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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 29390 | . . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ) | |
2 | sshjval 29455 | . . 3 ⊢ ((𝐴 ⊆ ℋ ∧ (⊥‘𝐴) ⊆ ℋ) → (𝐴 ∨ℋ (⊥‘𝐴)) = (⊥‘(⊥‘(𝐴 ∪ (⊥‘𝐴))))) | |
3 | 1, 2 | mpdan 687 | . 2 ⊢ (𝐴 ⊆ ℋ → (𝐴 ∨ℋ (⊥‘𝐴)) = (⊥‘(⊥‘(𝐴 ∪ (⊥‘𝐴))))) |
4 | ssun1 4100 | . . . . . . . 8 ⊢ 𝐴 ⊆ (𝐴 ∪ (⊥‘𝐴)) | |
5 | 1 | ancli 552 | . . . . . . . . . 10 ⊢ (𝐴 ⊆ ℋ → (𝐴 ⊆ ℋ ∧ (⊥‘𝐴) ⊆ ℋ)) |
6 | unss 4112 | . . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℋ ∧ (⊥‘𝐴) ⊆ ℋ) ↔ (𝐴 ∪ (⊥‘𝐴)) ⊆ ℋ) | |
7 | 5, 6 | sylib 221 | . . . . . . . . 9 ⊢ (𝐴 ⊆ ℋ → (𝐴 ∪ (⊥‘𝐴)) ⊆ ℋ) |
8 | occon 29392 | . . . . . . . . 9 ⊢ ((𝐴 ⊆ ℋ ∧ (𝐴 ∪ (⊥‘𝐴)) ⊆ ℋ) → (𝐴 ⊆ (𝐴 ∪ (⊥‘𝐴)) → (⊥‘(𝐴 ∪ (⊥‘𝐴))) ⊆ (⊥‘𝐴))) | |
9 | 7, 8 | mpdan 687 | . . . . . . . 8 ⊢ (𝐴 ⊆ ℋ → (𝐴 ⊆ (𝐴 ∪ (⊥‘𝐴)) → (⊥‘(𝐴 ∪ (⊥‘𝐴))) ⊆ (⊥‘𝐴))) |
10 | 4, 9 | mpi 20 | . . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (⊥‘(𝐴 ∪ (⊥‘𝐴))) ⊆ (⊥‘𝐴)) |
11 | ssun2 4101 | . . . . . . . 8 ⊢ (⊥‘𝐴) ⊆ (𝐴 ∪ (⊥‘𝐴)) | |
12 | occon 29392 | . . . . . . . . 9 ⊢ (((⊥‘𝐴) ⊆ ℋ ∧ (𝐴 ∪ (⊥‘𝐴)) ⊆ ℋ) → ((⊥‘𝐴) ⊆ (𝐴 ∪ (⊥‘𝐴)) → (⊥‘(𝐴 ∪ (⊥‘𝐴))) ⊆ (⊥‘(⊥‘𝐴)))) | |
13 | 1, 7, 12 | syl2anc 587 | . . . . . . . 8 ⊢ (𝐴 ⊆ ℋ → ((⊥‘𝐴) ⊆ (𝐴 ∪ (⊥‘𝐴)) → (⊥‘(𝐴 ∪ (⊥‘𝐴))) ⊆ (⊥‘(⊥‘𝐴)))) |
14 | 11, 13 | mpi 20 | . . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (⊥‘(𝐴 ∪ (⊥‘𝐴))) ⊆ (⊥‘(⊥‘𝐴))) |
15 | 10, 14 | ssind 4161 | . . . . . 6 ⊢ (𝐴 ⊆ ℋ → (⊥‘(𝐴 ∪ (⊥‘𝐴))) ⊆ ((⊥‘𝐴) ∩ (⊥‘(⊥‘𝐴)))) |
16 | ocsh 29388 | . . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ Sℋ ) | |
17 | ocin 29401 | . . . . . . 7 ⊢ ((⊥‘𝐴) ∈ Sℋ → ((⊥‘𝐴) ∩ (⊥‘(⊥‘𝐴))) = 0ℋ) | |
18 | 16, 17 | syl 17 | . . . . . 6 ⊢ (𝐴 ⊆ ℋ → ((⊥‘𝐴) ∩ (⊥‘(⊥‘𝐴))) = 0ℋ) |
19 | 15, 18 | sseqtrd 3955 | . . . . 5 ⊢ (𝐴 ⊆ ℋ → (⊥‘(𝐴 ∪ (⊥‘𝐴))) ⊆ 0ℋ) |
20 | ocsh 29388 | . . . . . 6 ⊢ ((𝐴 ∪ (⊥‘𝐴)) ⊆ ℋ → (⊥‘(𝐴 ∪ (⊥‘𝐴))) ∈ Sℋ ) | |
21 | sh0le 29545 | . . . . . 6 ⊢ ((⊥‘(𝐴 ∪ (⊥‘𝐴))) ∈ Sℋ → 0ℋ ⊆ (⊥‘(𝐴 ∪ (⊥‘𝐴)))) | |
22 | 7, 20, 21 | 3syl 18 | . . . . 5 ⊢ (𝐴 ⊆ ℋ → 0ℋ ⊆ (⊥‘(𝐴 ∪ (⊥‘𝐴)))) |
23 | 19, 22 | eqssd 3932 | . . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘(𝐴 ∪ (⊥‘𝐴))) = 0ℋ) |
24 | 23 | fveq2d 6739 | . . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘(𝐴 ∪ (⊥‘𝐴)))) = (⊥‘0ℋ)) |
25 | choc0 29431 | . . 3 ⊢ (⊥‘0ℋ) = ℋ | |
26 | 24, 25 | eqtrdi 2795 | . 2 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘(𝐴 ∪ (⊥‘𝐴)))) = ℋ) |
27 | 3, 26 | eqtrd 2778 | 1 ⊢ (𝐴 ⊆ ℋ → (𝐴 ∨ℋ (⊥‘𝐴)) = ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 ∪ cun 3878 ∩ cin 3879 ⊆ wss 3880 ‘cfv 6397 (class class class)co 7231 ℋchba 29024 Sℋ csh 29033 ⊥cort 29035 ∨ℋ chj 29038 0ℋc0h 29040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 ax-pre-sup 10831 ax-addf 10832 ax-mulf 10833 ax-hilex 29104 ax-hfvadd 29105 ax-hvcom 29106 ax-hvass 29107 ax-hv0cl 29108 ax-hvaddid 29109 ax-hfvmul 29110 ax-hvmulid 29111 ax-hvmulass 29112 ax-hvdistr1 29113 ax-hvdistr2 29114 ax-hvmul0 29115 ax-hfi 29184 ax-his1 29187 ax-his2 29188 ax-his3 29189 ax-his4 29190 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-om 7663 df-1st 7779 df-2nd 7780 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-er 8411 df-map 8530 df-pm 8531 df-en 8647 df-dom 8648 df-sdom 8649 df-sup 9082 df-inf 9083 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-div 11514 df-nn 11855 df-2 11917 df-3 11918 df-4 11919 df-n0 12115 df-z 12201 df-uz 12463 df-q 12569 df-rp 12611 df-xneg 12728 df-xadd 12729 df-xmul 12730 df-icc 12966 df-seq 13599 df-exp 13660 df-cj 14686 df-re 14687 df-im 14688 df-sqrt 14822 df-abs 14823 df-topgen 16972 df-psmet 20379 df-xmet 20380 df-met 20381 df-bl 20382 df-mopn 20383 df-top 21815 df-topon 21832 df-bases 21867 df-lm 22150 df-haus 22236 df-grpo 28598 df-gid 28599 df-ginv 28600 df-gdiv 28601 df-ablo 28650 df-vc 28664 df-nv 28697 df-va 28700 df-ba 28701 df-sm 28702 df-0v 28703 df-vs 28704 df-nmcv 28705 df-ims 28706 df-hnorm 29073 df-hvsub 29076 df-hlim 29077 df-sh 29312 df-ch 29326 df-oc 29357 df-ch0 29358 df-chj 29415 |
This theorem is referenced by: chjoi 29593 |
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