ipasslem8 - Metamath Proof Explorer

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Theorem ipasslem8 30857

Description: Lemma for ipassi 30861. By ipasslem5 30855, 𝐹 is 0 for all ℚ; since it is continuous and ℚ is dense in ℝ by qdensere2 24819, we conclude 𝐹 is 0 for all ℝ. (Contributed by NM, 24-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))))

**Assertion**
Ref	Expression
ipasslem8	⊢ 𝐹:ℝ⟶{0}

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

Proof of Theorem ipasslem8 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0cn 11254	. 2 ⊢ 0 ∈ ℂ
2		qre 12996	. . . . . 6 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ)
3		oveq1 7439	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤𝑆𝐴) = (𝑥𝑆𝐴))
4	3	oveq1d 7447	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((𝑤𝑆𝐴)𝑃𝐵) = ((𝑥𝑆𝐴)𝑃𝐵))
5		oveq1 7439	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤 · (𝐴𝑃𝐵)) = (𝑥 · (𝐴𝑃𝐵)))
6	4, 5	oveq12d 7450	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))) = (((𝑥𝑆𝐴)𝑃𝐵) − (𝑥 · (𝐴𝑃𝐵))))
7		ipasslem7.f	. . . . . . 7 ⊢ 𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))))
8		ovex 7465	. . . . . . 7 ⊢ (((𝑥𝑆𝐴)𝑃𝐵) − (𝑥 · (𝐴𝑃𝐵))) ∈ V
9	6, 7, 8	fvmpt 7015	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) = (((𝑥𝑆𝐴)𝑃𝐵) − (𝑥 · (𝐴𝑃𝐵))))
10	2, 9	syl 17	. . . . 5 ⊢ (𝑥 ∈ ℚ → (𝐹‘𝑥) = (((𝑥𝑆𝐴)𝑃𝐵) − (𝑥 · (𝐴𝑃𝐵))))
11		ipasslem7.a	. . . . . 6 ⊢ 𝐴 ∈ 𝑋
12		qcn 13006	. . . . . . . . 9 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℂ)
13		ip1i.9	. . . . . . . . . . 11 ⊢ 𝑈 ∈ CPreHil_OLD
14	13	phnvi 30836	. . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec
15		ip1i.1	. . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈)
16		ip1i.4	. . . . . . . . . . 11 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
17	15, 16	nvscl 30646	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑥𝑆𝐴) ∈ 𝑋)
18	14, 17	mp3an1 1449	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑥𝑆𝐴) ∈ 𝑋)
19	12, 18	sylan 580	. . . . . . . 8 ⊢ ((𝑥 ∈ ℚ ∧ 𝐴 ∈ 𝑋) → (𝑥𝑆𝐴) ∈ 𝑋)
20		ipasslem7.b	. . . . . . . . 9 ⊢ 𝐵 ∈ 𝑋
21		ip1i.7	. . . . . . . . . 10 ⊢ 𝑃 = (·_𝑖OLD‘𝑈)
22	15, 21	dipcl 30732	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥𝑆𝐴) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑥𝑆𝐴)𝑃𝐵) ∈ ℂ)
23	14, 20, 22	mp3an13 1453	. . . . . . . 8 ⊢ ((𝑥𝑆𝐴) ∈ 𝑋 → ((𝑥𝑆𝐴)𝑃𝐵) ∈ ℂ)
24	19, 23	syl 17	. . . . . . 7 ⊢ ((𝑥 ∈ ℚ ∧ 𝐴 ∈ 𝑋) → ((𝑥𝑆𝐴)𝑃𝐵) ∈ ℂ)
25		ip1i.2	. . . . . . . 8 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
26	15, 25, 16, 21, 13, 20	ipasslem5 30855	. . . . . . 7 ⊢ ((𝑥 ∈ ℚ ∧ 𝐴 ∈ 𝑋) → ((𝑥𝑆𝐴)𝑃𝐵) = (𝑥 · (𝐴𝑃𝐵)))
27	24, 26	subeq0bd 11690	. . . . . 6 ⊢ ((𝑥 ∈ ℚ ∧ 𝐴 ∈ 𝑋) → (((𝑥𝑆𝐴)𝑃𝐵) − (𝑥 · (𝐴𝑃𝐵))) = 0)
28	11, 27	mpan2 691	. . . . 5 ⊢ (𝑥 ∈ ℚ → (((𝑥𝑆𝐴)𝑃𝐵) − (𝑥 · (𝐴𝑃𝐵))) = 0)
29	10, 28	eqtrd 2776	. . . 4 ⊢ (𝑥 ∈ ℚ → (𝐹‘𝑥) = 0)
30	29	rgen 3062	. . 3 ⊢ ∀𝑥 ∈ ℚ (𝐹‘𝑥) = 0
31	7	funmpt2 6604	. . . 4 ⊢ Fun 𝐹
32		qssre 13002	. . . . 5 ⊢ ℚ ⊆ ℝ
33		ovex 7465	. . . . . 6 ⊢ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))) ∈ V
34	33, 7	dmmpti 6711	. . . . 5 ⊢ dom 𝐹 = ℝ
35	32, 34	sseqtrri 4032	. . . 4 ⊢ ℚ ⊆ dom 𝐹
36		funconstss 7075	. . . 4 ⊢ ((Fun 𝐹 ∧ ℚ ⊆ dom 𝐹) → (∀𝑥 ∈ ℚ (𝐹‘𝑥) = 0 ↔ ℚ ⊆ (^◡𝐹 “ {0})))
37	31, 35, 36	mp2an 692	. . 3 ⊢ (∀𝑥 ∈ ℚ (𝐹‘𝑥) = 0 ↔ ℚ ⊆ (^◡𝐹 “ {0}))
38	30, 37	mpbi 230	. 2 ⊢ ℚ ⊆ (^◡𝐹 “ {0})
39		qdensere 24791	. 2 ⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ
40		eqid 2736	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
41	40	cnfldhaus 24806	. . . 4 ⊢ (TopOpen‘ℂ_fld) ∈ Haus
42		haust1 23361	. . . 4 ⊢ ((TopOpen‘ℂ_fld) ∈ Haus → (TopOpen‘ℂ_fld) ∈ Fre)
43	41, 42	ax-mp 5	. . 3 ⊢ (TopOpen‘ℂ_fld) ∈ Fre
44		eqid 2736	. . . 4 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
45	15, 25, 16, 21, 13, 11, 20, 7, 44, 40	ipasslem7 30856	. . 3 ⊢ 𝐹 ∈ ((topGen‘ran (,)) Cn (TopOpen‘ℂ_fld))
46		uniretop 24784	. . . 4 ⊢ ℝ = ∪ (topGen‘ran (,))
47	40	cnfldtopon 24804	. . . . 5 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
48	47	toponunii 22923	. . . 4 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
49	46, 48	dnsconst 23387	. . 3 ⊢ ((((TopOpen‘ℂ_fld) ∈ Fre ∧ 𝐹 ∈ ((topGen‘ran (,)) Cn (TopOpen‘ℂ_fld))) ∧ (0 ∈ ℂ ∧ ℚ ⊆ (^◡𝐹 “ {0}) ∧ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ)) → 𝐹:ℝ⟶{0})
50	43, 45, 49	mpanl12 702	. 2 ⊢ ((0 ∈ ℂ ∧ ℚ ⊆ (^◡𝐹 “ {0}) ∧ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ) → 𝐹:ℝ⟶{0})
51	1, 38, 39, 50	mp3an 1462	1 ⊢ 𝐹:ℝ⟶{0}

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ⊆ wss 3950 {csn 4625 ↦ cmpt 5224 ^◡ccnv 5683 dom cdm 5684 ran crn 5685 “ cima 5687 Fun wfun 6554 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 ℝcr 11155 0cc0 11156 · cmul 11161 − cmin 11493 ℚcq 12991 (,)cioo 13388 TopOpenctopn 17467 topGenctg 17483 ℂ_fldccnfld 21365 clsccl 23027 Cn ccn 23233 Frect1 23316 Hauscha 23317 NrmCVeccnv 30604 +_𝑣 cpv 30605 BaseSetcba 30606 ·_𝑠OLD cns 30607 ·_𝑖OLDcdip 30720 CPreHil_OLDccphlo 30832

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-icc 13395 df-fz 13549 df-fzo 13696 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-sum 15724 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-cn 23236 df-cnp 23237 df-t1 23323 df-haus 23324 df-tx 23571 df-hmeo 23764 df-xms 24331 df-ms 24332 df-tms 24333 df-grpo 30513 df-gid 30514 df-ginv 30515 df-gdiv 30516 df-ablo 30565 df-vc 30579 df-nv 30612 df-va 30615 df-ba 30616 df-sm 30617 df-0v 30618 df-vs 30619 df-nmcv 30620 df-ims 30621 df-dip 30721 df-ph 30833

This theorem is referenced by: ipasslem9 30858

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