df-imas 17522 df-xps 17524 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-mulg 19051 df-cntz 19300 df-cmn 19763 df-psmet 21307 df-xmet 21308 df-bl 21310 df-mopn 21311 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cn 23165 df-cnp 23166 df-tx 23500 df-hmeo 23693 df-xms 24259 df-tms 24261

This theorem is referenced by: ngptgp 24575 nlmvscn 24626 xmetdcn2 24777 addcnlem 24804 ipcn 25198 vacn 30675 smcnlem 30678

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