ipasslem7 - Metamath Proof Explorer

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Theorem ipasslem7 30929

Description: Lemma for ipassi 30934. 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 24790	. . . . 5 ⊢ (topGen‘ran (,)) = (𝐾 ↾_t ℝ)
5	2, 4	eqtri 2764	. . . 4 ⊢ 𝐽 = (𝐾 ↾_t ℝ)
6	3	cnfldtopon 24769	. . . . 5 ⊢ 𝐾 ∈ (TopOn‘ℂ)
7	6	a1i 11	. . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ))
8		ax-resscn 11090	. . . . 5 ⊢ ℝ ⊆ ℂ
9	8	a1i 11	. . . 4 ⊢ (⊤ → ℝ ⊆ ℂ)
10	7	cnmptid 23648	. . . . . . 7 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝑤) ∈ (𝐾 Cn 𝐾))
11		ip1i.9	. . . . . . . . . . 11 ⊢ 𝑈 ∈ CPreHil_OLD
12	11	phnvi 30909	. . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec
13		ip1i.1	. . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈)
14		eqid 2741	. . . . . . . . . . 11 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈)
15	13, 14	imsxmet 30785	. . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) ∈ (∞Met‘𝑋))
16	12, 15	ax-mp 5	. . . . . . . . 9 ⊢ (IndMet‘𝑈) ∈ (∞Met‘𝑋)
17		eqid 2741	. . . . . . . . . 10 ⊢ (MetOpen‘(IndMet‘𝑈)) = (MetOpen‘(IndMet‘𝑈))
18	17	mopntopon 24426	. . . . . . . . 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 23649	. . . . . . 7 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝐴) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈))))
23		ip1i.4	. . . . . . . . 9 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
24	14, 17, 23, 3	smcn 30791	. . . . . . . 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 23653	. . . . . 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 23649	. . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ 𝐵) ∈ (𝐾 Cn (MetOpen‘(IndMet‘𝑈))))
30		ip1i.7	. . . . . . . 8 ⊢ 𝑃 = (·_𝑖OLD‘𝑈)
31	30, 14, 17, 3	dipcn 30813	. . . . . . 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 23653	. . . . 5 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ ((𝑤𝑆𝐴)𝑃𝐵)) ∈ (𝐾 Cn 𝐾))
34	13, 30	dipcl 30805	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ)
35	12, 20, 27, 34	mp3an 1470	. . . . . . . 8 ⊢ (𝐴𝑃𝐵) ∈ ℂ
36	35	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝐴𝑃𝐵) ∈ ℂ)
37	7, 7, 36	cnmptc 23649	. . . . . 6 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝐴𝑃𝐵)) ∈ (𝐾 Cn 𝐾))
38	3	mulcn 24855	. . . . . . 7 ⊢ · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
39	38	a1i 11	. . . . . 6 ⊢ (⊤ → · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
40	7, 10, 37, 39	cnmpt12f 23653	. . . . 5 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (𝑤 · (𝐴𝑃𝐵))) ∈ (𝐾 Cn 𝐾))
41	3	subcn 24854	. . . . . 6 ⊢ − ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
42	41	a1i 11	. . . . 5 ⊢ (⊤ → − ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
43	7, 33, 40, 42	cnmpt12f 23653	. . . 4 ⊢ (⊤ → (𝑤 ∈ ℂ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐾 Cn 𝐾))
44	5, 7, 9, 43	cnmpt1res 23663	. . 3 ⊢ (⊤ → (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ∈ (𝐽 Cn 𝐾))
45	44	mptru 1555	. 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 1548 ⊤wtru 1549 ∈ wcel 2121 ⊆ wss 3885 ↦ cmpt 5156 ran crn 5622 ‘cfv 6489 (class class class)co 7360 ℂcc 11031 ℝcr 11032 · cmul 11038 − cmin 11372 (,)cioo 13293 ↾_t crest 17378 TopOpenctopn 17379 topGenctg 17395 ∞Metcxmet 21336 MetOpencmopn 21341 ℂ_fldccnfld 21351 TopOnctopon 22897 Cn ccn 23211 ×_t ctx 23547 NrmCVeccnv 30677 +_𝑣 cpv 30678 BaseSetcba 30679 ·_𝑠OLD cns 30680 IndMetcims 30684 ·_𝑖OLDcdip 30793 CPreHil_OLDccphlo 30905

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-inf2 9557 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112 ax-mulf 11113

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-iin 4927 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-ioo 13297 df-icc 13300 df-fz 13457 df-fzo 13604 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-clim 15445 df-sum 15644 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-pt 17402 df-prds 17405 df-xrs 17461 df-qtop 17466 df-imas 17467 df-xps 17469 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-mulg 19039 df-cntz 19287 df-cmn 19752 df-psmet 21343 df-xmet 21344 df-met 21345 df-bl 21346 df-mopn 21347 df-cnfld 21352 df-top 22881 df-topon 22898 df-topsp 22920 df-bases 22933 df-cn 23214 df-cnp 23215 df-tx 23549 df-hmeo 23742 df-xms 24307 df-ms 24308 df-tms 24309 df-grpo 30586 df-gid 30587 df-ginv 30588 df-gdiv 30589 df-ablo 30638 df-vc 30652 df-nv 30685 df-va 30688 df-ba 30689 df-sm 30690 df-0v 30691 df-vs 30692 df-nmcv 30693 df-ims 30694 df-dip 30794 df-ph 30906

This theorem is referenced by: ipasslem8 30930

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