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Theorem vmcn 30727

Description: Vector subtraction is jointly continuous in both arguments. (Contributed by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
vmcn.c	⊢ 𝐶 = (IndMet‘𝑈)
vmcn.j	⊢ 𝐽 = (MetOpen‘𝐶)
vmcn.m	⊢ 𝑀 = ( −_𝑣 ‘𝑈)

**Assertion**
Ref	Expression
vmcn	⊢ (𝑈 ∈ NrmCVec → 𝑀 ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))

Proof of Theorem vmcn Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2734	. . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		eqid 2734	. . 3 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
3		eqid 2734	. . 3 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
4		vmcn.m	. . 3 ⊢ 𝑀 = ( −_𝑣 ‘𝑈)
5	1, 2, 3, 4	nvmfval 30672	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑀 = (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ (𝑥( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝑦))))
6		vmcn.c	. . . . 5 ⊢ 𝐶 = (IndMet‘𝑈)
7	1, 6	imsxmet 30720	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝐶 ∈ (∞Met‘(BaseSet‘𝑈)))
8		vmcn.j	. . . . 5 ⊢ 𝐽 = (MetOpen‘𝐶)
9	8	mopntopon 24464	. . . 4 ⊢ (𝐶 ∈ (∞Met‘(BaseSet‘𝑈)) → 𝐽 ∈ (TopOn‘(BaseSet‘𝑈)))
10	7, 9	syl 17	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐽 ∈ (TopOn‘(BaseSet‘𝑈)))
11	10, 10	cnmpt1st 23691	. . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ 𝑥) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
12		eqid 2734	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
13	12	cnfldtopon 24818	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
14	13	a1i 11	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
15		neg1cn 12377	. . . . . 6 ⊢ -1 ∈ ℂ
16	15	a1i 11	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → -1 ∈ ℂ)
17	10, 10, 14, 16	cnmpt2c 23693	. . . 4 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ -1) ∈ ((𝐽 ×_t 𝐽) Cn (TopOpen‘ℂ_fld)))
18	10, 10	cnmpt2nd 23692	. . . 4 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ 𝑦) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
19	6, 8, 3, 12	smcn 30726	. . . 4 ⊢ (𝑈 ∈ NrmCVec → ( ·_𝑠OLD ‘𝑈) ∈ (((TopOpen‘ℂ_fld) ×_t 𝐽) Cn 𝐽))
20	10, 10, 17, 18, 19	cnmpt22f 23698	. . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ (-1( ·_𝑠OLD ‘𝑈)𝑦)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
21	6, 8, 2	vacn 30722	. . 3 ⊢ (𝑈 ∈ NrmCVec → ( +_𝑣 ‘𝑈) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
22	10, 10, 11, 20, 21	cnmpt22f 23698	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ (𝑥( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝑦))) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
23	5, 22	eqeltrd 2838	1 ⊢ (𝑈 ∈ NrmCVec → 𝑀 ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 ∈ cmpo 7432 ℂcc 11150 1c1 11153 -cneg 11490 TopOpenctopn 17467 ∞Metcxmet 21366 MetOpencmopn 21371 ℂ_fldccnfld 21381 TopOnctopon 22931 Cn ccn 23247 ×_t ctx 23583 NrmCVeccnv 30612 +_𝑣 cpv 30613 BaseSetcba 30614 ·_𝑠OLD cns 30615 −_𝑣 cnsb 30617 IndMetcims 30619

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 ax-mulf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-icc 13390 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cn 23250 df-cnp 23251 df-tx 23585 df-hmeo 23778 df-xms 24345 df-ms 24346 df-tms 24347 df-grpo 30521 df-gid 30522 df-ginv 30523 df-gdiv 30524 df-ablo 30573 df-vc 30587 df-nv 30620 df-va 30623 df-ba 30624 df-sm 30625 df-0v 30626 df-vs 30627 df-nmcv 30628 df-ims 30629

This theorem is referenced by: hmopidmchi 32179

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