ipasslem7 - Metamath Proof Explorer

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Theorem ipasslem7 30913

Description: Lemma for ipassi 30918. 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 24749	. . . . 5 ⊢ (topGen‘ran (,)) = (𝐾 ↾_t ℝ)
5	2, 4	eqtri 2759	. . . 4 ⊢ 𝐽 = (𝐾 ↾_t ℝ)
6	3	cnfldtopon 24728	. . . . 5 ⊢ 𝐾 ∈ (TopOn‘ℂ)
7	6	a1i 11	. . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ))
8		ax-resscn 11085	. . . . 5 ⊢ ℝ ⊆ ℂ
9	8	a1i 11	. . . 4 ⊢ (⊤ → ℝ ⊆ ℂ)
10	7	cnmptid 23607	. . . . . . 7 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝑤) ∈ (𝐾 Cn 𝐾))
11		ip1i.9	. . . . . . . . . . 11 ⊢ 𝑈 ∈ CPreHil_OLD
12	11	phnvi 30893	. . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec
13		ip1i.1	. . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈)
14		eqid 2736	. . . . . . . . . . 11 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈)
15	13, 14	imsxmet 30769	. . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) ∈ (∞Met‘𝑋))
16	12, 15	ax-mp 5	. . . . . . . . 9 ⊢ (IndMet‘𝑈) ∈ (∞Met‘𝑋)
17		eqid 2736	. . . . . . . . . 10 ⊢ (MetOpen‘(IndMet‘𝑈)) = (MetOpen‘(IndMet‘𝑈))
18	17	mopntopon 24385	. . . . . . . . 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 23608	. . . . . . 7 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝐴) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈))))
23		ip1i.4	. . . . . . . . 9 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
24	14, 17, 23, 3	smcn 30775	. . . . . . . 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 23612	. . . . . 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 23608	. . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝐵) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈))))
30		ip1i.7	. . . . . . . 8 ⊢ 𝑃 = (·_𝑖OLD‘𝑈)
31	30, 14, 17, 3	dipcn 30797	. . . . . . 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 23612	. . . . 5 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ ((𝑤𝑆𝐴)𝑃𝐵)) ∈ (𝐾 Cn 𝐾))
34	13, 30	dipcl 30789	. . . . . . . . 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 23608	. . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝐴𝑃𝐵)) ∈ (𝐾 Cn 𝐾))
38	3	mulcn 24814	. . . . . . 7 ⊢ · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
39	38	a1i 11	. . . . . 6 ⊢ (⊤ → · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
40	7, 10, 37, 39	cnmpt12f 23612	. . . . 5 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝑤 · (𝐴𝑃𝐵))) ∈ (𝐾 Cn 𝐾))
41	3	subcn 24813	. . . . . 6 ⊢ − ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
42	41	a1i 11	. . . . 5 ⊢ (⊤ → − ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
43	7, 33, 40, 42	cnmpt12f 23612	. . . 4 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐾 Cn 𝐾))
44	5, 7, 9, 43	cnmpt1res 23622	. . 3 ⊢ (⊤ → (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐽 Cn 𝐾))
45	44	mptru 1548	. 2 ⊢ (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐽 Cn 𝐾)
46	1, 45	eqeltri 2832	1 ⊢ 𝐹 ∈ (𝐽 Cn 𝐾)

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ⊤wtru 1542 ∈ wcel 2113 ⊆ wss 3901 ↦ cmpt 5179 ran crn 5625 ‘cfv 6492 (class class class)co 7358 ℂcc 11026 ℝcr 11027 · cmul 11033 − cmin 11366 (,)cioo 13263 ↾_t crest 17342 TopOpenctopn 17343 topGenctg 17359 ∞Metcxmet 21296 MetOpencmopn 21301 ℂ_fldccnfld 21311 TopOnctopon 22856 Cn ccn 23170 ×_t ctx 23506 NrmCVeccnv 30661 +_𝑣 cpv 30662 BaseSetcba 30663 ·_𝑠OLD cns 30664 IndMetcims 30668 ·_𝑖OLDcdip 30777 CPreHil_OLDccphlo 30889

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9552 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107 ax-mulf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8767 df-ixp 8838 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fsupp 9267 df-fi 9316 df-sup 9347 df-inf 9348 df-oi 9417 df-card 9853 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-ioo 13267 df-icc 13270 df-fz 13426 df-fzo 13573 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-sum 15612 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17425 df-qtop 17430 df-imas 17431 df-xps 17433 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19248 df-cmn 19713 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-cnfld 21312 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22892 df-cn 23173 df-cnp 23174 df-tx 23508 df-hmeo 23701 df-xms 24266 df-ms 24267 df-tms 24268 df-grpo 30570 df-gid 30571 df-ginv 30572 df-gdiv 30573 df-ablo 30622 df-vc 30636 df-nv 30669 df-va 30672 df-ba 30673 df-sm 30674 df-0v 30675 df-vs 30676 df-nmcv 30677 df-ims 30678 df-dip 30778 df-ph 30890

This theorem is referenced by: ipasslem8 30914

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