cnmpt1ip - Metamath Proof Explorer

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Theorem cnmpt1ip 25201

Description: Continuity of inner product; analogue of cnmpt12f 23608 which cannot be used directly because ·_𝑖 is not a function. (Contributed by Mario Carneiro, 13-Oct-2015.)

**Hypotheses**
Ref	Expression
cnmpt1ip.j	⊢ 𝐽 = (TopOpen‘𝑊)
cnmpt1ip.c	⊢ 𝐶 = (TopOpen‘ℂ_fld)
cnmpt1ip.h	⊢ , = (·_𝑖‘𝑊)
cnmpt1ip.r	⊢ (𝜑 → 𝑊 ∈ ℂPreHil)
cnmpt1ip.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))
cnmpt1ip.a	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))
cnmpt1ip.b	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽))

**Assertion**
Ref	Expression
cnmpt1ip	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 , 𝐵)) ∈ (𝐾 Cn 𝐶))

Distinct variable groups: 𝑥,𝐶 𝑥,𝐽 𝑥,𝐾 𝜑,𝑥 𝑥,𝑊 𝑥,𝑋 Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥) , (𝑥)

**Proof of Theorem cnmpt1ip**
Step	Hyp	Ref	Expression
1		cnmpt1ip.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))
2		cnmpt1ip.r	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℂPreHil)
3		cphngp 25127	. . . . . . . 8 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
4		ngptps 24544	. . . . . . . 8 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ TopSp)
5	2, 3, 4	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ TopSp)
6		eqid 2734	. . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊)
7		cnmpt1ip.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝑊)
8	6, 7	istps 22876	. . . . . . 7 ⊢ (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑊)))
9	5, 8	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘𝑊)))
10		cnmpt1ip.a	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))
11		cnf2 23191	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝑊)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝑊))
12	1, 9, 10, 11	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝑊))
13	12	fvmptelcdm 7056	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ (Base‘𝑊))
14		cnmpt1ip.b	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽))
15		cnf2 23191	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝑊)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝑊))
16	1, 9, 14, 15	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝑊))
17	16	fvmptelcdm 7056	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (Base‘𝑊))
18		cnmpt1ip.h	. . . . 5 ⊢ , = (·_𝑖‘𝑊)
19		eqid 2734	. . . . 5 ⊢ (·_if‘𝑊) = (·_if‘𝑊)
20	6, 18, 19	ipfval 21602	. . . 4 ⊢ ((𝐴 ∈ (Base‘𝑊) ∧ 𝐵 ∈ (Base‘𝑊)) → (𝐴(·_if‘𝑊)𝐵) = (𝐴 , 𝐵))
21	13, 17, 20	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴(·_if‘𝑊)𝐵) = (𝐴 , 𝐵))
22	21	mpteq2dva 5189	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(·_if‘𝑊)𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 , 𝐵)))
23		cnmpt1ip.c	. . . . 5 ⊢ 𝐶 = (TopOpen‘ℂ_fld)
24	19, 7, 23	ipcn 25200	. . . 4 ⊢ (𝑊 ∈ ℂPreHil → (·_if‘𝑊) ∈ ((𝐽 ×_t 𝐽) Cn 𝐶))
25	2, 24	syl 17	. . 3 ⊢ (𝜑 → (·_if‘𝑊) ∈ ((𝐽 ×_t 𝐽) Cn 𝐶))
26	1, 10, 14, 25	cnmpt12f 23608	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(·_if‘𝑊)𝐵)) ∈ (𝐾 Cn 𝐶))
27	22, 26	eqeltrrd 2835	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 , 𝐵)) ∈ (𝐾 Cn 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ↦ cmpt 5177 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 Basecbs 17134 ·_𝑖cip 17180 TopOpenctopn 17339 ℂ_fldccnfld 21307 ·_ifcipf 21578 TopOnctopon 22852 TopSpctps 22874 Cn ccn 23166 ×_t ctx 23502 NrmGrpcngp 24519 ℂPreHilccph 25120

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103 ax-mulf 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8763 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-seq 13923 df-exp 13983 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-pt 17362 df-prds 17365 df-xrs 17421 df-qtop 17426 df-imas 17427 df-xps 17429 df-mre 17503 df-mrc 17504 df-acs 17506 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18706 df-submnd 18707 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18996 df-subg 19051 df-ghm 19140 df-cntz 19244 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-oppr 20271 df-dvdsr 20291 df-unit 20292 df-invr 20322 df-dvr 20335 df-rhm 20406 df-subrng 20477 df-subrg 20501 df-drng 20662 df-staf 20770 df-srng 20771 df-lmod 20811 df-lmhm 20972 df-lvec 21053 df-sra 21123 df-rgmod 21124 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-cnfld 21308 df-phl 21579 df-ipf 21580 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cn 23169 df-cnp 23170 df-tx 23504 df-hmeo 23697 df-xms 24262 df-ms 24263 df-tms 24264 df-nm 24524 df-ngp 24525 df-tng 24526 df-nlm 24528 df-clm 25017 df-cph 25122 df-tcph 25123

This theorem is referenced by: csscld 25203 clsocv 25204

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