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Mirrors > Home > MPE Home > Th. List > ipasslem7 | Structured version Visualization version GIF version |
Description: Lemma for ipassi 28620. 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 23413 | . . . . 5 ⊢ (topGen‘ran (,)) = (𝐾 ↾t ℝ) |
5 | 2, 4 | eqtri 2846 | . . . 4 ⊢ 𝐽 = (𝐾 ↾t ℝ) |
6 | 3 | cnfldtopon 23393 | . . . . 5 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
7 | 6 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ)) |
8 | ax-resscn 10596 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
9 | 8 | a1i 11 | . . . 4 ⊢ (⊤ → ℝ ⊆ ℂ) |
10 | 7 | cnmptid 22271 | . . . . . . 7 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝑤) ∈ (𝐾 Cn 𝐾)) |
11 | ip1i.9 | . . . . . . . . . . 11 ⊢ 𝑈 ∈ CPreHilOLD | |
12 | 11 | phnvi 28595 | . . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec |
13 | ip1i.1 | . . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈) | |
14 | eqid 2823 | . . . . . . . . . . 11 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈) | |
15 | 13, 14 | imsxmet 28471 | . . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) ∈ (∞Met‘𝑋)) |
16 | 12, 15 | ax-mp 5 | . . . . . . . . 9 ⊢ (IndMet‘𝑈) ∈ (∞Met‘𝑋) |
17 | eqid 2823 | . . . . . . . . . 10 ⊢ (MetOpen‘(IndMet‘𝑈)) = (MetOpen‘(IndMet‘𝑈)) | |
18 | 17 | mopntopon 23051 | . . . . . . . . 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 22272 | . . . . . . 7 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝐴) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈)))) |
23 | ip1i.4 | . . . . . . . . 9 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
24 | 14, 17, 23, 3 | smcn 28477 | . . . . . . . 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 22276 | . . . . . 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 22272 | . . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝐵) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈)))) |
30 | ip1i.7 | . . . . . . . 8 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
31 | 30, 14, 17, 3 | dipcn 28499 | . . . . . . 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 22276 | . . . . 5 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ ((𝑤𝑆𝐴)𝑃𝐵)) ∈ (𝐾 Cn 𝐾)) |
34 | 13, 30 | dipcl 28491 | . . . . . . . . 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 22272 | . . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝐴𝑃𝐵)) ∈ (𝐾 Cn 𝐾)) |
38 | 3 | mulcn 23477 | . . . . . . 7 ⊢ · ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
39 | 38 | a1i 11 | . . . . . 6 ⊢ (⊤ → · ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
40 | 7, 10, 37, 39 | cnmpt12f 22276 | . . . . 5 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝑤 · (𝐴𝑃𝐵))) ∈ (𝐾 Cn 𝐾)) |
41 | 3 | subcn 23476 | . . . . . 6 ⊢ − ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
42 | 41 | a1i 11 | . . . . 5 ⊢ (⊤ → − ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
43 | 7, 33, 40, 42 | cnmpt12f 22276 | . . . 4 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐾 Cn 𝐾)) |
44 | 5, 7, 9, 43 | cnmpt1res 22286 | . . 3 ⊢ (⊤ → (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐽 Cn 𝐾)) |
45 | 44 | mptru 1544 | . 2 ⊢ (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐽 Cn 𝐾) |
46 | 1, 45 | eqeltri 2911 | 1 ⊢ 𝐹 ∈ (𝐽 Cn 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ⊆ wss 3938 ↦ cmpt 5148 ran crn 5558 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 · cmul 10544 − cmin 10872 (,)cioo 12741 ↾t crest 16696 TopOpenctopn 16697 topGenctg 16713 ∞Metcxmet 20532 MetOpencmopn 20537 ℂfldccnfld 20547 TopOnctopon 21520 Cn ccn 21834 ×t ctx 22170 NrmCVeccnv 28363 +𝑣 cpv 28364 BaseSetcba 28365 ·𝑠OLD cns 28366 IndMetcims 28370 ·𝑖OLDcdip 28479 CPreHilOLDccphlo 28591 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cn 21837 df-cnp 21838 df-tx 22172 df-hmeo 22365 df-xms 22932 df-ms 22933 df-tms 22934 df-grpo 28272 df-gid 28273 df-ginv 28274 df-gdiv 28275 df-ablo 28324 df-vc 28338 df-nv 28371 df-va 28374 df-ba 28375 df-sm 28376 df-0v 28377 df-vs 28378 df-nmcv 28379 df-ims 28380 df-dip 28480 df-ph 28592 |
This theorem is referenced by: ipasslem8 28616 |
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