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Theorem vmcn 30718

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 2737	. . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		eqid 2737	. . 3 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
3		eqid 2737	. . 3 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
4		vmcn.m	. . 3 ⊢ 𝑀 = ( −_𝑣 ‘𝑈)
5	1, 2, 3, 4	nvmfval 30663	. 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 30711	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝐶 ∈ (∞Met‘(BaseSet‘𝑈)))
8		vmcn.j	. . . . 5 ⊢ 𝐽 = (MetOpen‘𝐶)
9	8	mopntopon 24449	. . . 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 23676	. . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ 𝑥) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
12		eqid 2737	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
13	12	cnfldtopon 24803	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
14	13	a1i 11	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
15		neg1cn 12380	. . . . . 6 ⊢ -1 ∈ ℂ
16	15	a1i 11	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → -1 ∈ ℂ)
17	10, 10, 14, 16	cnmpt2c 23678	. . . 4 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ -1) ∈ ((𝐽 ×_t 𝐽) Cn (TopOpen‘ℂ_fld)))
18	10, 10	cnmpt2nd 23677	. . . 4 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ 𝑦) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
19	6, 8, 3, 12	smcn 30717	. . . 4 ⊢ (𝑈 ∈ NrmCVec → ( ·_𝑠OLD ‘𝑈) ∈ (((TopOpen‘ℂ_fld) ×_t 𝐽) Cn 𝐽))
20	10, 10, 17, 18, 19	cnmpt22f 23683	. . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ (-1( ·_𝑠OLD ‘𝑈)𝑦)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
21	6, 8, 2	vacn 30713	. . 3 ⊢ (𝑈 ∈ NrmCVec → ( +_𝑣 ‘𝑈) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
22	10, 10, 11, 20, 21	cnmpt22f 23683	. 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 2841	1 ⊢ (𝑈 ∈ NrmCVec → 𝑀 ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ℂcc 11153 1c1 11156 -cneg 11493 TopOpenctopn 17466 ∞Metcxmet 21349 MetOpencmopn 21354 ℂ_fldccnfld 21364 TopOnctopon 22916 Cn ccn 23232 ×_t ctx 23568 NrmCVeccnv 30603 +_𝑣 cpv 30604 BaseSetcba 30605 ·_𝑠OLD cns 30606 −_𝑣 cnsb 30608 IndMetcims 30610

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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 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 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cn 23235 df-cnp 23236 df-tx 23570 df-hmeo 23763 df-xms 24330 df-ms 24331 df-tms 24332 df-grpo 30512 df-gid 30513 df-ginv 30514 df-gdiv 30515 df-ablo 30564 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-vs 30618 df-nmcv 30619 df-ims 30620

This theorem is referenced by: hmopidmchi 32170

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