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Mirrors > Home > HSE Home > Th. List > chlejb1 | Structured version Visualization version GIF version |
Description: Hilbert lattice ordering in terms of join. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chlejb1 | ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∨ℋ 𝐵) = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 3951 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → (𝐴 ⊆ 𝐵 ↔ if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ⊆ 𝐵)) | |
2 | oveq1 7276 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → (𝐴 ∨ℋ 𝐵) = (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ 𝐵)) | |
3 | 2 | eqeq1d 2742 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → ((𝐴 ∨ℋ 𝐵) = 𝐵 ↔ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ 𝐵) = 𝐵)) |
4 | 1, 3 | bibi12d 346 | . 2 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → ((𝐴 ⊆ 𝐵 ↔ (𝐴 ∨ℋ 𝐵) = 𝐵) ↔ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ⊆ 𝐵 ↔ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ 𝐵) = 𝐵))) |
5 | sseq2 3952 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) → (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ⊆ 𝐵 ↔ if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ⊆ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ))) | |
6 | oveq2 7277 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) → (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ 𝐵) = (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ))) | |
7 | id 22 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) → 𝐵 = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ)) | |
8 | 6, 7 | eqeq12d 2756 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) → ((if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ 𝐵) = 𝐵 ↔ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ)) = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ))) |
9 | 5, 8 | bibi12d 346 | . 2 ⊢ (𝐵 = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) → ((if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ⊆ 𝐵 ↔ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ 𝐵) = 𝐵) ↔ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ⊆ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) ↔ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ)) = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ)))) |
10 | h0elch 29611 | . . . 4 ⊢ 0ℋ ∈ Cℋ | |
11 | 10 | elimel 4534 | . . 3 ⊢ if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∈ Cℋ |
12 | 10 | elimel 4534 | . . 3 ⊢ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) ∈ Cℋ |
13 | 11, 12 | chlejb1i 29832 | . 2 ⊢ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ⊆ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) ↔ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ)) = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ)) |
14 | 4, 9, 13 | dedth2h 4524 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∨ℋ 𝐵) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ⊆ wss 3892 ifcif 4465 (class class class)co 7269 Cℋ cch 29285 ∨ℋ chj 29289 0ℋc0h 29291 |
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 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-inf2 9375 ax-cc 10190 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-pre-sup 10948 ax-addf 10949 ax-mulf 10950 ax-hilex 29355 ax-hfvadd 29356 ax-hvcom 29357 ax-hvass 29358 ax-hv0cl 29359 ax-hvaddid 29360 ax-hfvmul 29361 ax-hvmulid 29362 ax-hvmulass 29363 ax-hvdistr1 29364 ax-hvdistr2 29365 ax-hvmul0 29366 ax-hfi 29435 ax-his1 29438 ax-his2 29439 ax-his3 29440 ax-his4 29441 ax-hcompl 29558 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-om 7705 df-1st 7822 df-2nd 7823 df-supp 7967 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-2o 8287 df-oadd 8290 df-omul 8291 df-er 8479 df-map 8598 df-pm 8599 df-ixp 8667 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-fsupp 9105 df-fi 9146 df-sup 9177 df-inf 9178 df-oi 9245 df-card 9696 df-acn 9699 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12435 df-uz 12580 df-q 12686 df-rp 12728 df-xneg 12845 df-xadd 12846 df-xmul 12847 df-ioo 13080 df-ico 13082 df-icc 13083 df-fz 13237 df-fzo 13380 df-fl 13508 df-seq 13718 df-exp 13779 df-hash 14041 df-cj 14806 df-re 14807 df-im 14808 df-sqrt 14942 df-abs 14943 df-clim 15193 df-rlim 15194 df-sum 15394 df-struct 16844 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-ress 16938 df-plusg 16971 df-mulr 16972 df-starv 16973 df-sca 16974 df-vsca 16975 df-ip 16976 df-tset 16977 df-ple 16978 df-ds 16980 df-unif 16981 df-hom 16982 df-cco 16983 df-rest 17129 df-topn 17130 df-0g 17148 df-gsum 17149 df-topgen 17150 df-pt 17151 df-prds 17154 df-xrs 17209 df-qtop 17214 df-imas 17215 df-xps 17217 df-mre 17291 df-mrc 17292 df-acs 17294 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-submnd 18427 df-mulg 18697 df-cntz 18919 df-cmn 19384 df-psmet 20585 df-xmet 20586 df-met 20587 df-bl 20588 df-mopn 20589 df-fbas 20590 df-fg 20591 df-cnfld 20594 df-top 22039 df-topon 22056 df-topsp 22078 df-bases 22092 df-cld 22166 df-ntr 22167 df-cls 22168 df-nei 22245 df-cn 22374 df-cnp 22375 df-lm 22376 df-haus 22462 df-tx 22709 df-hmeo 22902 df-fil 22993 df-fm 23085 df-flim 23086 df-flf 23087 df-xms 23469 df-ms 23470 df-tms 23471 df-cfil 24415 df-cau 24416 df-cmet 24417 df-grpo 28849 df-gid 28850 df-ginv 28851 df-gdiv 28852 df-ablo 28901 df-vc 28915 df-nv 28948 df-va 28951 df-ba 28952 df-sm 28953 df-0v 28954 df-vs 28955 df-nmcv 28956 df-ims 28957 df-dip 29057 df-ssp 29078 df-ph 29169 df-cbn 29219 df-hnorm 29324 df-hba 29325 df-hvsub 29327 df-hlim 29328 df-hcau 29329 df-sh 29563 df-ch 29577 df-oc 29608 df-ch0 29609 df-shs 29664 df-chj 29666 |
This theorem is referenced by: chlejb2 29869 mdsymlem1 30759 |
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