ipasslem8 - Metamath Proof Explorer

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Theorem ipasslem8 30927

Description: Lemma for ipassi 30931. By ipasslem5 30925, 𝐹 is 0 for all ℚ; since it is continuous and ℚ is dense in ℝ by qdensere2 24776, 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 11131	. 2 ⊢ 0 ∈ ℂ
2		qre 12898	. . . . . 6 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ)
3		oveq1 7369	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤𝑆𝐴) = (𝑥𝑆𝐴))
4	3	oveq1d 7377	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((𝑤𝑆𝐴)𝑃𝐵) = ((𝑥𝑆𝐴)𝑃𝐵))
5		oveq1 7369	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤 · (𝐴𝑃𝐵)) = (𝑥 · (𝐴𝑃𝐵)))
6	4, 5	oveq12d 7380	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))) = (((𝑥𝑆𝐴)𝑃𝐵) − (𝑥 · (𝐴𝑃𝐵))))
7		ipasslem7.f	. . . . . . 7 ⊢ 𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))))
8		ovex 7395	. . . . . . 7 ⊢ (((𝑥𝑆𝐴)𝑃𝐵) − (𝑥 · (𝐴𝑃𝐵))) ∈ V
9	6, 7, 8	fvmpt 6943	. . . . . 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 12908	. . . . . . . . 9 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℂ)
13		ip1i.9	. . . . . . . . . . 11 ⊢ 𝑈 ∈ CPreHil_OLD
14	13	phnvi 30906	. . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec
15		ip1i.1	. . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈)
16		ip1i.4	. . . . . . . . . . 11 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
17	15, 16	nvscl 30716	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑥𝑆𝐴) ∈ 𝑋)
18	14, 17	mp3an1 1451	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑥𝑆𝐴) ∈ 𝑋)
19	12, 18	sylan 581	. . . . . . . 8 ⊢ ((𝑥 ∈ ℚ ∧ 𝐴 ∈ 𝑋) → (𝑥𝑆𝐴) ∈ 𝑋)
20		ipasslem7.b	. . . . . . . . 9 ⊢ 𝐵 ∈ 𝑋
21		ip1i.7	. . . . . . . . . 10 ⊢ 𝑃 = (·_𝑖OLD‘𝑈)
22	15, 21	dipcl 30802	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥𝑆𝐴) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑥𝑆𝐴)𝑃𝐵) ∈ ℂ)
23	14, 20, 22	mp3an13 1455	. . . . . . . 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 30925	. . . . . . 7 ⊢ ((𝑥 ∈ ℚ ∧ 𝐴 ∈ 𝑋) → ((𝑥𝑆𝐴)𝑃𝐵) = (𝑥 · (𝐴𝑃𝐵)))
27	24, 26	subeq0bd 11571	. . . . . 6 ⊢ ((𝑥 ∈ ℚ ∧ 𝐴 ∈ 𝑋) → (((𝑥𝑆𝐴)𝑃𝐵) − (𝑥 · (𝐴𝑃𝐵))) = 0)
28	11, 27	mpan2 692	. . . . 5 ⊢ (𝑥 ∈ ℚ → (((𝑥𝑆𝐴)𝑃𝐵) − (𝑥 · (𝐴𝑃𝐵))) = 0)
29	10, 28	eqtrd 2772	. . . 4 ⊢ (𝑥 ∈ ℚ → (𝐹‘𝑥) = 0)
30	29	rgen 3054	. . 3 ⊢ ∀𝑥 ∈ ℚ (𝐹‘𝑥) = 0
31	7	funmpt2 6533	. . . 4 ⊢ Fun 𝐹
32		qssre 12904	. . . . 5 ⊢ ℚ ⊆ ℝ
33		ovex 7395	. . . . . 6 ⊢ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))) ∈ V
34	33, 7	dmmpti 6638	. . . . 5 ⊢ dom 𝐹 = ℝ
35	32, 34	sseqtrri 3972	. . . 4 ⊢ ℚ ⊆ dom 𝐹
36		funconstss 7004	. . . 4 ⊢ ((Fun 𝐹 ∧ ℚ ⊆ dom 𝐹) → (∀𝑥 ∈ ℚ (𝐹‘𝑥) = 0 ↔ ℚ ⊆ (^◡𝐹 “ {0})))
37	31, 35, 36	mp2an 693	. . 3 ⊢ (∀𝑥 ∈ ℚ (𝐹‘𝑥) = 0 ↔ ℚ ⊆ (^◡𝐹 “ {0}))
38	30, 37	mpbi 230	. 2 ⊢ ℚ ⊆ (^◡𝐹 “ {0})
39		qdensere 24748	. 2 ⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ
40		eqid 2737	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
41	40	cnfldhaus 24763	. . . 4 ⊢ (TopOpen‘ℂ_fld) ∈ Haus
42		haust1 23331	. . . 4 ⊢ ((TopOpen‘ℂ_fld) ∈ Haus → (TopOpen‘ℂ_fld) ∈ Fre)
43	41, 42	ax-mp 5	. . 3 ⊢ (TopOpen‘ℂ_fld) ∈ Fre
44		eqid 2737	. . . 4 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
45	15, 25, 16, 21, 13, 11, 20, 7, 44, 40	ipasslem7 30926	. . 3 ⊢ 𝐹 ∈ ((topGen‘ran (,)) Cn (TopOpen‘ℂ_fld))
46		uniretop 24741	. . . 4 ⊢ ℝ = ∪ (topGen‘ran (,))
47	40	cnfldtopon 24761	. . . . 5 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
48	47	toponunii 22895	. . . 4 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
49	46, 48	dnsconst 23357	. . 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 703	. 2 ⊢ ((0 ∈ ℂ ∧ ℚ ⊆ (^◡𝐹 “ {0}) ∧ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ) → 𝐹:ℝ⟶{0})
51	1, 38, 39, 50	mp3an 1464	1 ⊢ 𝐹:ℝ⟶{0}

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ⊆ wss 3890 {csn 4568 ↦ cmpt 5167 ^◡ccnv 5625 dom cdm 5626 ran crn 5627 “ cima 5629 Fun wfun 6488 ⟶wf 6490 ‘cfv 6494 (class class class)co 7362 ℂcc 11031 ℝcr 11032 0cc0 11033 · cmul 11038 − cmin 11372 ℚcq 12893 (,)cioo 13293 TopOpenctopn 17379 topGenctg 17395 ℂ_fldccnfld 21348 clsccl 22997 Cn ccn 23203 Frect1 23286 Hauscha 23287 NrmCVeccnv 30674 +_𝑣 cpv 30675 BaseSetcba 30676 ·_𝑠OLD cns 30677 ·_𝑖OLDcdip 30790 CPreHil_OLDccphlo 30902

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 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 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-se 5580 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7626 df-om 7813 df-1st 7937 df-2nd 7938 df-supp 8106 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-er 8638 df-map 8770 df-ixp 8841 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fsupp 9270 df-fi 9319 df-sup 9350 df-inf 9351 df-oi 9420 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 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-cnfld 21349 df-top 22873 df-topon 22890 df-topsp 22912 df-bases 22925 df-cld 22998 df-ntr 22999 df-cls 23000 df-cn 23206 df-cnp 23207 df-t1 23293 df-haus 23294 df-tx 23541 df-hmeo 23734 df-xms 24299 df-ms 24300 df-tms 24301 df-grpo 30583 df-gid 30584 df-ginv 30585 df-gdiv 30586 df-ablo 30635 df-vc 30649 df-nv 30682 df-va 30685 df-ba 30686 df-sm 30687 df-0v 30688 df-vs 30689 df-nmcv 30690 df-ims 30691 df-dip 30791 df-ph 30903

This theorem is referenced by: ipasslem9 30928

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