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Theorem txmetcn 24494

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 24385	. . . . 5 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
3		metcn.4	. . . . . 6 ⊢ 𝐾 = (MetOpen‘𝐷)
4	3	mopntopon 24385	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌))
5		txtopon 23537	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×_t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6	2, 4, 5	syl2an 597	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) → (𝐽 ×_t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
7	6	3adant3 1133	. . 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 24385	. . . 4 ⊢ (𝐸 ∈ (∞Met‘𝑍) → 𝐿 ∈ (TopOn‘𝑍))
10	9	3ad2ant3 1136	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) → 𝐿 ∈ (TopOn‘𝑍))
11		cncnp 23226	. . 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 585	. 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 6833	. . . . . 6 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (((𝐽 ×_t 𝐾) CnP 𝐿)‘𝑡) = (((𝐽 ×_t 𝐾) CnP 𝐿)‘⟨𝑥, 𝑦⟩))
14	13	eleq2d 2821	. . . . 5 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (𝐹 ∈ (((𝐽 ×_t 𝐾) CnP 𝐿)‘𝑡) ↔ 𝐹 ∈ (((𝐽 ×_t 𝐾) CnP 𝐿)‘⟨𝑥, 𝑦⟩)))
15	14	ralxp 5789	. . . 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 769	. . . . . 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 24493	. . . . . . 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 716	. . . . . 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 708	. . . . 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 3197	. . . 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 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3050 ∃wrex 3059 ⟨cop 4585 class class class wbr 5097 × cxp 5621 ⟶wf 6487 ‘cfv 6491 (class class class)co 7358 < clt 11168 ℝ⁺crp 12907 ∞Metcxmet 21296 MetOpencmopn 21301 TopOnctopon 22856 Cn ccn 23170 CnP ccnp 23171 ×_t ctx 23506

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2183 ax-ext 2707 ax-rep 5223 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3349 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4902 df-iun 4947 df-iin 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6258 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8767 df-ixp 8838 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fsupp 9267 df-fi 9316 df-sup 9347 df-inf 9348 df-oi 9417 df-card 9853 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-icc 13270 df-fz 13426 df-fzo 13573 df-seq 13927 df-hash 14256 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17425 df-qtop 17430 df-imas 17431 df-xps 17433 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19248 df-cmn 19713 df-psmet 21303 df-xmet 21304 df-bl 21306 df-mopn 21307 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22892 df-cn 23173 df-cnp 23174 df-tx 23508 df-hmeo 23701 df-xms 24266 df-tms 24268

This theorem is referenced by: ngptgp 24582 nlmvscn 24633 xmetdcn2 24784 addcnlem 24811 ipcn 25204 vacn 30750 smcnlem 30753

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