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Mirrors > Home > HSE Home > Th. List > axhcompl-zf | Structured version Visualization version GIF version |
Description: Derive Axiom ax-hcompl 29069 from Hilbert space under ZF set theory. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axhil.1 | ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 |
axhil.2 | ⊢ 𝑈 ∈ CHilOLD |
Ref | Expression |
---|---|
axhcompl-zf | ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axhil.2 | . . . . . 6 ⊢ 𝑈 ∈ CHilOLD | |
2 | simpl 487 | . . . . . 6 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ)) → 𝐹 ∈ (Cau‘(IndMet‘𝑈))) | |
3 | eqid 2759 | . . . . . . 7 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈) | |
4 | eqid 2759 | . . . . . . 7 ⊢ (MetOpen‘(IndMet‘𝑈)) = (MetOpen‘(IndMet‘𝑈)) | |
5 | 3, 4 | hlcompl 28782 | . . . . . 6 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐹 ∈ (Cau‘(IndMet‘𝑈))) → 𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))) |
6 | 1, 2, 5 | sylancr 591 | . . . . 5 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ)) → 𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))) |
7 | eldm2g 5732 | . . . . . 6 ⊢ (𝐹 ∈ (Cau‘(IndMet‘𝑈)) → (𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(IndMet‘𝑈))) ↔ ∃𝑥〈𝐹, 𝑥〉 ∈ (⇝𝑡‘(MetOpen‘(IndMet‘𝑈))))) | |
8 | 7 | adantr 485 | . . . . 5 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ)) → (𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(IndMet‘𝑈))) ↔ ∃𝑥〈𝐹, 𝑥〉 ∈ (⇝𝑡‘(MetOpen‘(IndMet‘𝑈))))) |
9 | 6, 8 | mpbid 235 | . . . 4 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ)) → ∃𝑥〈𝐹, 𝑥〉 ∈ (⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))) |
10 | df-br 5026 | . . . . . 6 ⊢ (𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥 ↔ 〈𝐹, 𝑥〉 ∈ (⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))) | |
11 | 1 | hlnvi 28759 | . . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec |
12 | df-hba 28836 | . . . . . . . . . . . 12 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
13 | axhil.1 | . . . . . . . . . . . . 13 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
14 | 13 | fveq2i 6654 | . . . . . . . . . . . 12 ⊢ (BaseSet‘𝑈) = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
15 | 12, 14 | eqtr4i 2785 | . . . . . . . . . . 11 ⊢ ℋ = (BaseSet‘𝑈) |
16 | 15, 3 | imsxmet 28559 | . . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) ∈ (∞Met‘ ℋ)) |
17 | 4 | mopntopon 23126 | . . . . . . . . . 10 ⊢ ((IndMet‘𝑈) ∈ (∞Met‘ ℋ) → (MetOpen‘(IndMet‘𝑈)) ∈ (TopOn‘ ℋ)) |
18 | 11, 16, 17 | mp2b 10 | . . . . . . . . 9 ⊢ (MetOpen‘(IndMet‘𝑈)) ∈ (TopOn‘ ℋ) |
19 | lmcl 21982 | . . . . . . . . 9 ⊢ (((MetOpen‘(IndMet‘𝑈)) ∈ (TopOn‘ ℋ) ∧ 𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥) → 𝑥 ∈ ℋ) | |
20 | 18, 19 | mpan 690 | . . . . . . . 8 ⊢ (𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥 → 𝑥 ∈ ℋ) |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ)) → (𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥 → 𝑥 ∈ ℋ)) |
22 | 13, 11, 15, 3, 4 | h2hlm 28847 | . . . . . . . . . . . 12 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘(IndMet‘𝑈))) ↾ ( ℋ ↑m ℕ)) |
23 | 22 | breqi 5031 | . . . . . . . . . . 11 ⊢ (𝐹 ⇝𝑣 𝑥 ↔ 𝐹((⇝𝑡‘(MetOpen‘(IndMet‘𝑈))) ↾ ( ℋ ↑m ℕ))𝑥) |
24 | brres 5823 | . . . . . . . . . . . 12 ⊢ (𝑥 ∈ V → (𝐹((⇝𝑡‘(MetOpen‘(IndMet‘𝑈))) ↾ ( ℋ ↑m ℕ))𝑥 ↔ (𝐹 ∈ ( ℋ ↑m ℕ) ∧ 𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥))) | |
25 | 24 | elv 3413 | . . . . . . . . . . 11 ⊢ (𝐹((⇝𝑡‘(MetOpen‘(IndMet‘𝑈))) ↾ ( ℋ ↑m ℕ))𝑥 ↔ (𝐹 ∈ ( ℋ ↑m ℕ) ∧ 𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥)) |
26 | 23, 25 | bitri 278 | . . . . . . . . . 10 ⊢ (𝐹 ⇝𝑣 𝑥 ↔ (𝐹 ∈ ( ℋ ↑m ℕ) ∧ 𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥)) |
27 | 26 | baib 540 | . . . . . . . . 9 ⊢ (𝐹 ∈ ( ℋ ↑m ℕ) → (𝐹 ⇝𝑣 𝑥 ↔ 𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥)) |
28 | 27 | adantl 486 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ)) → (𝐹 ⇝𝑣 𝑥 ↔ 𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥)) |
29 | 28 | biimprd 251 | . . . . . . 7 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ)) → (𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥 → 𝐹 ⇝𝑣 𝑥)) |
30 | 21, 29 | jcad 517 | . . . . . 6 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ)) → (𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥 → (𝑥 ∈ ℋ ∧ 𝐹 ⇝𝑣 𝑥))) |
31 | 10, 30 | syl5bir 246 | . . . . 5 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ)) → (〈𝐹, 𝑥〉 ∈ (⇝𝑡‘(MetOpen‘(IndMet‘𝑈))) → (𝑥 ∈ ℋ ∧ 𝐹 ⇝𝑣 𝑥))) |
32 | 31 | eximdv 1919 | . . . 4 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ)) → (∃𝑥〈𝐹, 𝑥〉 ∈ (⇝𝑡‘(MetOpen‘(IndMet‘𝑈))) → ∃𝑥(𝑥 ∈ ℋ ∧ 𝐹 ⇝𝑣 𝑥))) |
33 | 9, 32 | mpd 15 | . . 3 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ)) → ∃𝑥(𝑥 ∈ ℋ ∧ 𝐹 ⇝𝑣 𝑥)) |
34 | elin 3870 | . . 3 ⊢ (𝐹 ∈ ((Cau‘(IndMet‘𝑈)) ∩ ( ℋ ↑m ℕ)) ↔ (𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ))) | |
35 | df-rex 3074 | . . 3 ⊢ (∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥 ↔ ∃𝑥(𝑥 ∈ ℋ ∧ 𝐹 ⇝𝑣 𝑥)) | |
36 | 33, 34, 35 | 3imtr4i 296 | . 2 ⊢ (𝐹 ∈ ((Cau‘(IndMet‘𝑈)) ∩ ( ℋ ↑m ℕ)) → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) |
37 | 13, 11, 15, 3 | h2hcau 28846 | . 2 ⊢ Cauchy = ((Cau‘(IndMet‘𝑈)) ∩ ( ℋ ↑m ℕ)) |
38 | 36, 37 | eleq2s 2869 | 1 ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1539 ∃wex 1782 ∈ wcel 2112 ∃wrex 3069 Vcvv 3407 ∩ cin 3853 〈cop 4521 class class class wbr 5025 dom cdm 5517 ↾ cres 5519 ‘cfv 6328 (class class class)co 7143 ↑m cmap 8409 ℕcn 11659 ∞Metcxmet 20136 MetOpencmopn 20141 TopOnctopon 21595 ⇝𝑡clm 21911 Cauccau 23938 NrmCVeccnv 28451 BaseSetcba 28453 IndMetcims 28458 CHilOLDchlo 28752 ℋchba 28786 +ℎ cva 28787 ·ℎ csm 28788 normℎcno 28790 Cauchyccauold 28793 ⇝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 |
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-n0 11920 df-z 12006 df-uz 12268 df-q 12374 df-rp 12416 df-xneg 12533 df-xadd 12534 df-xmul 12535 df-ico 12770 df-seq 13404 df-exp 13465 df-cj 14491 df-re 14492 df-im 14493 df-sqrt 14627 df-abs 14628 df-rest 16739 df-topgen 16760 df-psmet 20143 df-xmet 20144 df-met 20145 df-bl 20146 df-mopn 20147 df-fbas 20148 df-fg 20149 df-top 21579 df-topon 21596 df-bases 21631 df-ntr 21705 df-nei 21783 df-lm 21914 df-fil 22531 df-fm 22623 df-flim 22624 df-flf 22625 df-cfil 23940 df-cau 23941 df-cmet 23942 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-cbn 28730 df-hlo 28753 df-hba 28836 df-hvsub 28838 df-hlim 28839 df-hcau 28840 |
This theorem is referenced by: (None) |
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