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Mirrors > Home > MPE Home > Th. List > ipasslem8 | Structured version Visualization version GIF version |
Description: Lemma for ipassi 28612. By ipasslem5 28606, 𝐹 is 0 for all ℚ; since it is continuous and ℚ is dense in ℝ by qdensere2 23399, 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.) |
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 ∈ CPreHilOLD |
ipasslem7.a | ⊢ 𝐴 ∈ 𝑋 |
ipasslem7.b | ⊢ 𝐵 ∈ 𝑋 |
ipasslem7.f | ⊢ 𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) |
Ref | Expression |
---|---|
ipasslem8 | ⊢ 𝐹:ℝ⟶{0} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 10627 | . 2 ⊢ 0 ∈ ℂ | |
2 | qre 12347 | . . . . . 6 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ) | |
3 | oveq1 7157 | . . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤𝑆𝐴) = (𝑥𝑆𝐴)) | |
4 | 3 | oveq1d 7165 | . . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((𝑤𝑆𝐴)𝑃𝐵) = ((𝑥𝑆𝐴)𝑃𝐵)) |
5 | oveq1 7157 | . . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤 · (𝐴𝑃𝐵)) = (𝑥 · (𝐴𝑃𝐵))) | |
6 | 4, 5 | oveq12d 7168 | . . . . . . 7 ⊢ (𝑤 = 𝑥 → (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))) = (((𝑥𝑆𝐴)𝑃𝐵) − (𝑥 · (𝐴𝑃𝐵)))) |
7 | ipasslem7.f | . . . . . . 7 ⊢ 𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) | |
8 | ovex 7183 | . . . . . . 7 ⊢ (((𝑥𝑆𝐴)𝑃𝐵) − (𝑥 · (𝐴𝑃𝐵))) ∈ V | |
9 | 6, 7, 8 | fvmpt 6762 | . . . . . 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 12356 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℂ) | |
13 | ip1i.9 | . . . . . . . . . . 11 ⊢ 𝑈 ∈ CPreHilOLD | |
14 | 13 | phnvi 28587 | . . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec |
15 | ip1i.1 | . . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈) | |
16 | ip1i.4 | . . . . . . . . . . 11 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
17 | 15, 16 | nvscl 28397 | . . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑥𝑆𝐴) ∈ 𝑋) |
18 | 14, 17 | mp3an1 1444 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑥𝑆𝐴) ∈ 𝑋) |
19 | 12, 18 | sylan 582 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℚ ∧ 𝐴 ∈ 𝑋) → (𝑥𝑆𝐴) ∈ 𝑋) |
20 | ipasslem7.b | . . . . . . . . 9 ⊢ 𝐵 ∈ 𝑋 | |
21 | ip1i.7 | . . . . . . . . . 10 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
22 | 15, 21 | dipcl 28483 | . . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥𝑆𝐴) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑥𝑆𝐴)𝑃𝐵) ∈ ℂ) |
23 | 14, 20, 22 | mp3an13 1448 | . . . . . . . 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 28606 | . . . . . . 7 ⊢ ((𝑥 ∈ ℚ ∧ 𝐴 ∈ 𝑋) → ((𝑥𝑆𝐴)𝑃𝐵) = (𝑥 · (𝐴𝑃𝐵))) |
27 | 24, 26 | subeq0bd 11060 | . . . . . 6 ⊢ ((𝑥 ∈ ℚ ∧ 𝐴 ∈ 𝑋) → (((𝑥𝑆𝐴)𝑃𝐵) − (𝑥 · (𝐴𝑃𝐵))) = 0) |
28 | 11, 27 | mpan2 689 | . . . . 5 ⊢ (𝑥 ∈ ℚ → (((𝑥𝑆𝐴)𝑃𝐵) − (𝑥 · (𝐴𝑃𝐵))) = 0) |
29 | 10, 28 | eqtrd 2856 | . . . 4 ⊢ (𝑥 ∈ ℚ → (𝐹‘𝑥) = 0) |
30 | 29 | rgen 3148 | . . 3 ⊢ ∀𝑥 ∈ ℚ (𝐹‘𝑥) = 0 |
31 | 7 | funmpt2 6388 | . . . 4 ⊢ Fun 𝐹 |
32 | qssre 12352 | . . . . 5 ⊢ ℚ ⊆ ℝ | |
33 | ovex 7183 | . . . . . 6 ⊢ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))) ∈ V | |
34 | 33, 7 | dmmpti 6486 | . . . . 5 ⊢ dom 𝐹 = ℝ |
35 | 32, 34 | sseqtrri 4003 | . . . 4 ⊢ ℚ ⊆ dom 𝐹 |
36 | funconstss 6820 | . . . 4 ⊢ ((Fun 𝐹 ∧ ℚ ⊆ dom 𝐹) → (∀𝑥 ∈ ℚ (𝐹‘𝑥) = 0 ↔ ℚ ⊆ (◡𝐹 “ {0}))) | |
37 | 31, 35, 36 | mp2an 690 | . . 3 ⊢ (∀𝑥 ∈ ℚ (𝐹‘𝑥) = 0 ↔ ℚ ⊆ (◡𝐹 “ {0})) |
38 | 30, 37 | mpbi 232 | . 2 ⊢ ℚ ⊆ (◡𝐹 “ {0}) |
39 | qdensere 23372 | . 2 ⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ | |
40 | eqid 2821 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
41 | 40 | cnfldhaus 23387 | . . . 4 ⊢ (TopOpen‘ℂfld) ∈ Haus |
42 | haust1 21954 | . . . 4 ⊢ ((TopOpen‘ℂfld) ∈ Haus → (TopOpen‘ℂfld) ∈ Fre) | |
43 | 41, 42 | ax-mp 5 | . . 3 ⊢ (TopOpen‘ℂfld) ∈ Fre |
44 | eqid 2821 | . . . 4 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
45 | 15, 25, 16, 21, 13, 11, 20, 7, 44, 40 | ipasslem7 28607 | . . 3 ⊢ 𝐹 ∈ ((topGen‘ran (,)) Cn (TopOpen‘ℂfld)) |
46 | uniretop 23365 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
47 | 40 | cnfldtopon 23385 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
48 | 47 | toponunii 21518 | . . . 4 ⊢ ℂ = ∪ (TopOpen‘ℂfld) |
49 | 46, 48 | dnsconst 21980 | . . 3 ⊢ ((((TopOpen‘ℂfld) ∈ Fre ∧ 𝐹 ∈ ((topGen‘ran (,)) Cn (TopOpen‘ℂfld))) ∧ (0 ∈ ℂ ∧ ℚ ⊆ (◡𝐹 “ {0}) ∧ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ)) → 𝐹:ℝ⟶{0}) |
50 | 43, 45, 49 | mpanl12 700 | . 2 ⊢ ((0 ∈ ℂ ∧ ℚ ⊆ (◡𝐹 “ {0}) ∧ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ) → 𝐹:ℝ⟶{0}) |
51 | 1, 38, 39, 50 | mp3an 1457 | 1 ⊢ 𝐹:ℝ⟶{0} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 {csn 4560 ↦ cmpt 5138 ◡ccnv 5548 dom cdm 5549 ran crn 5550 “ cima 5552 Fun wfun 6343 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 · cmul 10536 − cmin 10864 ℚcq 12342 (,)cioo 12732 TopOpenctopn 16689 topGenctg 16705 ℂfldccnfld 20539 clsccl 21620 Cn ccn 21826 Frect1 21909 Hauscha 21910 NrmCVeccnv 28355 +𝑣 cpv 28356 BaseSetcba 28357 ·𝑠OLD cns 28358 ·𝑖OLDcdip 28471 CPreHilOLDccphlo 28583 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-icc 12739 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-cn 21829 df-cnp 21830 df-t1 21916 df-haus 21917 df-tx 22164 df-hmeo 22357 df-xms 22924 df-ms 22925 df-tms 22926 df-grpo 28264 df-gid 28265 df-ginv 28266 df-gdiv 28267 df-ablo 28316 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-vs 28370 df-nmcv 28371 df-ims 28372 df-dip 28472 df-ph 28584 |
This theorem is referenced by: ipasslem9 28609 |
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