ipasslem7 - Metamath Proof Explorer

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Theorem ipasslem7 29099

Description: Lemma for ipassi 29104. 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.)

**Hypotheses**
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 ∈ CPreHil_OLD
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‘ℂ_fld)

**Assertion**
Ref	Expression
ipasslem7	⊢ 𝐹 ∈ (𝐽 Cn 𝐾)

Distinct variable groups: 𝑤,𝐵 𝑤,𝐾 𝑤,𝑃 𝑤,𝑆 𝑤,𝑈 𝑤,𝑋 𝑤,𝐴 Allowed substitution hints: 𝐹(𝑤) 𝐺(𝑤) 𝐽(𝑤)

**Proof of Theorem ipasslem7**
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‘ℂ_fld)
4	3	tgioo2 23872	. . . . 5 ⊢ (topGen‘ran (,)) = (𝐾 ↾_t ℝ)
5	2, 4	eqtri 2766	. . . 4 ⊢ 𝐽 = (𝐾 ↾_t ℝ)
6	3	cnfldtopon 23852	. . . . 5 ⊢ 𝐾 ∈ (TopOn‘ℂ)
7	6	a1i 11	. . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ))
8		ax-resscn 10859	. . . . 5 ⊢ ℝ ⊆ ℂ
9	8	a1i 11	. . . 4 ⊢ (⊤ → ℝ ⊆ ℂ)
10	7	cnmptid 22720	. . . . . . 7 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝑤) ∈ (𝐾 Cn 𝐾))
11		ip1i.9	. . . . . . . . . . 11 ⊢ 𝑈 ∈ CPreHil_OLD
12	11	phnvi 29079	. . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec
13		ip1i.1	. . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈)
14		eqid 2738	. . . . . . . . . . 11 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈)
15	13, 14	imsxmet 28955	. . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) ∈ (∞Met‘𝑋))
16	12, 15	ax-mp 5	. . . . . . . . 9 ⊢ (IndMet‘𝑈) ∈ (∞Met‘𝑋)
17		eqid 2738	. . . . . . . . . 10 ⊢ (MetOpen‘(IndMet‘𝑈)) = (MetOpen‘(IndMet‘𝑈))
18	17	mopntopon 23500	. . . . . . . . 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 22721	. . . . . . 7 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝐴) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈))))
23		ip1i.4	. . . . . . . . 9 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
24	14, 17, 23, 3	smcn 28961	. . . . . . . 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 22725	. . . . . 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 22721	. . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝐵) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈))))
30		ip1i.7	. . . . . . . 8 ⊢ 𝑃 = (·_𝑖OLD‘𝑈)
31	30, 14, 17, 3	dipcn 28983	. . . . . . 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 22725	. . . . 5 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ ((𝑤𝑆𝐴)𝑃𝐵)) ∈ (𝐾 Cn 𝐾))
34	13, 30	dipcl 28975	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ)
35	12, 20, 27, 34	mp3an 1459	. . . . . . . 8 ⊢ (𝐴𝑃𝐵) ∈ ℂ
36	35	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝐴𝑃𝐵) ∈ ℂ)
37	7, 7, 36	cnmptc 22721	. . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝐴𝑃𝐵)) ∈ (𝐾 Cn 𝐾))
38	3	mulcn 23936	. . . . . . 7 ⊢ · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
39	38	a1i 11	. . . . . 6 ⊢ (⊤ → · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
40	7, 10, 37, 39	cnmpt12f 22725	. . . . 5 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝑤 · (𝐴𝑃𝐵))) ∈ (𝐾 Cn 𝐾))
41	3	subcn 23935	. . . . . 6 ⊢ − ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
42	41	a1i 11	. . . . 5 ⊢ (⊤ → − ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
43	7, 33, 40, 42	cnmpt12f 22725	. . . 4 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐾 Cn 𝐾))
44	5, 7, 9, 43	cnmpt1res 22735	. . 3 ⊢ (⊤ → (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐽 Cn 𝐾))
45	44	mptru 1546	. 2 ⊢ (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐽 Cn 𝐾)
46	1, 45	eqeltri 2835	1 ⊢ 𝐹 ∈ (𝐽 Cn 𝐾)

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ⊤wtru 1540 ∈ wcel 2108 ⊆ wss 3883 ↦ cmpt 5153 ran crn 5581 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 · cmul 10807 − cmin 11135 (,)cioo 13008 ↾_t crest 17048 TopOpenctopn 17049 topGenctg 17065 ∞Metcxmet 20495 MetOpencmopn 20500 ℂ_fldccnfld 20510 TopOnctopon 21967 Cn ccn 22283 ×_t ctx 22619 NrmCVeccnv 28847 +_𝑣 cpv 28848 BaseSetcba 28849 ·_𝑠OLD cns 28850 IndMetcims 28854 ·_𝑖OLDcdip 28963 CPreHil_OLDccphlo 29075

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-icc 13015 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cn 22286 df-cnp 22287 df-tx 22621 df-hmeo 22814 df-xms 23381 df-ms 23382 df-tms 23383 df-grpo 28756 df-gid 28757 df-ginv 28758 df-gdiv 28759 df-ablo 28808 df-vc 28822 df-nv 28855 df-va 28858 df-ba 28859 df-sm 28860 df-0v 28861 df-vs 28862 df-nmcv 28863 df-ims 28864 df-dip 28964 df-ph 29076

This theorem is referenced by: ipasslem8 29100

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