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Mirrors > Home > MPE Home > Th. List > ipasslem7 | Structured version Visualization version GIF version |
Description: Lemma for ipassi 27824. Show that ((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)) is continuous on ℝ. (Contributed by NM, 23-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ip1i.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
ip1i.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
ip1i.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
ip1i.7 | ⊢ 𝑃 = (·𝑖OLD‘𝑈) |
ip1i.9 | ⊢ 𝑈 ∈ CPreHilOLD |
ipasslem7.a | ⊢ 𝐴 ∈ 𝑋 |
ipasslem7.b | ⊢ 𝐵 ∈ 𝑋 |
ipasslem7.f | ⊢ 𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) |
ipasslem7.j | ⊢ 𝐽 = (topGen‘ran (,)) |
ipasslem7.k | ⊢ 𝐾 = (TopOpen‘ℂfld) |
Ref | Expression |
---|---|
ipasslem7 | ⊢ 𝐹 ∈ (𝐽 Cn 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipasslem7.f | . 2 ⊢ 𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) | |
2 | ipasslem7.j | . . . . 5 ⊢ 𝐽 = (topGen‘ran (,)) | |
3 | ipasslem7.k | . . . . . 6 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
4 | 3 | tgioo2 22653 | . . . . 5 ⊢ (topGen‘ran (,)) = (𝐾 ↾t ℝ) |
5 | 2, 4 | eqtri 2673 | . . . 4 ⊢ 𝐽 = (𝐾 ↾t ℝ) |
6 | 3 | cnfldtopon 22633 | . . . . 5 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
7 | 6 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ)) |
8 | ax-resscn 10031 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
9 | 8 | a1i 11 | . . . 4 ⊢ (⊤ → ℝ ⊆ ℂ) |
10 | 7 | cnmptid 21512 | . . . . . . 7 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝑤) ∈ (𝐾 Cn 𝐾)) |
11 | ip1i.9 | . . . . . . . . . . 11 ⊢ 𝑈 ∈ CPreHilOLD | |
12 | 11 | phnvi 27799 | . . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec |
13 | ip1i.1 | . . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈) | |
14 | eqid 2651 | . . . . . . . . . . 11 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈) | |
15 | 13, 14 | imsxmet 27675 | . . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) ∈ (∞Met‘𝑋)) |
16 | 12, 15 | ax-mp 5 | . . . . . . . . 9 ⊢ (IndMet‘𝑈) ∈ (∞Met‘𝑋) |
17 | eqid 2651 | . . . . . . . . . 10 ⊢ (MetOpen‘(IndMet‘𝑈)) = (MetOpen‘(IndMet‘𝑈)) | |
18 | 17 | mopntopon 22291 | . . . . . . . . 9 ⊢ ((IndMet‘𝑈) ∈ (∞Met‘𝑋) → (MetOpen‘(IndMet‘𝑈)) ∈ (TopOn‘𝑋)) |
19 | 16, 18 | mp1i 13 | . . . . . . . 8 ⊢ (⊤ → (MetOpen‘(IndMet‘𝑈)) ∈ (TopOn‘𝑋)) |
20 | ipasslem7.a | . . . . . . . . 9 ⊢ 𝐴 ∈ 𝑋 | |
21 | 20 | a1i 11 | . . . . . . . 8 ⊢ (⊤ → 𝐴 ∈ 𝑋) |
22 | 7, 19, 21 | cnmptc 21513 | . . . . . . 7 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝐴) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈)))) |
23 | ip1i.4 | . . . . . . . . 9 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
24 | 14, 17, 23, 3 | smcn 27681 | . . . . . . . 8 ⊢ (𝑈 ∈ NrmCVec → 𝑆 ∈ ((𝐾 ×t (MetOpen‘(IndMet‘𝑈))) Cn (MetOpen‘(IndMet‘𝑈)))) |
25 | 12, 24 | mp1i 13 | . . . . . . 7 ⊢ (⊤ → 𝑆 ∈ ((𝐾 ×t (MetOpen‘(IndMet‘𝑈))) Cn (MetOpen‘(IndMet‘𝑈)))) |
26 | 7, 10, 22, 25 | cnmpt12f 21517 | . . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝑤𝑆𝐴)) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈)))) |
27 | ipasslem7.b | . . . . . . . 8 ⊢ 𝐵 ∈ 𝑋 | |
28 | 27 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 𝐵 ∈ 𝑋) |
29 | 7, 19, 28 | cnmptc 21513 | . . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝐵) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈)))) |
30 | ip1i.7 | . . . . . . . 8 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
31 | 30, 14, 17, 3 | dipcn 27703 | . . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑃 ∈ (((MetOpen‘(IndMet‘𝑈)) ×t (MetOpen‘(IndMet‘𝑈))) Cn 𝐾)) |
32 | 12, 31 | mp1i 13 | . . . . . 6 ⊢ (⊤ → 𝑃 ∈ (((MetOpen‘(IndMet‘𝑈)) ×t (MetOpen‘(IndMet‘𝑈))) Cn 𝐾)) |
33 | 7, 26, 29, 32 | cnmpt12f 21517 | . . . . 5 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ ((𝑤𝑆𝐴)𝑃𝐵)) ∈ (𝐾 Cn 𝐾)) |
34 | 13, 30 | dipcl 27695 | . . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ) |
35 | 12, 20, 27, 34 | mp3an 1464 | . . . . . . . 8 ⊢ (𝐴𝑃𝐵) ∈ ℂ |
36 | 35 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (𝐴𝑃𝐵) ∈ ℂ) |
37 | 7, 7, 36 | cnmptc 21513 | . . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝐴𝑃𝐵)) ∈ (𝐾 Cn 𝐾)) |
38 | 3 | mulcn 22717 | . . . . . . 7 ⊢ · ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
39 | 38 | a1i 11 | . . . . . 6 ⊢ (⊤ → · ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
40 | 7, 10, 37, 39 | cnmpt12f 21517 | . . . . 5 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝑤 · (𝐴𝑃𝐵))) ∈ (𝐾 Cn 𝐾)) |
41 | 3 | subcn 22716 | . . . . . 6 ⊢ − ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
42 | 41 | a1i 11 | . . . . 5 ⊢ (⊤ → − ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
43 | 7, 33, 40, 42 | cnmpt12f 21517 | . . . 4 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐾 Cn 𝐾)) |
44 | 5, 7, 9, 43 | cnmpt1res 21527 | . . 3 ⊢ (⊤ → (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐽 Cn 𝐾)) |
45 | 44 | trud 1533 | . 2 ⊢ (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐽 Cn 𝐾) |
46 | 1, 45 | eqeltri 2726 | 1 ⊢ 𝐹 ∈ (𝐽 Cn 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 ⊤wtru 1524 ∈ wcel 2030 ⊆ wss 3607 ↦ cmpt 4762 ran crn 5144 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 ℝcr 9973 · cmul 9979 − cmin 10304 (,)cioo 12213 ↾t crest 16128 TopOpenctopn 16129 topGenctg 16145 ∞Metcxmt 19779 MetOpencmopn 19784 ℂfldccnfld 19794 TopOnctopon 20763 Cn ccn 21076 ×t ctx 21411 NrmCVeccnv 27567 +𝑣 cpv 27568 BaseSetcba 27569 ·𝑠OLD cns 27570 IndMetcims 27574 ·𝑖OLDcdip 27683 CPreHilOLDccphlo 27795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-icc 12220 df-fz 12365 df-fzo 12505 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-clim 14263 df-sum 14461 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-xrs 16209 df-qtop 16214 df-imas 16215 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-mulg 17588 df-cntz 17796 df-cmn 18241 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-cnfld 19795 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cn 21079 df-cnp 21080 df-tx 21413 df-hmeo 21606 df-xms 22172 df-ms 22173 df-tms 22174 df-grpo 27475 df-gid 27476 df-ginv 27477 df-gdiv 27478 df-ablo 27527 df-vc 27542 df-nv 27575 df-va 27578 df-ba 27579 df-sm 27580 df-0v 27581 df-vs 27582 df-nmcv 27583 df-ims 27584 df-dip 27684 df-ph 27796 |
This theorem is referenced by: ipasslem8 27820 |
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