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Theorem txmetcn 24716

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 24607	. . . . 5 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
3		metcn.4	. . . . . 6 ⊢ 𝐾 = (MetOpen‘𝐷)
4	3	mopntopon 24607	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌))
5		txtopon 23759	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×_t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6	2, 4, 5	syl2an 607	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) → (𝐽 ×_t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
7	6	3adant3 1149	. . 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 24607	. . . 4 ⊢ (𝐸 ∈ (∞Met‘𝑍) → 𝐿 ∈ (TopOn‘𝑍))
10	9	3ad2ant3 1152	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) → 𝐿 ∈ (TopOn‘𝑍))
11		cncnp 23448	. . 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 595	. 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 6881	. . . . . 6 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (((𝐽 ×_t 𝐾) CnP 𝐿)‘𝑡) = (((𝐽 ×_t 𝐾) CnP 𝐿)‘⟨𝑥, 𝑦⟩))
14	13	eleq2d 2848	. . . . 5 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (𝐹 ∈ (((𝐽 ×_t 𝐾) CnP 𝐿)‘𝑡) ↔ 𝐹 ∈ (((𝐽 ×_t 𝐾) CnP 𝐿)‘⟨𝑥, 𝑦⟩)))
15	14	ralxp 5826	. . . 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 780	. . . . . 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 24715	. . . . . . 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 727	. . . . . 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 719	. . . . 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 3226	. . . 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 286	. . 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 589	. 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 282	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1102 = wceq 1569 ∈ wcel 2142 ∀wral 3078 ∃wrex 3088 ⟨cop 4594 class class class wbr 5108 × cxp 5658 ⟶wf 6532 ‘cfv 6536 (class class class)co 7412 < clt 11249 ℝ⁺crp 13022 ∞Metcxmet 21518 MetOpencmopn 21523 TopOnctopon 23078 Cn ccn 23392 CnP ccnp 23393 ×_t ctx 23728

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-er 8692 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-icc 13385 df-fz 13542 df-fzo 13690 df-seq 14045 df-hash 14374 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-hom 17340 df-cco 17341 df-rest 17481 df-topn 17482 df-0g 17500 df-gsum 17501 df-topgen 17502 df-pt 17503 df-prds 17506 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-submnd 18848 df-mulg 19140 df-cntz 19393 df-cmn 19858 df-psmet 21525 df-xmet 21526 df-bl 21528 df-mopn 21529 df-top 23062 df-topon 23079 df-topsp 23101 df-bases 23114 df-cn 23395 df-cnp 23396 df-tx 23730 df-hmeo 23923 df-xms 24488 df-tms 24490

This theorem is used by: ngptgp 24804 nlmvscn 24855 xmetdcn2 25006 addcnlem 25033 ipcn 25416 vacn 31057 smcnlem 31060

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