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Mirrors > Home > MPE Home > Th. List > ipasslem8 | Structured version Visualization version GIF version |
Description: Lemma for ipassi 28621. By ipasslem5 28615, 𝐹 is 0 for all ℚ; since it is continuous and ℚ is dense in ℝ by qdensere2 23408, 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 10636 | . 2 ⊢ 0 ∈ ℂ | |
2 | qre 12356 | . . . . . 6 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ) | |
3 | oveq1 7166 | . . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤𝑆𝐴) = (𝑥𝑆𝐴)) | |
4 | 3 | oveq1d 7174 | . . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((𝑤𝑆𝐴)𝑃𝐵) = ((𝑥𝑆𝐴)𝑃𝐵)) |
5 | oveq1 7166 | . . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤 · (𝐴𝑃𝐵)) = (𝑥 · (𝐴𝑃𝐵))) | |
6 | 4, 5 | oveq12d 7177 | . . . . . . 7 ⊢ (𝑤 = 𝑥 → (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))) = (((𝑥𝑆𝐴)𝑃𝐵) − (𝑥 · (𝐴𝑃𝐵)))) |
7 | ipasslem7.f | . . . . . . 7 ⊢ 𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) | |
8 | ovex 7192 | . . . . . . 7 ⊢ (((𝑥𝑆𝐴)𝑃𝐵) − (𝑥 · (𝐴𝑃𝐵))) ∈ V | |
9 | 6, 7, 8 | fvmpt 6771 | . . . . . 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 12365 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℂ) | |
13 | ip1i.9 | . . . . . . . . . . 11 ⊢ 𝑈 ∈ CPreHilOLD | |
14 | 13 | phnvi 28596 | . . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec |
15 | ip1i.1 | . . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈) | |
16 | ip1i.4 | . . . . . . . . . . 11 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
17 | 15, 16 | nvscl 28406 | . . . . . . . . . 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 28492 | . . . . . . . . 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 28615 | . . . . . . 7 ⊢ ((𝑥 ∈ ℚ ∧ 𝐴 ∈ 𝑋) → ((𝑥𝑆𝐴)𝑃𝐵) = (𝑥 · (𝐴𝑃𝐵))) |
27 | 24, 26 | subeq0bd 11069 | . . . . . 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 2859 | . . . 4 ⊢ (𝑥 ∈ ℚ → (𝐹‘𝑥) = 0) |
30 | 29 | rgen 3151 | . . 3 ⊢ ∀𝑥 ∈ ℚ (𝐹‘𝑥) = 0 |
31 | 7 | funmpt2 6397 | . . . 4 ⊢ Fun 𝐹 |
32 | qssre 12361 | . . . . 5 ⊢ ℚ ⊆ ℝ | |
33 | ovex 7192 | . . . . . 6 ⊢ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))) ∈ V | |
34 | 33, 7 | dmmpti 6495 | . . . . 5 ⊢ dom 𝐹 = ℝ |
35 | 32, 34 | sseqtrri 4007 | . . . 4 ⊢ ℚ ⊆ dom 𝐹 |
36 | funconstss 6829 | . . . 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 23381 | . 2 ⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ | |
40 | eqid 2824 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
41 | 40 | cnfldhaus 23396 | . . . 4 ⊢ (TopOpen‘ℂfld) ∈ Haus |
42 | haust1 21963 | . . . 4 ⊢ ((TopOpen‘ℂfld) ∈ Haus → (TopOpen‘ℂfld) ∈ Fre) | |
43 | 41, 42 | ax-mp 5 | . . 3 ⊢ (TopOpen‘ℂfld) ∈ Fre |
44 | eqid 2824 | . . . 4 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
45 | 15, 25, 16, 21, 13, 11, 20, 7, 44, 40 | ipasslem7 28616 | . . 3 ⊢ 𝐹 ∈ ((topGen‘ran (,)) Cn (TopOpen‘ℂfld)) |
46 | uniretop 23374 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
47 | 40 | cnfldtopon 23394 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
48 | 47 | toponunii 21527 | . . . 4 ⊢ ℂ = ∪ (TopOpen‘ℂfld) |
49 | 46, 48 | dnsconst 21989 | . . 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 1536 ∈ wcel 2113 ∀wral 3141 ⊆ wss 3939 {csn 4570 ↦ cmpt 5149 ◡ccnv 5557 dom cdm 5558 ran crn 5559 “ cima 5561 Fun wfun 6352 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 ℝcr 10539 0cc0 10540 · cmul 10545 − cmin 10873 ℚcq 12351 (,)cioo 12741 TopOpenctopn 16698 topGenctg 16714 ℂfldccnfld 20548 clsccl 21629 Cn ccn 21835 Frect1 21918 Hauscha 21919 NrmCVeccnv 28364 +𝑣 cpv 28365 BaseSetcba 28366 ·𝑠OLD cns 28367 ·𝑖OLDcdip 28480 CPreHilOLDccphlo 28592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-sum 15046 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-hom 16592 df-cco 16593 df-rest 16699 df-topn 16700 df-0g 16718 df-gsum 16719 df-topgen 16720 df-pt 16721 df-prds 16724 df-xrs 16778 df-qtop 16783 df-imas 16784 df-xps 16786 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-mulg 18228 df-cntz 18450 df-cmn 18911 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-cnfld 20549 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cld 21630 df-ntr 21631 df-cls 21632 df-cn 21838 df-cnp 21839 df-t1 21925 df-haus 21926 df-tx 22173 df-hmeo 22366 df-xms 22933 df-ms 22934 df-tms 22935 df-grpo 28273 df-gid 28274 df-ginv 28275 df-gdiv 28276 df-ablo 28325 df-vc 28339 df-nv 28372 df-va 28375 df-ba 28376 df-sm 28377 df-0v 28378 df-vs 28379 df-nmcv 28380 df-ims 28381 df-dip 28481 df-ph 28593 |
This theorem is referenced by: ipasslem9 28618 |
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