Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > chj0 | Structured version Visualization version GIF version |
Description: Join with Hilbert lattice zero. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chj0 | ⊢ (𝐴 ∈ Cℋ → (𝐴 ∨ℋ 0ℋ) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7190 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → (𝐴 ∨ℋ 0ℋ) = (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ 0ℋ)) | |
2 | id 22 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → 𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ)) | |
3 | 1, 2 | eqeq12d 2755 | . 2 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → ((𝐴 ∨ℋ 0ℋ) = 𝐴 ↔ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ 0ℋ) = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ))) |
4 | h0elch 29203 | . . . 4 ⊢ 0ℋ ∈ Cℋ | |
5 | 4 | elimel 4493 | . . 3 ⊢ if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∈ Cℋ |
6 | 5 | chj0i 29403 | . 2 ⊢ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ 0ℋ) = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) |
7 | 3, 6 | dedth 4482 | 1 ⊢ (𝐴 ∈ Cℋ → (𝐴 ∨ℋ 0ℋ) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ifcif 4424 (class class class)co 7183 Cℋ cch 28877 ∨ℋ chj 28881 0ℋc0h 28883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-inf2 9190 ax-cc 9948 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 ax-pre-sup 10706 ax-addf 10707 ax-mulf 10708 ax-hilex 28947 ax-hfvadd 28948 ax-hvcom 28949 ax-hvass 28950 ax-hv0cl 28951 ax-hvaddid 28952 ax-hfvmul 28953 ax-hvmulid 28954 ax-hvmulass 28955 ax-hvdistr1 28956 ax-hvdistr2 28957 ax-hvmul0 28958 ax-hfi 29027 ax-his1 29030 ax-his2 29031 ax-his3 29032 ax-his4 29033 ax-hcompl 29150 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-oadd 8148 df-omul 8149 df-er 8333 df-map 8452 df-pm 8453 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-fi 8961 df-sup 8992 df-inf 8993 df-oi 9060 df-card 9454 df-acn 9457 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-div 11389 df-nn 11730 df-2 11792 df-3 11793 df-4 11794 df-n0 11990 df-z 12076 df-uz 12338 df-q 12444 df-rp 12486 df-xneg 12603 df-xadd 12604 df-xmul 12605 df-ico 12840 df-icc 12841 df-fz 12995 df-fl 13266 df-seq 13474 df-exp 13535 df-cj 14561 df-re 14562 df-im 14563 df-sqrt 14697 df-abs 14698 df-clim 14948 df-rlim 14949 df-rest 16812 df-topgen 16833 df-psmet 20222 df-xmet 20223 df-met 20224 df-bl 20225 df-mopn 20226 df-fbas 20227 df-fg 20228 df-top 21658 df-topon 21675 df-bases 21710 df-cld 21783 df-ntr 21784 df-cls 21785 df-nei 21862 df-lm 21993 df-haus 22079 df-fil 22610 df-fm 22702 df-flim 22703 df-flf 22704 df-cfil 24020 df-cau 24021 df-cmet 24022 df-grpo 28441 df-gid 28442 df-ginv 28443 df-gdiv 28444 df-ablo 28493 df-vc 28507 df-nv 28540 df-va 28543 df-ba 28544 df-sm 28545 df-0v 28546 df-vs 28547 df-nmcv 28548 df-ims 28549 df-ssp 28670 df-ph 28761 df-cbn 28811 df-hnorm 28916 df-hba 28917 df-hvsub 28919 df-hlim 28920 df-hcau 28921 df-sh 29155 df-ch 29169 df-oc 29200 df-ch0 29201 df-chj 29258 |
This theorem is referenced by: mdsl0 30258 atordi 30332 chirredlem2 30339 chirredlem3 30340 |
Copyright terms: Public domain | W3C validator |