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Mirrors > Home > HSE Home > Th. List > axhcompl-zf | Structured version Visualization version GIF version |
Description: Derive axiom ax-hcompl 27443 from Hilbert space under ZF set theory. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (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 472 | . . . . . 6 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑𝑚 ℕ)) → 𝐹 ∈ (Cau‘(IndMet‘𝑈))) | |
3 | eqid 2610 | . . . . . . 7 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈) | |
4 | eqid 2610 | . . . . . . 7 ⊢ (MetOpen‘(IndMet‘𝑈)) = (MetOpen‘(IndMet‘𝑈)) | |
5 | 3, 4 | hlcompl 27155 | . . . . . 6 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐹 ∈ (Cau‘(IndMet‘𝑈))) → 𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))) |
6 | 1, 2, 5 | sylancr 694 | . . . . 5 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑𝑚 ℕ)) → 𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))) |
7 | eldm2g 5242 | . . . . . 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 221 | . . . 4 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑𝑚 ℕ)) → ∃𝑥〈𝐹, 𝑥〉 ∈ (⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))) |
10 | df-br 4584 | . . . . . 6 ⊢ (𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥 ↔ 〈𝐹, 𝑥〉 ∈ (⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))) | |
11 | 1 | hlnvi 27132 | . . . . . . . . . 10 ⊢ 𝑈 ∈ NrmCVec |
12 | df-hba 27210 | . . . . . . . . . . . 12 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
13 | axhil.1 | . . . . . . . . . . . . 13 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
14 | 13 | fveq2i 6106 | . . . . . . . . . . . 12 ⊢ (BaseSet‘𝑈) = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
15 | 12, 14 | eqtr4i 2635 | . . . . . . . . . . 11 ⊢ ℋ = (BaseSet‘𝑈) |
16 | 15, 3 | imsxmet 26931 | . . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) ∈ (∞Met‘ ℋ)) |
17 | 4 | mopntopon 22054 | . . . . . . . . . 10 ⊢ ((IndMet‘𝑈) ∈ (∞Met‘ ℋ) → (MetOpen‘(IndMet‘𝑈)) ∈ (TopOn‘ ℋ)) |
18 | 11, 16, 17 | mp2b 10 | . . . . . . . . 9 ⊢ (MetOpen‘(IndMet‘𝑈)) ∈ (TopOn‘ ℋ) |
19 | lmcl 20911 | . . . . . . . . 9 ⊢ (((MetOpen‘(IndMet‘𝑈)) ∈ (TopOn‘ ℋ) ∧ 𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥) → 𝑥 ∈ ℋ) | |
20 | 18, 19 | mpan 702 | . . . . . . . 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 27221 | . . . . . . . . . . . 12 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘(IndMet‘𝑈))) ↾ ( ℋ ↑𝑚 ℕ)) |
23 | 22 | breqi 4589 | . . . . . . . . . . 11 ⊢ (𝐹 ⇝𝑣 𝑥 ↔ 𝐹((⇝𝑡‘(MetOpen‘(IndMet‘𝑈))) ↾ ( ℋ ↑𝑚 ℕ))𝑥) |
24 | vex 3176 | . . . . . . . . . . . 12 ⊢ 𝑥 ∈ V | |
25 | 24 | brres 5323 | . . . . . . . . . . 11 ⊢ (𝐹((⇝𝑡‘(MetOpen‘(IndMet‘𝑈))) ↾ ( ℋ ↑𝑚 ℕ))𝑥 ↔ (𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥 ∧ 𝐹 ∈ ( ℋ ↑𝑚 ℕ))) |
26 | ancom 465 | . . . . . . . . . . 11 ⊢ ((𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥 ∧ 𝐹 ∈ ( ℋ ↑𝑚 ℕ)) ↔ (𝐹 ∈ ( ℋ ↑𝑚 ℕ) ∧ 𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥)) | |
27 | 23, 25, 26 | 3bitri 285 | . . . . . . . . . 10 ⊢ (𝐹 ⇝𝑣 𝑥 ↔ (𝐹 ∈ ( ℋ ↑𝑚 ℕ) ∧ 𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥)) |
28 | 27 | baib 942 | . . . . . . . . 9 ⊢ (𝐹 ∈ ( ℋ ↑𝑚 ℕ) → (𝐹 ⇝𝑣 𝑥 ↔ 𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥)) |
29 | 28 | adantl 481 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑𝑚 ℕ)) → (𝐹 ⇝𝑣 𝑥 ↔ 𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥)) |
30 | 29 | biimprd 237 | . . . . . . 7 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑𝑚 ℕ)) → (𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥 → 𝐹 ⇝𝑣 𝑥)) |
31 | 21, 30 | jcad 554 | . . . . . 6 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑𝑚 ℕ)) → (𝐹(⇝𝑡‘(MetOpen‘(IndMet‘𝑈)))𝑥 → (𝑥 ∈ ℋ ∧ 𝐹 ⇝𝑣 𝑥))) |
32 | 10, 31 | syl5bir 232 | . . . . 5 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑𝑚 ℕ)) → (〈𝐹, 𝑥〉 ∈ (⇝𝑡‘(MetOpen‘(IndMet‘𝑈))) → (𝑥 ∈ ℋ ∧ 𝐹 ⇝𝑣 𝑥))) |
33 | 32 | eximdv 1833 | . . . 4 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑𝑚 ℕ)) → (∃𝑥〈𝐹, 𝑥〉 ∈ (⇝𝑡‘(MetOpen‘(IndMet‘𝑈))) → ∃𝑥(𝑥 ∈ ℋ ∧ 𝐹 ⇝𝑣 𝑥))) |
34 | 9, 33 | mpd 15 | . . 3 ⊢ ((𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑𝑚 ℕ)) → ∃𝑥(𝑥 ∈ ℋ ∧ 𝐹 ⇝𝑣 𝑥)) |
35 | elin 3758 | . . 3 ⊢ (𝐹 ∈ ((Cau‘(IndMet‘𝑈)) ∩ ( ℋ ↑𝑚 ℕ)) ↔ (𝐹 ∈ (Cau‘(IndMet‘𝑈)) ∧ 𝐹 ∈ ( ℋ ↑𝑚 ℕ))) | |
36 | df-rex 2902 | . . 3 ⊢ (∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥 ↔ ∃𝑥(𝑥 ∈ ℋ ∧ 𝐹 ⇝𝑣 𝑥)) | |
37 | 34, 35, 36 | 3imtr4i 280 | . 2 ⊢ (𝐹 ∈ ((Cau‘(IndMet‘𝑈)) ∩ ( ℋ ↑𝑚 ℕ)) → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) |
38 | 13, 11, 15, 3 | h2hcau 27220 | . 2 ⊢ Cauchy = ((Cau‘(IndMet‘𝑈)) ∩ ( ℋ ↑𝑚 ℕ)) |
39 | 37, 38 | eleq2s 2706 | 1 ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∃wrex 2897 ∩ cin 3539 〈cop 4131 class class class wbr 4583 dom cdm 5038 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 ℕcn 10897 ∞Metcxmt 19552 MetOpencmopn 19557 TopOnctopon 20518 ⇝𝑡clm 20840 Caucca 22859 NrmCVeccnv 26823 BaseSetcba 26825 IndMetcims 26830 CHilOLDchlo 27125 ℋchil 27160 +ℎ cva 27161 ·ℎ csm 27162 normℎcno 27164 Cauchyccau 27167 ⇝𝑣 chli 27168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ico 12052 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-rest 15906 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-top 20521 df-bases 20522 df-topon 20523 df-ntr 20634 df-nei 20712 df-lm 20843 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-cfil 22861 df-cau 22862 df-cmet 22863 df-grpo 26731 df-gid 26732 df-ginv 26733 df-gdiv 26734 df-ablo 26783 df-vc 26798 df-nv 26831 df-va 26834 df-ba 26835 df-sm 26836 df-0v 26837 df-vs 26838 df-nmcv 26839 df-ims 26840 df-cbn 27103 df-hlo 27126 df-hba 27210 df-hvsub 27212 df-hlim 27213 df-hcau 27214 |
This theorem is referenced by: (None) |
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