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Mirrors > Home > MPE Home > Th. List > ipasslem7 | Structured version Visualization version GIF version |
Description: Lemma for ipassi 27696. 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 22606 | . . . . 5 ⊢ (topGen‘ran (,)) = (𝐾 ↾t ℝ) |
5 | 2, 4 | eqtri 2644 | . . . 4 ⊢ 𝐽 = (𝐾 ↾t ℝ) |
6 | 3 | cnfldtopon 22586 | . . . . 5 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
7 | 6 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ)) |
8 | ax-resscn 9993 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
9 | 8 | a1i 11 | . . . 4 ⊢ (⊤ → ℝ ⊆ ℂ) |
10 | 7 | cnmptid 21464 | . . . . . . 7 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝑤) ∈ (𝐾 Cn 𝐾)) |
11 | ip1i.9 | . . . . . . . . . . 11 ⊢ 𝑈 ∈ CPreHilOLD | |
12 | 11 | phnvi 27671 | . . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec |
13 | ip1i.1 | . . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈) | |
14 | eqid 2622 | . . . . . . . . . . 11 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈) | |
15 | 13, 14 | imsxmet 27547 | . . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) ∈ (∞Met‘𝑋)) |
16 | 12, 15 | ax-mp 5 | . . . . . . . . 9 ⊢ (IndMet‘𝑈) ∈ (∞Met‘𝑋) |
17 | eqid 2622 | . . . . . . . . . 10 ⊢ (MetOpen‘(IndMet‘𝑈)) = (MetOpen‘(IndMet‘𝑈)) | |
18 | 17 | mopntopon 22244 | . . . . . . . . 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 21465 | . . . . . . 7 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝐴) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈)))) |
23 | ip1i.4 | . . . . . . . . 9 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
24 | 14, 17, 23, 3 | smcn 27553 | . . . . . . . 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 21469 | . . . . . 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 21465 | . . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝐵) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈)))) |
30 | ip1i.7 | . . . . . . . 8 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
31 | 30, 14, 17, 3 | dipcn 27575 | . . . . . . 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 21469 | . . . . 5 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ ((𝑤𝑆𝐴)𝑃𝐵)) ∈ (𝐾 Cn 𝐾)) |
34 | 13, 30 | dipcl 27567 | . . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ) |
35 | 12, 20, 27, 34 | mp3an 1424 | . . . . . . . 8 ⊢ (𝐴𝑃𝐵) ∈ ℂ |
36 | 35 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (𝐴𝑃𝐵) ∈ ℂ) |
37 | 7, 7, 36 | cnmptc 21465 | . . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝐴𝑃𝐵)) ∈ (𝐾 Cn 𝐾)) |
38 | 3 | mulcn 22670 | . . . . . . 7 ⊢ · ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
39 | 38 | a1i 11 | . . . . . 6 ⊢ (⊤ → · ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
40 | 7, 10, 37, 39 | cnmpt12f 21469 | . . . . 5 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝑤 · (𝐴𝑃𝐵))) ∈ (𝐾 Cn 𝐾)) |
41 | 3 | subcn 22669 | . . . . . 6 ⊢ − ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
42 | 41 | a1i 11 | . . . . 5 ⊢ (⊤ → − ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
43 | 7, 33, 40, 42 | cnmpt12f 21469 | . . . 4 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐾 Cn 𝐾)) |
44 | 5, 7, 9, 43 | cnmpt1res 21479 | . . 3 ⊢ (⊤ → (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐽 Cn 𝐾)) |
45 | 44 | trud 1493 | . 2 ⊢ (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐽 Cn 𝐾) |
46 | 1, 45 | eqeltri 2697 | 1 ⊢ 𝐹 ∈ (𝐽 Cn 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 ⊆ wss 3574 ↦ cmpt 4729 ran crn 5115 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 · cmul 9941 − cmin 10266 (,)cioo 12175 ↾t crest 16081 TopOpenctopn 16082 topGenctg 16098 ∞Metcxmt 19731 MetOpencmopn 19736 ℂfldccnfld 19746 TopOnctopon 20715 Cn ccn 21028 ×t ctx 21363 NrmCVeccnv 27439 +𝑣 cpv 27440 BaseSetcba 27441 ·𝑠OLD cns 27442 IndMetcims 27446 ·𝑖OLDcdip 27555 CPreHilOLDccphlo 27667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cn 21031 df-cnp 21032 df-tx 21365 df-hmeo 21558 df-xms 22125 df-ms 22126 df-tms 22127 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 df-ims 27456 df-dip 27556 df-ph 27668 |
This theorem is referenced by: ipasslem8 27692 |
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