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Mirrors > Home > MPE Home > Th. List > ipasslem8 | Structured version Visualization version GIF version |
Description: Lemma for ipassi 30870. By ipasslem5 30864, 𝐹 is 0 for all ℚ; since it is continuous and ℚ is dense in ℝ by qdensere2 24833, 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 11251 | . 2 ⊢ 0 ∈ ℂ | |
2 | qre 12993 | . . . . . 6 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ) | |
3 | oveq1 7438 | . . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤𝑆𝐴) = (𝑥𝑆𝐴)) | |
4 | 3 | oveq1d 7446 | . . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((𝑤𝑆𝐴)𝑃𝐵) = ((𝑥𝑆𝐴)𝑃𝐵)) |
5 | oveq1 7438 | . . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤 · (𝐴𝑃𝐵)) = (𝑥 · (𝐴𝑃𝐵))) | |
6 | 4, 5 | oveq12d 7449 | . . . . . . 7 ⊢ (𝑤 = 𝑥 → (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))) = (((𝑥𝑆𝐴)𝑃𝐵) − (𝑥 · (𝐴𝑃𝐵)))) |
7 | ipasslem7.f | . . . . . . 7 ⊢ 𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) | |
8 | ovex 7464 | . . . . . . 7 ⊢ (((𝑥𝑆𝐴)𝑃𝐵) − (𝑥 · (𝐴𝑃𝐵))) ∈ V | |
9 | 6, 7, 8 | fvmpt 7016 | . . . . . 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 13003 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℂ) | |
13 | ip1i.9 | . . . . . . . . . . 11 ⊢ 𝑈 ∈ CPreHilOLD | |
14 | 13 | phnvi 30845 | . . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec |
15 | ip1i.1 | . . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈) | |
16 | ip1i.4 | . . . . . . . . . . 11 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
17 | 15, 16 | nvscl 30655 | . . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑥𝑆𝐴) ∈ 𝑋) |
18 | 14, 17 | mp3an1 1447 | . . . . . . . . 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 30741 | . . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥𝑆𝐴) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑥𝑆𝐴)𝑃𝐵) ∈ ℂ) |
23 | 14, 20, 22 | mp3an13 1451 | . . . . . . . 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 30864 | . . . . . . 7 ⊢ ((𝑥 ∈ ℚ ∧ 𝐴 ∈ 𝑋) → ((𝑥𝑆𝐴)𝑃𝐵) = (𝑥 · (𝐴𝑃𝐵))) |
27 | 24, 26 | subeq0bd 11687 | . . . . . 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 2775 | . . . 4 ⊢ (𝑥 ∈ ℚ → (𝐹‘𝑥) = 0) |
30 | 29 | rgen 3061 | . . 3 ⊢ ∀𝑥 ∈ ℚ (𝐹‘𝑥) = 0 |
31 | 7 | funmpt2 6607 | . . . 4 ⊢ Fun 𝐹 |
32 | qssre 12999 | . . . . 5 ⊢ ℚ ⊆ ℝ | |
33 | ovex 7464 | . . . . . 6 ⊢ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))) ∈ V | |
34 | 33, 7 | dmmpti 6713 | . . . . 5 ⊢ dom 𝐹 = ℝ |
35 | 32, 34 | sseqtrri 4033 | . . . 4 ⊢ ℚ ⊆ dom 𝐹 |
36 | funconstss 7076 | . . . 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 24806 | . 2 ⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ | |
40 | eqid 2735 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
41 | 40 | cnfldhaus 24821 | . . . 4 ⊢ (TopOpen‘ℂfld) ∈ Haus |
42 | haust1 23376 | . . . 4 ⊢ ((TopOpen‘ℂfld) ∈ Haus → (TopOpen‘ℂfld) ∈ Fre) | |
43 | 41, 42 | ax-mp 5 | . . 3 ⊢ (TopOpen‘ℂfld) ∈ Fre |
44 | eqid 2735 | . . . 4 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
45 | 15, 25, 16, 21, 13, 11, 20, 7, 44, 40 | ipasslem7 30865 | . . 3 ⊢ 𝐹 ∈ ((topGen‘ran (,)) Cn (TopOpen‘ℂfld)) |
46 | uniretop 24799 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
47 | 40 | cnfldtopon 24819 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
48 | 47 | toponunii 22938 | . . . 4 ⊢ ℂ = ∪ (TopOpen‘ℂfld) |
49 | 46, 48 | dnsconst 23402 | . . 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 702 | . 2 ⊢ ((0 ∈ ℂ ∧ ℚ ⊆ (◡𝐹 “ {0}) ∧ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ) → 𝐹:ℝ⟶{0}) |
51 | 1, 38, 39, 50 | mp3an 1460 | 1 ⊢ 𝐹:ℝ⟶{0} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 {csn 4631 ↦ cmpt 5231 ◡ccnv 5688 dom cdm 5689 ran crn 5690 “ cima 5692 Fun wfun 6557 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 · cmul 11158 − cmin 11490 ℚcq 12988 (,)cioo 13384 TopOpenctopn 17468 topGenctg 17484 ℂfldccnfld 21382 clsccl 23042 Cn ccn 23248 Frect1 23331 Hauscha 23332 NrmCVeccnv 30613 +𝑣 cpv 30614 BaseSetcba 30615 ·𝑠OLD cns 30616 ·𝑖OLDcdip 30729 CPreHilOLDccphlo 30841 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 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 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-cn 23251 df-cnp 23252 df-t1 23338 df-haus 23339 df-tx 23586 df-hmeo 23779 df-xms 24346 df-ms 24347 df-tms 24348 df-grpo 30522 df-gid 30523 df-ginv 30524 df-gdiv 30525 df-ablo 30574 df-vc 30588 df-nv 30621 df-va 30624 df-ba 30625 df-sm 30626 df-0v 30627 df-vs 30628 df-nmcv 30629 df-ims 30630 df-dip 30730 df-ph 30842 |
This theorem is referenced by: ipasslem9 30867 |
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