| Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > HSE Home > Th. List > axhcompl-zf | Structured version Visualization version GIF version | ||
| Description: Derive Axiom ax-hcompl 31221 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 482 | . . . . . 6 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ)) → 𝐹 ∈ (Cau‘(IndMet‘𝑈))) | |
| 3 | eqid 2737 | . . . . . . 7 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈) | |
| 4 | eqid 2737 | . . . . . . 7 ⊢ (MetOpen‘(IndMet‘𝑈)) = (MetOpen‘(IndMet‘𝑈)) | |
| 5 | 3, 4 | hlcompl 30934 | . . . . . 6 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐹 ∈ (Cau‘(IndMet‘𝑈))) → 𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))) |
| 6 | 1, 2, 5 | sylancr 587 | . . . . 5 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ)) → 𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))) |
| 7 | eldm2g 5910 | . . . . . 6 ⊢ (𝐹 ∈ (Cau‘(IndMet‘𝑈)) → (𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(IndMet‘𝑈))) ↔ ∃𝑥〈𝐹, 𝑥〉 ∈ (⇝𝑡‘(MetOpen‘(IndMet‘𝑈))))) | |
| 8 | 7 | adantr 480 | . . . . 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 232 | . . . 4 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ)) → ∃𝑥〈𝐹, 𝑥〉 ∈ (⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))) |
| 10 | df-br 5144 | . . . . . 6 ⊢ (𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥 ↔ 〈𝐹, 𝑥〉 ∈ (⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))) | |
| 11 | 1 | hlnvi 30911 | . . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec |
| 12 | df-hba 30988 | . . . . . . . . . . . 12 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
| 13 | axhil.1 | . . . . . . . . . . . . 13 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
| 14 | 13 | fveq2i 6909 | . . . . . . . . . . . 12 ⊢ (BaseSet‘𝑈) = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
| 15 | 12, 14 | eqtr4i 2768 | . . . . . . . . . . 11 ⊢ ℋ = (BaseSet‘𝑈) |
| 16 | 15, 3 | imsxmet 30711 | . . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) ∈ (∞Met‘ ℋ)) |
| 17 | 4 | mopntopon 24449 | . . . . . . . . . 10 ⊢ ((IndMet‘𝑈) ∈ (∞Met‘ ℋ) → (MetOpen‘(IndMet‘𝑈)) ∈ (TopOn‘ ℋ)) |
| 18 | 11, 16, 17 | mp2b 10 | . . . . . . . . 9 ⊢ (MetOpen‘(IndMet‘𝑈)) ∈ (TopOn‘ ℋ) |
| 19 | lmcl 23305 | . . . . . . . . 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 30999 | . . . . . . . . . . . 12 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘(IndMet‘𝑈))) ↾ ( ℋ ↑m ℕ)) |
| 23 | 22 | breqi 5149 | . . . . . . . . . . 11 ⊢ (𝐹 ⇝𝑣 𝑥 ↔ 𝐹((⇝𝑡‘(MetOpen‘(IndMet‘𝑈))) ↾ ( ℋ ↑m ℕ))𝑥) |
| 24 | brres 6004 | . . . . . . . . . . . 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 3485 | . . . . . . . . . . 11 ⊢ (𝐹((⇝𝑡‘(MetOpen‘(IndMet‘𝑈))) ↾ ( ℋ ↑m ℕ))𝑥 ↔ (𝐹 ∈ ( ℋ ↑m ℕ) ∧ 𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥)) |
| 26 | 23, 25 | bitri 275 | . . . . . . . . . 10 ⊢ (𝐹 ⇝𝑣 𝑥 ↔ (𝐹 ∈ ( ℋ ↑m ℕ) ∧ 𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥)) |
| 27 | 26 | baib 535 | . . . . . . . . 9 ⊢ (𝐹 ∈ ( ℋ ↑m ℕ) → (𝐹 ⇝𝑣 𝑥 ↔ 𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥)) |
| 28 | 27 | adantl 481 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ)) → (𝐹 ⇝𝑣 𝑥 ↔ 𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥)) |
| 29 | 28 | biimprd 248 | . . . . . . 7 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ)) → (𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥 → 𝐹 ⇝𝑣 𝑥)) |
| 30 | 21, 29 | jcad 512 | . . . . . 6 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ)) → (𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥 → (𝑥 ∈ ℋ ∧ 𝐹 ⇝𝑣 𝑥))) |
| 31 | 10, 30 | biimtrrid 243 | . . . . 5 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ)) → (〈𝐹, 𝑥〉 ∈ (⇝𝑡‘(MetOpen‘(IndMet‘𝑈))) → (𝑥 ∈ ℋ ∧ 𝐹 ⇝𝑣 𝑥))) |
| 32 | 31 | eximdv 1917 | . . . 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 3967 | . . 3 ⊢ (𝐹 ∈ ((Cau‘(IndMet‘𝑈)) ∩ ( ℋ ↑m ℕ)) ↔ (𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑m ℕ))) | |
| 35 | df-rex 3071 | . . 3 ⊢ (∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥 ↔ ∃𝑥(𝑥 ∈ ℋ ∧ 𝐹 ⇝𝑣 𝑥)) | |
| 36 | 33, 34, 35 | 3imtr4i 292 | . 2 ⊢ (𝐹 ∈ ((Cau‘(IndMet‘𝑈)) ∩ ( ℋ ↑m ℕ)) → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) |
| 37 | 13, 11, 15, 3 | h2hcau 30998 | . 2 ⊢ Cauchy = ((Cau‘(IndMet‘𝑈)) ∩ ( ℋ ↑m ℕ)) |
| 38 | 36, 37 | eleq2s 2859 | 1 ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ∩ cin 3950 〈cop 4632 class class class wbr 5143 dom cdm 5685 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 ℕcn 12266 ∞Metcxmet 21349 MetOpencmopn 21354 TopOnctopon 22916 ⇝𝑡clm 23234 Cauccau 25287 NrmCVeccnv 30603 BaseSetcba 30605 IndMetcims 30610 CHilOLDchlo 30904 ℋchba 30938 +ℎ cva 30939 ·ℎ csm 30940 normℎcno 30942 Cauchyccauold 30945 ⇝𝑣 chli 30946 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ico 13393 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-rest 17467 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-top 22900 df-topon 22917 df-bases 22953 df-ntr 23028 df-nei 23106 df-lm 23237 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-cfil 25289 df-cau 25290 df-cmet 25291 df-grpo 30512 df-gid 30513 df-ginv 30514 df-gdiv 30515 df-ablo 30564 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-vs 30618 df-nmcv 30619 df-ims 30620 df-cbn 30882 df-hlo 30905 df-hba 30988 df-hvsub 30990 df-hlim 30991 df-hcau 30992 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |