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Theorem dipcn 27915

Description: Inner product is jointly continuous in both arguments. (Contributed by NM, 21-Aug-2007.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
dipcn.p	⊢ 𝑃 = (·_𝑖OLD‘𝑈)
dipcn.c	⊢ 𝐶 = (IndMet‘𝑈)
dipcn.j	⊢ 𝐽 = (MetOpen‘𝐶)
dipcn.k	⊢ 𝐾 = (TopOpen‘ℂ_fld)

**Assertion**
Ref	Expression
dipcn	⊢ (𝑈 ∈ NrmCVec → 𝑃 ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))

Proof of Theorem dipcn Dummy variables 𝑥 𝑘 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2771	. . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		eqid 2771	. . 3 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
3		eqid 2771	. . 3 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
4		eqid 2771	. . 3 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
5		dipcn.p	. . 3 ⊢ 𝑃 = (·_𝑖OLD‘𝑈)
6	1, 2, 3, 4, 5	dipfval 27897	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑃 = (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((norm_CV‘𝑈)‘(𝑥( +_𝑣 ‘𝑈)((i↑𝑘)( ·_𝑠OLD ‘𝑈)𝑦)))↑2)) / 4)))
7		dipcn.c	. . . . 5 ⊢ 𝐶 = (IndMet‘𝑈)
8	1, 7	imsxmet 27887	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝐶 ∈ (∞Met‘(BaseSet‘𝑈)))
9		dipcn.j	. . . . 5 ⊢ 𝐽 = (MetOpen‘𝐶)
10	9	mopntopon 22464	. . . 4 ⊢ (𝐶 ∈ (∞Met‘(BaseSet‘𝑈)) → 𝐽 ∈ (TopOn‘(BaseSet‘𝑈)))
11	8, 10	syl 17	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐽 ∈ (TopOn‘(BaseSet‘𝑈)))
12		dipcn.k	. . . 4 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
13		fzfid 12980	. . . 4 ⊢ (𝑈 ∈ NrmCVec → (1...4) ∈ Fin)
14	11	adantr 466	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → 𝐽 ∈ (TopOn‘(BaseSet‘𝑈)))
15	12	cnfldtopon 22806	. . . . . . 7 ⊢ 𝐾 ∈ (TopOn‘ℂ)
16	15	a1i 11	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → 𝐾 ∈ (TopOn‘ℂ))
17		ax-icn 10201	. . . . . . 7 ⊢ i ∈ ℂ
18		elfznn 12577	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...4) → 𝑘 ∈ ℕ)
19	18	adantl 467	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → 𝑘 ∈ ℕ)
20	19	nnnn0d 11558	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → 𝑘 ∈ ℕ₀)
21		expcl 13085	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (i↑𝑘) ∈ ℂ)
22	17, 20, 21	sylancr 575	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (i↑𝑘) ∈ ℂ)
23	14, 14, 16, 22	cnmpt2c 21694	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ (i↑𝑘)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
24	14, 14	cnmpt1st 21692	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ 𝑥) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
25	14, 14	cnmpt2nd 21693	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ 𝑦) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
26	7, 9, 3, 12	smcn 27893	. . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → ( ·_𝑠OLD ‘𝑈) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
27	26	adantr 466	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → ( ·_𝑠OLD ‘𝑈) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
28	14, 14, 23, 25, 27	cnmpt22f 21699	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ ((i↑𝑘)( ·_𝑠OLD ‘𝑈)𝑦)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
29	7, 9, 2	vacn 27889	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → ( +_𝑣 ‘𝑈) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
30	29	adantr 466	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → ( +_𝑣 ‘𝑈) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
31	14, 14, 24, 28, 30	cnmpt22f 21699	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ (𝑥( +_𝑣 ‘𝑈)((i↑𝑘)( ·_𝑠OLD ‘𝑈)𝑦))) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
32	4, 7, 9, 12	nmcnc 27891	. . . . . . . 8 ⊢ (𝑈 ∈ NrmCVec → (norm_CV‘𝑈) ∈ (𝐽 Cn 𝐾))
33	32	adantr 466	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (norm_CV‘𝑈) ∈ (𝐽 Cn 𝐾))
34	14, 14, 31, 33	cnmpt21f 21696	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ ((norm_CV‘𝑈)‘(𝑥( +_𝑣 ‘𝑈)((i↑𝑘)( ·_𝑠OLD ‘𝑈)𝑦)))) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
35	12	sqcn 22897	. . . . . . 7 ⊢ (𝑧 ∈ ℂ ↦ (𝑧↑2)) ∈ (𝐾 Cn 𝐾)
36	35	a1i 11	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (𝑧 ∈ ℂ ↦ (𝑧↑2)) ∈ (𝐾 Cn 𝐾))
37		oveq1 6803	. . . . . 6 ⊢ (𝑧 = ((norm_CV‘𝑈)‘(𝑥( +_𝑣 ‘𝑈)((i↑𝑘)( ·_𝑠OLD ‘𝑈)𝑦))) → (𝑧↑2) = (((norm_CV‘𝑈)‘(𝑥( +_𝑣 ‘𝑈)((i↑𝑘)( ·_𝑠OLD ‘𝑈)𝑦)))↑2))
38	14, 14, 34, 16, 36, 37	cnmpt21 21695	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ (((norm_CV‘𝑈)‘(𝑥( +_𝑣 ‘𝑈)((i↑𝑘)( ·_𝑠OLD ‘𝑈)𝑦)))↑2)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
39	12	mulcn 22890	. . . . . 6 ⊢ · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
40	39	a1i 11	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
41	14, 14, 23, 38, 40	cnmpt22f 21699	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑘 ∈ (1...4)) → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ ((i↑𝑘) · (((norm_CV‘𝑈)‘(𝑥( +_𝑣 ‘𝑈)((i↑𝑘)( ·_𝑠OLD ‘𝑈)𝑦)))↑2))) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
42	12, 11, 13, 11, 41	fsum2cn 22894	. . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ Σ𝑘 ∈ (1...4)((i↑𝑘) · (((norm_CV‘𝑈)‘(𝑥( +_𝑣 ‘𝑈)((i↑𝑘)( ·_𝑠OLD ‘𝑈)𝑦)))↑2))) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
43	15	a1i 11	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐾 ∈ (TopOn‘ℂ))
44		4cn 11304	. . . . 5 ⊢ 4 ∈ ℂ
45		4ne0 11323	. . . . 5 ⊢ 4 ≠ 0
46	12	divccn 22896	. . . . 5 ⊢ ((4 ∈ ℂ ∧ 4 ≠ 0) → (𝑧 ∈ ℂ ↦ (𝑧 / 4)) ∈ (𝐾 Cn 𝐾))
47	44, 45, 46	mp2an 672	. . . 4 ⊢ (𝑧 ∈ ℂ ↦ (𝑧 / 4)) ∈ (𝐾 Cn 𝐾)
48	47	a1i 11	. . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑧 ∈ ℂ ↦ (𝑧 / 4)) ∈ (𝐾 Cn 𝐾))
49		oveq1 6803	. . 3 ⊢ (𝑧 = Σ𝑘 ∈ (1...4)((i↑𝑘) · (((norm_CV‘𝑈)‘(𝑥( +_𝑣 ‘𝑈)((i↑𝑘)( ·_𝑠OLD ‘𝑈)𝑦)))↑2)) → (𝑧 / 4) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((norm_CV‘𝑈)‘(𝑥( +_𝑣 ‘𝑈)((i↑𝑘)( ·_𝑠OLD ‘𝑈)𝑦)))↑2)) / 4))
50	11, 11, 42, 43, 48, 49	cnmpt21 21695	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ (BaseSet‘𝑈), 𝑦 ∈ (BaseSet‘𝑈) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((norm_CV‘𝑈)‘(𝑥( +_𝑣 ‘𝑈)((i↑𝑘)( ·_𝑠OLD ‘𝑈)𝑦)))↑2)) / 4)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
51	6, 50	eqeltrd 2850	1 ⊢ (𝑈 ∈ NrmCVec → 𝑃 ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ↦ cmpt 4864 ‘cfv 6030 (class class class)co 6796 ↦ cmpt2 6798 ℂcc 10140 0cc0 10142 1c1 10143 ici 10144 · cmul 10147 / cdiv 10890 ℕcn 11226 2c2 11276 4c4 11278 ℕ₀cn0 11499 ...cfz 12533 ↑cexp 13067 Σcsu 14624 TopOpenctopn 16290 ∞Metcxmt 19946 MetOpencmopn 19951 ℂ_fldccnfld 19961 TopOnctopon 20935 Cn ccn 21249 ×_t ctx 21584 NrmCVeccnv 27779 +_𝑣 cpv 27780 BaseSetcba 27781 ·_𝑠OLD cns 27782 norm_CVcnmcv 27785 IndMetcims 27786 ·_𝑖OLDcdip 27895

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-inf2 8706 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220 ax-addf 10221 ax-mulf 10222

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-iin 4658 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-of 7048 df-om 7217 df-1st 7319 df-2nd 7320 df-supp 7451 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-2o 7718 df-oadd 7721 df-er 7900 df-map 8015 df-ixp 8067 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-fsupp 8436 df-fi 8477 df-sup 8508 df-inf 8509 df-oi 8575 df-card 8969 df-cda 9196 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-2 11285 df-3 11286 df-4 11287 df-5 11288 df-6 11289 df-7 11290 df-8 11291 df-9 11292 df-n0 11500 df-z 11585 df-dec 11701 df-uz 11894 df-q 11997 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-ioo 12384 df-icc 12387 df-fz 12534 df-fzo 12674 df-seq 13009 df-exp 13068 df-hash 13322 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-clim 14427 df-sum 14625 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-starv 16164 df-sca 16165 df-vsca 16166 df-ip 16167 df-tset 16168 df-ple 16169 df-ds 16172 df-unif 16173 df-hom 16174 df-cco 16175 df-rest 16291 df-topn 16292 df-0g 16310 df-gsum 16311 df-topgen 16312 df-pt 16313 df-prds 16316 df-xrs 16370 df-qtop 16375 df-imas 16376 df-xps 16378 df-mre 16454 df-mrc 16455 df-acs 16457 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-mulg 17749 df-cntz 17957 df-cmn 18402 df-psmet 19953 df-xmet 19954 df-met 19955 df-bl 19956 df-mopn 19957 df-cnfld 19962 df-top 20919 df-topon 20936 df-topsp 20958 df-bases 20971 df-cn 21252 df-cnp 21253 df-tx 21586 df-hmeo 21779 df-xms 22345 df-ms 22346 df-tms 22347 df-grpo 27687 df-gid 27688 df-ginv 27689 df-gdiv 27690 df-ablo 27739 df-vc 27754 df-nv 27787 df-va 27790 df-ba 27791 df-sm 27792 df-0v 27793 df-vs 27794 df-nmcv 27795 df-ims 27796 df-dip 27896

This theorem is referenced by: ipasslem7 28031 occllem 28502

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