ipasslem7 - Metamath Proof Explorer

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Theorem ipasslem7 28619

Description: Lemma for ipassi 28624. 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 23408	. . . . 5 ⊢ (topGen‘ran (,)) = (𝐾 ↾_t ℝ)
5	2, 4	eqtri 2821	. . . 4 ⊢ 𝐽 = (𝐾 ↾_t ℝ)
6	3	cnfldtopon 23388	. . . . 5 ⊢ 𝐾 ∈ (TopOn‘ℂ)
7	6	a1i 11	. . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ))
8		ax-resscn 10583	. . . . 5 ⊢ ℝ ⊆ ℂ
9	8	a1i 11	. . . 4 ⊢ (⊤ → ℝ ⊆ ℂ)
10	7	cnmptid 22266	. . . . . . 7 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝑤) ∈ (𝐾 Cn 𝐾))
11		ip1i.9	. . . . . . . . . . 11 ⊢ 𝑈 ∈ CPreHil_OLD
12	11	phnvi 28599	. . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec
13		ip1i.1	. . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈)
14		eqid 2798	. . . . . . . . . . 11 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈)
15	13, 14	imsxmet 28475	. . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) ∈ (∞Met‘𝑋))
16	12, 15	ax-mp 5	. . . . . . . . 9 ⊢ (IndMet‘𝑈) ∈ (∞Met‘𝑋)
17		eqid 2798	. . . . . . . . . 10 ⊢ (MetOpen‘(IndMet‘𝑈)) = (MetOpen‘(IndMet‘𝑈))
18	17	mopntopon 23046	. . . . . . . . 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 22267	. . . . . . 7 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝐴) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈))))
23		ip1i.4	. . . . . . . . 9 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
24	14, 17, 23, 3	smcn 28481	. . . . . . . 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 22271	. . . . . 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 22267	. . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝐵) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈))))
30		ip1i.7	. . . . . . . 8 ⊢ 𝑃 = (·_𝑖OLD‘𝑈)
31	30, 14, 17, 3	dipcn 28503	. . . . . . 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 22271	. . . . 5 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ ((𝑤𝑆𝐴)𝑃𝐵)) ∈ (𝐾 Cn 𝐾))
34	13, 30	dipcl 28495	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ)
35	12, 20, 27, 34	mp3an 1458	. . . . . . . 8 ⊢ (𝐴𝑃𝐵) ∈ ℂ
36	35	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝐴𝑃𝐵) ∈ ℂ)
37	7, 7, 36	cnmptc 22267	. . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝐴𝑃𝐵)) ∈ (𝐾 Cn 𝐾))
38	3	mulcn 23472	. . . . . . 7 ⊢ · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
39	38	a1i 11	. . . . . 6 ⊢ (⊤ → · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
40	7, 10, 37, 39	cnmpt12f 22271	. . . . 5 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝑤 · (𝐴𝑃𝐵))) ∈ (𝐾 Cn 𝐾))
41	3	subcn 23471	. . . . . 6 ⊢ − ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
42	41	a1i 11	. . . . 5 ⊢ (⊤ → − ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
43	7, 33, 40, 42	cnmpt12f 22271	. . . 4 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐾 Cn 𝐾))
44	5, 7, 9, 43	cnmpt1res 22281	. . 3 ⊢ (⊤ → (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐽 Cn 𝐾))
45	44	mptru 1545	. 2 ⊢ (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐽 Cn 𝐾)
46	1, 45	eqeltri 2886	1 ⊢ 𝐹 ∈ (𝐽 Cn 𝐾)

Colors of variables: wff setvar class

Syntax hints: = wceq 1538 ⊤wtru 1539 ∈ wcel 2111 ⊆ wss 3881 ↦ cmpt 5110 ran crn 5520 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 · cmul 10531 − cmin 10859 (,)cioo 12726 ↾_t crest 16686 TopOpenctopn 16687 topGenctg 16703 ∞Metcxmet 20076 MetOpencmopn 20081 ℂ_fldccnfld 20091 TopOnctopon 21515 Cn ccn 21829 ×_t ctx 22165 NrmCVeccnv 28367 +_𝑣 cpv 28368 BaseSetcba 28369 ·_𝑠OLD cns 28370 IndMetcims 28374 ·_𝑖OLDcdip 28483 CPreHil_OLDccphlo 28595

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cn 21832 df-cnp 21833 df-tx 22167 df-hmeo 22360 df-xms 22927 df-ms 22928 df-tms 22929 df-grpo 28276 df-gid 28277 df-ginv 28278 df-gdiv 28279 df-ablo 28328 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-vs 28382 df-nmcv 28383 df-ims 28384 df-dip 28484 df-ph 28596

This theorem is referenced by: ipasslem8 28620

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