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Mirrors > Home > HSE Home > Th. List > hlimreui | Structured version Visualization version GIF version |
Description: The limit of a Hilbert space sequence is unique. (Contributed by NM, 19-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlimreui | ⊢ (∃𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ↔ ∃!𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlimuni 29105 | . . . 4 ⊢ ((𝐹 ⇝𝑣 𝑥 ∧ 𝐹 ⇝𝑣 𝑦) → 𝑥 = 𝑦) | |
2 | 1 | rgen2w 3081 | . . 3 ⊢ ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 ((𝐹 ⇝𝑣 𝑥 ∧ 𝐹 ⇝𝑣 𝑦) → 𝑥 = 𝑦) |
3 | 2 | biantru 534 | . 2 ⊢ (∃𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ↔ (∃𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ∧ ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 ((𝐹 ⇝𝑣 𝑥 ∧ 𝐹 ⇝𝑣 𝑦) → 𝑥 = 𝑦))) |
4 | breq2 5029 | . . 3 ⊢ (𝑥 = 𝑦 → (𝐹 ⇝𝑣 𝑥 ↔ 𝐹 ⇝𝑣 𝑦)) | |
5 | 4 | reu4 3642 | . 2 ⊢ (∃!𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ↔ (∃𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ∧ ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 ((𝐹 ⇝𝑣 𝑥 ∧ 𝐹 ⇝𝑣 𝑦) → 𝑥 = 𝑦))) |
6 | 3, 5 | bitr4i 281 | 1 ⊢ (∃𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ↔ ∃!𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∀wral 3068 ∃wrex 3069 ∃!wreu 3070 class class class wbr 5025 ⇝𝑣 chli 28794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 ax-pre-sup 10638 ax-addf 10639 ax-mulf 10640 ax-hilex 28866 ax-hfvadd 28867 ax-hvcom 28868 ax-hvass 28869 ax-hv0cl 28870 ax-hvaddid 28871 ax-hfvmul 28872 ax-hvmulid 28873 ax-hvmulass 28874 ax-hvdistr1 28875 ax-hvdistr2 28876 ax-hvmul0 28877 ax-hfi 28946 ax-his1 28949 ax-his2 28950 ax-his3 28951 ax-his4 28952 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-map 8411 df-pm 8412 df-en 8521 df-dom 8522 df-sdom 8523 df-sup 8924 df-inf 8925 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-div 11321 df-nn 11660 df-2 11722 df-3 11723 df-4 11724 df-n0 11920 df-z 12006 df-uz 12268 df-q 12374 df-rp 12416 df-xneg 12533 df-xadd 12534 df-xmul 12535 df-icc 12771 df-seq 13404 df-exp 13465 df-cj 14491 df-re 14492 df-im 14493 df-sqrt 14627 df-abs 14628 df-topgen 16760 df-psmet 20143 df-xmet 20144 df-met 20145 df-bl 20146 df-mopn 20147 df-top 21579 df-topon 21596 df-bases 21631 df-lm 21914 df-haus 22000 df-grpo 28360 df-gid 28361 df-ginv 28362 df-gdiv 28363 df-ablo 28412 df-vc 28426 df-nv 28459 df-va 28462 df-ba 28463 df-sm 28464 df-0v 28465 df-vs 28466 df-nmcv 28467 df-ims 28468 df-hnorm 28835 df-hvsub 28838 df-hlim 28839 |
This theorem is referenced by: hlimeui 29107 |
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