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Mirrors > Home > MPE Home > Th. List > ipasslem7 | Structured version Visualization version GIF version |
Description: Lemma for ipassi 28617. 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 23410 | . . . . 5 ⊢ (topGen‘ran (,)) = (𝐾 ↾t ℝ) |
5 | 2, 4 | eqtri 2844 | . . . 4 ⊢ 𝐽 = (𝐾 ↾t ℝ) |
6 | 3 | cnfldtopon 23390 | . . . . 5 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
7 | 6 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ)) |
8 | ax-resscn 10593 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
9 | 8 | a1i 11 | . . . 4 ⊢ (⊤ → ℝ ⊆ ℂ) |
10 | 7 | cnmptid 22268 | . . . . . . 7 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝑤) ∈ (𝐾 Cn 𝐾)) |
11 | ip1i.9 | . . . . . . . . . . 11 ⊢ 𝑈 ∈ CPreHilOLD | |
12 | 11 | phnvi 28592 | . . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec |
13 | ip1i.1 | . . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈) | |
14 | eqid 2821 | . . . . . . . . . . 11 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈) | |
15 | 13, 14 | imsxmet 28468 | . . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) ∈ (∞Met‘𝑋)) |
16 | 12, 15 | ax-mp 5 | . . . . . . . . 9 ⊢ (IndMet‘𝑈) ∈ (∞Met‘𝑋) |
17 | eqid 2821 | . . . . . . . . . 10 ⊢ (MetOpen‘(IndMet‘𝑈)) = (MetOpen‘(IndMet‘𝑈)) | |
18 | 17 | mopntopon 23048 | . . . . . . . . 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 22269 | . . . . . . 7 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝐴) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈)))) |
23 | ip1i.4 | . . . . . . . . 9 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
24 | 14, 17, 23, 3 | smcn 28474 | . . . . . . . 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 22273 | . . . . . 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 22269 | . . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝐵) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈)))) |
30 | ip1i.7 | . . . . . . . 8 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
31 | 30, 14, 17, 3 | dipcn 28496 | . . . . . . 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 22273 | . . . . 5 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ ((𝑤𝑆𝐴)𝑃𝐵)) ∈ (𝐾 Cn 𝐾)) |
34 | 13, 30 | dipcl 28488 | . . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ) |
35 | 12, 20, 27, 34 | mp3an 1457 | . . . . . . . 8 ⊢ (𝐴𝑃𝐵) ∈ ℂ |
36 | 35 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (𝐴𝑃𝐵) ∈ ℂ) |
37 | 7, 7, 36 | cnmptc 22269 | . . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝐴𝑃𝐵)) ∈ (𝐾 Cn 𝐾)) |
38 | 3 | mulcn 23474 | . . . . . . 7 ⊢ · ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
39 | 38 | a1i 11 | . . . . . 6 ⊢ (⊤ → · ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
40 | 7, 10, 37, 39 | cnmpt12f 22273 | . . . . 5 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝑤 · (𝐴𝑃𝐵))) ∈ (𝐾 Cn 𝐾)) |
41 | 3 | subcn 23473 | . . . . . 6 ⊢ − ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
42 | 41 | a1i 11 | . . . . 5 ⊢ (⊤ → − ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
43 | 7, 33, 40, 42 | cnmpt12f 22273 | . . . 4 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐾 Cn 𝐾)) |
44 | 5, 7, 9, 43 | cnmpt1res 22283 | . . 3 ⊢ (⊤ → (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐽 Cn 𝐾)) |
45 | 44 | mptru 1540 | . 2 ⊢ (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐽 Cn 𝐾) |
46 | 1, 45 | eqeltri 2909 | 1 ⊢ 𝐹 ∈ (𝐽 Cn 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 ⊆ wss 3935 ↦ cmpt 5145 ran crn 5555 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 ℝcr 10535 · cmul 10541 − cmin 10869 (,)cioo 12737 ↾t crest 16693 TopOpenctopn 16694 topGenctg 16710 ∞Metcxmet 20529 MetOpencmopn 20534 ℂfldccnfld 20544 TopOnctopon 21517 Cn ccn 21831 ×t ctx 22167 NrmCVeccnv 28360 +𝑣 cpv 28361 BaseSetcba 28362 ·𝑠OLD cns 28363 IndMetcims 28367 ·𝑖OLDcdip 28476 CPreHilOLDccphlo 28588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-icc 12744 df-fz 12892 df-fzo 13033 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-sum 15042 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-cnfld 20545 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cn 21834 df-cnp 21835 df-tx 22169 df-hmeo 22362 df-xms 22929 df-ms 22930 df-tms 22931 df-grpo 28269 df-gid 28270 df-ginv 28271 df-gdiv 28272 df-ablo 28321 df-vc 28335 df-nv 28368 df-va 28371 df-ba 28372 df-sm 28373 df-0v 28374 df-vs 28375 df-nmcv 28376 df-ims 28377 df-dip 28477 df-ph 28589 |
This theorem is referenced by: ipasslem8 28613 |
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