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Mirrors > Home > HSE Home > Th. List > axpjcl | Structured version Visualization version GIF version |
Description: Closure of a projection in its subspace. If we consider this together with axpjpj 29287 to be axioms, the need for the ax-hcompl 29069 can often be avoided for the kinds of theorems we are interested in here. An interesting project is to see how far we can go by using them in place of it. In particular, we can prove the orthomodular law pjomli 29302.) (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axpjcl | ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘𝐻)‘𝐴) ∈ 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2759 | . . 3 ⊢ ((projℎ‘𝐻)‘𝐴) = ((projℎ‘𝐻)‘𝐴) | |
2 | pjeq 29266 | . . 3 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (((projℎ‘𝐻)‘𝐴) = ((projℎ‘𝐻)‘𝐴) ↔ (((projℎ‘𝐻)‘𝐴) ∈ 𝐻 ∧ ∃𝑥 ∈ (⊥‘𝐻)𝐴 = (((projℎ‘𝐻)‘𝐴) +ℎ 𝑥)))) | |
3 | 1, 2 | mpbii 236 | . 2 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (((projℎ‘𝐻)‘𝐴) ∈ 𝐻 ∧ ∃𝑥 ∈ (⊥‘𝐻)𝐴 = (((projℎ‘𝐻)‘𝐴) +ℎ 𝑥))) |
4 | 3 | simpld 499 | 1 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘𝐻)‘𝐴) ∈ 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 ∃wrex 3069 ‘cfv 6328 (class class class)co 7143 ℋchba 28786 +ℎ cva 28787 Cℋ cch 28796 ⊥cort 28797 projℎcpjh 28804 |
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-inf2 9122 ax-cc 9880 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 ax-hilex 28866 ax-hfvadd 28867 ax-hvcom 28868 ax-hvass 28869 ax-hv0cl 28870 ax-hvaddid 28871 ax-hfvmul 28872 ax-hvmulid 28873 ax-hvmulass 28874 ax-hvdistr1 28875 ax-hvdistr2 28876 ax-hvmul0 28877 ax-hfi 28946 ax-his1 28949 ax-his2 28950 ax-his3 28951 ax-his4 28952 ax-hcompl 29069 |
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-int 4832 df-iun 4878 df-iin 4879 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-se 5477 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-isom 6337 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-1o 8105 df-oadd 8109 df-omul 8110 df-er 8292 df-map 8411 df-pm 8412 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-fi 8893 df-sup 8924 df-inf 8925 df-oi 8992 df-card 9386 df-acn 9389 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-4 11724 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-icc 12771 df-fz 12925 df-fl 13196 df-seq 13404 df-exp 13465 df-cj 14491 df-re 14492 df-im 14493 df-sqrt 14627 df-abs 14628 df-clim 14878 df-rlim 14879 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-cld 21704 df-ntr 21705 df-cls 21706 df-nei 21783 df-lm 21914 df-haus 22000 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-ssp 28589 df-ph 28680 df-cbn 28730 df-hnorm 28835 df-hba 28836 df-hvsub 28838 df-hlim 28839 df-hcau 28840 df-sh 29074 df-ch 29088 df-oc 29119 df-ch0 29120 df-shs 29175 df-pjh 29262 |
This theorem is referenced by: pjhcl 29268 pjcli 29284 pjpjhth 29292 pjoccl 29300 pjspansn 29444 pjorthi 29536 pjcompi 29539 |
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