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Theorem vmcn 30770

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 2736	. . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		eqid 2736	. . 3 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
3		eqid 2736	. . 3 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
4		vmcn.m	. . 3 ⊢ 𝑀 = ( −_𝑣 ‘𝑈)
5	1, 2, 3, 4	nvmfval 30715	. 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 30763	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝐶 ∈ (∞Met‘(BaseSet‘𝑈)))
8		vmcn.j	. . . . 5 ⊢ 𝐽 = (MetOpen‘𝐶)
9	8	mopntopon 24404	. . . 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 23633	. . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ 𝑥) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
12		eqid 2736	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
13	12	cnfldtopon 24747	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
14	13	a1i 11	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
15		neg1cn 12144	. . . . . 6 ⊢ -1 ∈ ℂ
16	15	a1i 11	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → -1 ∈ ℂ)
17	10, 10, 14, 16	cnmpt2c 23635	. . . 4 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ -1) ∈ ((𝐽 ×_t 𝐽) Cn (TopOpen‘ℂ_fld)))
18	10, 10	cnmpt2nd 23634	. . . 4 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ 𝑦) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
19	6, 8, 3, 12	smcn 30769	. . . 4 ⊢ (𝑈 ∈ NrmCVec → ( ·_𝑠OLD ‘𝑈) ∈ (((TopOpen‘ℂ_fld) ×_t 𝐽) Cn 𝐽))
20	10, 10, 17, 18, 19	cnmpt22f 23640	. . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ (-1( ·_𝑠OLD ‘𝑈)𝑦)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
21	6, 8, 2	vacn 30765	. . 3 ⊢ (𝑈 ∈ NrmCVec → ( +_𝑣 ‘𝑈) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
22	10, 10, 11, 20, 21	cnmpt22f 23640	. 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 2836	1 ⊢ (𝑈 ∈ NrmCVec → 𝑀 ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ‘cfv 6498 (class class class)co 7367 ∈ cmpo 7369 ℂcc 11036 1c1 11039 -cneg 11378 TopOpenctopn 17384 ∞Metcxmet 21337 MetOpencmopn 21342 ℂ_fldccnfld 21352 TopOnctopon 22875 Cn ccn 23189 ×_t ctx 23525 NrmCVeccnv 30655 +_𝑣 cpv 30656 BaseSetcba 30657 ·_𝑠OLD cns 30658 −_𝑣 cnsb 30660 IndMetcims 30662

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 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-icc 13305 df-fz 13462 df-fzo 13609 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cn 23192 df-cnp 23193 df-tx 23527 df-hmeo 23720 df-xms 24285 df-ms 24286 df-tms 24287 df-grpo 30564 df-gid 30565 df-ginv 30566 df-gdiv 30567 df-ablo 30616 df-vc 30630 df-nv 30663 df-va 30666 df-ba 30667 df-sm 30668 df-0v 30669 df-vs 30670 df-nmcv 30671 df-ims 30672

This theorem is referenced by: hmopidmchi 32222

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