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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 30167 to be axioms, the need for the ax-hcompl 29949 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 30182.) (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 2738 | . . 3 ⊢ ((projℎ‘𝐻)‘𝐴) = ((projℎ‘𝐻)‘𝐴) | |
2 | pjeq 30146 | . . 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 232 | . 2 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (((projℎ‘𝐻)‘𝐴) ∈ 𝐻 ∧ ∃𝑥 ∈ (⊥‘𝐻)𝐴 = (((projℎ‘𝐻)‘𝐴) +ℎ 𝑥))) |
4 | 3 | simpld 496 | 1 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘𝐻)‘𝐴) ∈ 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3072 ‘cfv 6492 (class class class)co 7350 ℋchba 29666 +ℎ cva 29667 Cℋ cch 29676 ⊥cort 29677 projℎcpjh 29684 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-inf2 9511 ax-cc 10305 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-pre-sup 11063 ax-addf 11064 ax-mulf 11065 ax-hilex 29746 ax-hfvadd 29747 ax-hvcom 29748 ax-hvass 29749 ax-hv0cl 29750 ax-hvaddid 29751 ax-hfvmul 29752 ax-hvmulid 29753 ax-hvmulass 29754 ax-hvdistr1 29755 ax-hvdistr2 29756 ax-hvmul0 29757 ax-hfi 29826 ax-his1 29829 ax-his2 29830 ax-his3 29831 ax-his4 29832 ax-hcompl 29949 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-oadd 8384 df-omul 8385 df-er 8582 df-map 8701 df-pm 8702 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-fi 9281 df-sup 9312 df-inf 9313 df-oi 9380 df-card 9809 df-acn 9812 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-n0 12348 df-z 12434 df-uz 12698 df-q 12804 df-rp 12846 df-xneg 12963 df-xadd 12964 df-xmul 12965 df-ico 13200 df-icc 13201 df-fz 13355 df-fl 13627 df-seq 13837 df-exp 13898 df-cj 14919 df-re 14920 df-im 14921 df-sqrt 15055 df-abs 15056 df-clim 15306 df-rlim 15307 df-rest 17240 df-topgen 17261 df-psmet 20717 df-xmet 20718 df-met 20719 df-bl 20720 df-mopn 20721 df-fbas 20722 df-fg 20723 df-top 22171 df-topon 22188 df-bases 22224 df-cld 22298 df-ntr 22299 df-cls 22300 df-nei 22377 df-lm 22508 df-haus 22594 df-fil 23125 df-fm 23217 df-flim 23218 df-flf 23219 df-cfil 24547 df-cau 24548 df-cmet 24549 df-grpo 29240 df-gid 29241 df-ginv 29242 df-gdiv 29243 df-ablo 29292 df-vc 29306 df-nv 29339 df-va 29342 df-ba 29343 df-sm 29344 df-0v 29345 df-vs 29346 df-nmcv 29347 df-ims 29348 df-ssp 29469 df-ph 29560 df-cbn 29610 df-hnorm 29715 df-hba 29716 df-hvsub 29718 df-hlim 29719 df-hcau 29720 df-sh 29954 df-ch 29968 df-oc 29999 df-ch0 30000 df-shs 30055 df-pjh 30142 |
This theorem is referenced by: pjhcl 30148 pjcli 30164 pjpjhth 30172 pjoccl 30180 pjspansn 30324 pjorthi 30416 pjcompi 30419 |
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