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Mirrors > Home > HSE Home > Th. List > ssjo | Structured version Visualization version GIF version |
Description: The lattice join of a subset with its orthocomplement is the whole space. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ssjo | ⊢ (𝐴 ⊆ ℋ → (𝐴 ∨ℋ (⊥‘𝐴)) = ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ocss 28990 | . . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ) | |
2 | sshjval 29055 | . . 3 ⊢ ((𝐴 ⊆ ℋ ∧ (⊥‘𝐴) ⊆ ℋ) → (𝐴 ∨ℋ (⊥‘𝐴)) = (⊥‘(⊥‘(𝐴 ∪ (⊥‘𝐴))))) | |
3 | 1, 2 | mpdan 683 | . 2 ⊢ (𝐴 ⊆ ℋ → (𝐴 ∨ℋ (⊥‘𝐴)) = (⊥‘(⊥‘(𝐴 ∪ (⊥‘𝐴))))) |
4 | ssun1 4147 | . . . . . . . 8 ⊢ 𝐴 ⊆ (𝐴 ∪ (⊥‘𝐴)) | |
5 | 1 | ancli 549 | . . . . . . . . . 10 ⊢ (𝐴 ⊆ ℋ → (𝐴 ⊆ ℋ ∧ (⊥‘𝐴) ⊆ ℋ)) |
6 | unss 4159 | . . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℋ ∧ (⊥‘𝐴) ⊆ ℋ) ↔ (𝐴 ∪ (⊥‘𝐴)) ⊆ ℋ) | |
7 | 5, 6 | sylib 219 | . . . . . . . . 9 ⊢ (𝐴 ⊆ ℋ → (𝐴 ∪ (⊥‘𝐴)) ⊆ ℋ) |
8 | occon 28992 | . . . . . . . . 9 ⊢ ((𝐴 ⊆ ℋ ∧ (𝐴 ∪ (⊥‘𝐴)) ⊆ ℋ) → (𝐴 ⊆ (𝐴 ∪ (⊥‘𝐴)) → (⊥‘(𝐴 ∪ (⊥‘𝐴))) ⊆ (⊥‘𝐴))) | |
9 | 7, 8 | mpdan 683 | . . . . . . . 8 ⊢ (𝐴 ⊆ ℋ → (𝐴 ⊆ (𝐴 ∪ (⊥‘𝐴)) → (⊥‘(𝐴 ∪ (⊥‘𝐴))) ⊆ (⊥‘𝐴))) |
10 | 4, 9 | mpi 20 | . . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (⊥‘(𝐴 ∪ (⊥‘𝐴))) ⊆ (⊥‘𝐴)) |
11 | ssun2 4148 | . . . . . . . 8 ⊢ (⊥‘𝐴) ⊆ (𝐴 ∪ (⊥‘𝐴)) | |
12 | occon 28992 | . . . . . . . . 9 ⊢ (((⊥‘𝐴) ⊆ ℋ ∧ (𝐴 ∪ (⊥‘𝐴)) ⊆ ℋ) → ((⊥‘𝐴) ⊆ (𝐴 ∪ (⊥‘𝐴)) → (⊥‘(𝐴 ∪ (⊥‘𝐴))) ⊆ (⊥‘(⊥‘𝐴)))) | |
13 | 1, 7, 12 | syl2anc 584 | . . . . . . . 8 ⊢ (𝐴 ⊆ ℋ → ((⊥‘𝐴) ⊆ (𝐴 ∪ (⊥‘𝐴)) → (⊥‘(𝐴 ∪ (⊥‘𝐴))) ⊆ (⊥‘(⊥‘𝐴)))) |
14 | 11, 13 | mpi 20 | . . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (⊥‘(𝐴 ∪ (⊥‘𝐴))) ⊆ (⊥‘(⊥‘𝐴))) |
15 | 10, 14 | ssind 4208 | . . . . . 6 ⊢ (𝐴 ⊆ ℋ → (⊥‘(𝐴 ∪ (⊥‘𝐴))) ⊆ ((⊥‘𝐴) ∩ (⊥‘(⊥‘𝐴)))) |
16 | ocsh 28988 | . . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ Sℋ ) | |
17 | ocin 29001 | . . . . . . 7 ⊢ ((⊥‘𝐴) ∈ Sℋ → ((⊥‘𝐴) ∩ (⊥‘(⊥‘𝐴))) = 0ℋ) | |
18 | 16, 17 | syl 17 | . . . . . 6 ⊢ (𝐴 ⊆ ℋ → ((⊥‘𝐴) ∩ (⊥‘(⊥‘𝐴))) = 0ℋ) |
19 | 15, 18 | sseqtrd 4006 | . . . . 5 ⊢ (𝐴 ⊆ ℋ → (⊥‘(𝐴 ∪ (⊥‘𝐴))) ⊆ 0ℋ) |
20 | ocsh 28988 | . . . . . 6 ⊢ ((𝐴 ∪ (⊥‘𝐴)) ⊆ ℋ → (⊥‘(𝐴 ∪ (⊥‘𝐴))) ∈ Sℋ ) | |
21 | sh0le 29145 | . . . . . 6 ⊢ ((⊥‘(𝐴 ∪ (⊥‘𝐴))) ∈ Sℋ → 0ℋ ⊆ (⊥‘(𝐴 ∪ (⊥‘𝐴)))) | |
22 | 7, 20, 21 | 3syl 18 | . . . . 5 ⊢ (𝐴 ⊆ ℋ → 0ℋ ⊆ (⊥‘(𝐴 ∪ (⊥‘𝐴)))) |
23 | 19, 22 | eqssd 3983 | . . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘(𝐴 ∪ (⊥‘𝐴))) = 0ℋ) |
24 | 23 | fveq2d 6668 | . . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘(𝐴 ∪ (⊥‘𝐴)))) = (⊥‘0ℋ)) |
25 | choc0 29031 | . . 3 ⊢ (⊥‘0ℋ) = ℋ | |
26 | 24, 25 | syl6eq 2872 | . 2 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘(𝐴 ∪ (⊥‘𝐴)))) = ℋ) |
27 | 3, 26 | eqtrd 2856 | 1 ⊢ (𝐴 ⊆ ℋ → (𝐴 ∨ℋ (⊥‘𝐴)) = ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∪ cun 3933 ∩ cin 3934 ⊆ wss 3935 ‘cfv 6349 (class class class)co 7145 ℋchba 28624 Sℋ csh 28633 ⊥cort 28635 ∨ℋ chj 28638 0ℋc0h 28640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 ax-hilex 28704 ax-hfvadd 28705 ax-hvcom 28706 ax-hvass 28707 ax-hv0cl 28708 ax-hvaddid 28709 ax-hfvmul 28710 ax-hvmulid 28711 ax-hvmulass 28712 ax-hvdistr1 28713 ax-hvdistr2 28714 ax-hvmul0 28715 ax-hfi 28784 ax-his1 28787 ax-his2 28788 ax-his3 28789 ax-his4 28790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-map 8398 df-pm 8399 df-en 8499 df-dom 8500 df-sdom 8501 df-sup 8895 df-inf 8896 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-n0 11887 df-z 11971 df-uz 12233 df-q 12338 df-rp 12380 df-xneg 12497 df-xadd 12498 df-xmul 12499 df-icc 12735 df-seq 13360 df-exp 13420 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-topgen 16707 df-psmet 20467 df-xmet 20468 df-met 20469 df-bl 20470 df-mopn 20471 df-top 21432 df-topon 21449 df-bases 21484 df-lm 21767 df-haus 21853 df-grpo 28198 df-gid 28199 df-ginv 28200 df-gdiv 28201 df-ablo 28250 df-vc 28264 df-nv 28297 df-va 28300 df-ba 28301 df-sm 28302 df-0v 28303 df-vs 28304 df-nmcv 28305 df-ims 28306 df-hnorm 28673 df-hvsub 28676 df-hlim 28677 df-sh 28912 df-ch 28926 df-oc 28957 df-ch0 28958 df-chj 29015 |
This theorem is referenced by: chjoi 29193 |
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