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| Mirrors > Home > MPE Home > Th. List > ipasslem7 | Structured version Visualization version GIF version | ||
| Description: Lemma for ipassi 30860. 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 24824 | . . . . 5 ⊢ (topGen‘ran (,)) = (𝐾 ↾t ℝ) |
| 5 | 2, 4 | eqtri 2765 | . . . 4 ⊢ 𝐽 = (𝐾 ↾t ℝ) |
| 6 | 3 | cnfldtopon 24803 | . . . . 5 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
| 7 | 6 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ)) |
| 8 | ax-resscn 11212 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
| 9 | 8 | a1i 11 | . . . 4 ⊢ (⊤ → ℝ ⊆ ℂ) |
| 10 | 7 | cnmptid 23669 | . . . . . . 7 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝑤) ∈ (𝐾 Cn 𝐾)) |
| 11 | ip1i.9 | . . . . . . . . . . 11 ⊢ 𝑈 ∈ CPreHilOLD | |
| 12 | 11 | phnvi 30835 | . . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec |
| 13 | ip1i.1 | . . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈) | |
| 14 | eqid 2737 | . . . . . . . . . . 11 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈) | |
| 15 | 13, 14 | imsxmet 30711 | . . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) ∈ (∞Met‘𝑋)) |
| 16 | 12, 15 | ax-mp 5 | . . . . . . . . 9 ⊢ (IndMet‘𝑈) ∈ (∞Met‘𝑋) |
| 17 | eqid 2737 | . . . . . . . . . 10 ⊢ (MetOpen‘(IndMet‘𝑈)) = (MetOpen‘(IndMet‘𝑈)) | |
| 18 | 17 | mopntopon 24449 | . . . . . . . . 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 23670 | . . . . . . 7 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝐴) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈)))) |
| 23 | ip1i.4 | . . . . . . . . 9 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
| 24 | 14, 17, 23, 3 | smcn 30717 | . . . . . . . 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 23674 | . . . . . 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 23670 | . . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝐵) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈)))) |
| 30 | ip1i.7 | . . . . . . . 8 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
| 31 | 30, 14, 17, 3 | dipcn 30739 | . . . . . . 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 23674 | . . . . 5 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ ((𝑤𝑆𝐴)𝑃𝐵)) ∈ (𝐾 Cn 𝐾)) |
| 34 | 13, 30 | dipcl 30731 | . . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ) |
| 35 | 12, 20, 27, 34 | mp3an 1463 | . . . . . . . 8 ⊢ (𝐴𝑃𝐵) ∈ ℂ |
| 36 | 35 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (𝐴𝑃𝐵) ∈ ℂ) |
| 37 | 7, 7, 36 | cnmptc 23670 | . . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝐴𝑃𝐵)) ∈ (𝐾 Cn 𝐾)) |
| 38 | 3 | mulcn 24889 | . . . . . . 7 ⊢ · ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
| 39 | 38 | a1i 11 | . . . . . 6 ⊢ (⊤ → · ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
| 40 | 7, 10, 37, 39 | cnmpt12f 23674 | . . . . 5 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝑤 · (𝐴𝑃𝐵))) ∈ (𝐾 Cn 𝐾)) |
| 41 | 3 | subcn 24888 | . . . . . 6 ⊢ − ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
| 42 | 41 | a1i 11 | . . . . 5 ⊢ (⊤ → − ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
| 43 | 7, 33, 40, 42 | cnmpt12f 23674 | . . . 4 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐾 Cn 𝐾)) |
| 44 | 5, 7, 9, 43 | cnmpt1res 23684 | . . 3 ⊢ (⊤ → (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐽 Cn 𝐾)) |
| 45 | 44 | mptru 1547 | . 2 ⊢ (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐽 Cn 𝐾) |
| 46 | 1, 45 | eqeltri 2837 | 1 ⊢ 𝐹 ∈ (𝐽 Cn 𝐾) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 ⊆ wss 3951 ↦ cmpt 5225 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 · cmul 11160 − cmin 11492 (,)cioo 13387 ↾t crest 17465 TopOpenctopn 17466 topGenctg 17482 ∞Metcxmet 21349 MetOpencmopn 21354 ℂfldccnfld 21364 TopOnctopon 22916 Cn ccn 23232 ×t ctx 23568 NrmCVeccnv 30603 +𝑣 cpv 30604 BaseSetcba 30605 ·𝑠OLD cns 30606 IndMetcims 30610 ·𝑖OLDcdip 30719 CPreHilOLDccphlo 30831 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cn 23235 df-cnp 23236 df-tx 23570 df-hmeo 23763 df-xms 24330 df-ms 24331 df-tms 24332 df-grpo 30512 df-gid 30513 df-ginv 30514 df-gdiv 30515 df-ablo 30564 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-vs 30618 df-nmcv 30619 df-ims 30620 df-dip 30720 df-ph 30832 |
| This theorem is referenced by: ipasslem8 30856 |
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