Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > fh4i | Structured version Visualization version GIF version |
Description: Variation of the Foulis-Holland Theorem. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
fh1.1 | ⊢ 𝐴 ∈ Cℋ |
fh1.2 | ⊢ 𝐵 ∈ Cℋ |
fh1.3 | ⊢ 𝐶 ∈ Cℋ |
fh1.4 | ⊢ 𝐴 𝐶ℋ 𝐵 |
fh1.5 | ⊢ 𝐴 𝐶ℋ 𝐶 |
Ref | Expression |
---|---|
fh4i | ⊢ (𝐵 ∨ℋ (𝐴 ∩ 𝐶)) = ((𝐵 ∨ℋ 𝐴) ∩ (𝐵 ∨ℋ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fh1.1 | . . . . . 6 ⊢ 𝐴 ∈ Cℋ | |
2 | 1 | choccli 29663 | . . . . 5 ⊢ (⊥‘𝐴) ∈ Cℋ |
3 | fh1.2 | . . . . . 6 ⊢ 𝐵 ∈ Cℋ | |
4 | 3 | choccli 29663 | . . . . 5 ⊢ (⊥‘𝐵) ∈ Cℋ |
5 | fh1.3 | . . . . . 6 ⊢ 𝐶 ∈ Cℋ | |
6 | 5 | choccli 29663 | . . . . 5 ⊢ (⊥‘𝐶) ∈ Cℋ |
7 | fh1.4 | . . . . . . 7 ⊢ 𝐴 𝐶ℋ 𝐵 | |
8 | 1, 3, 7 | cmcm3ii 29955 | . . . . . 6 ⊢ (⊥‘𝐴) 𝐶ℋ 𝐵 |
9 | 2, 3, 8 | cmcm2ii 29954 | . . . . 5 ⊢ (⊥‘𝐴) 𝐶ℋ (⊥‘𝐵) |
10 | fh1.5 | . . . . . . 7 ⊢ 𝐴 𝐶ℋ 𝐶 | |
11 | 1, 5, 10 | cmcm3ii 29955 | . . . . . 6 ⊢ (⊥‘𝐴) 𝐶ℋ 𝐶 |
12 | 2, 5, 11 | cmcm2ii 29954 | . . . . 5 ⊢ (⊥‘𝐴) 𝐶ℋ (⊥‘𝐶) |
13 | 2, 4, 6, 9, 12 | fh2i 29978 | . . . 4 ⊢ ((⊥‘𝐵) ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐶))) = (((⊥‘𝐵) ∩ (⊥‘𝐴)) ∨ℋ ((⊥‘𝐵) ∩ (⊥‘𝐶))) |
14 | 1, 5 | chdmm1i 29833 | . . . . 5 ⊢ (⊥‘(𝐴 ∩ 𝐶)) = ((⊥‘𝐴) ∨ℋ (⊥‘𝐶)) |
15 | 14 | ineq2i 4149 | . . . 4 ⊢ ((⊥‘𝐵) ∩ (⊥‘(𝐴 ∩ 𝐶))) = ((⊥‘𝐵) ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐶))) |
16 | 3, 1 | chdmj1i 29837 | . . . . 5 ⊢ (⊥‘(𝐵 ∨ℋ 𝐴)) = ((⊥‘𝐵) ∩ (⊥‘𝐴)) |
17 | 3, 5 | chdmj1i 29837 | . . . . 5 ⊢ (⊥‘(𝐵 ∨ℋ 𝐶)) = ((⊥‘𝐵) ∩ (⊥‘𝐶)) |
18 | 16, 17 | oveq12i 7281 | . . . 4 ⊢ ((⊥‘(𝐵 ∨ℋ 𝐴)) ∨ℋ (⊥‘(𝐵 ∨ℋ 𝐶))) = (((⊥‘𝐵) ∩ (⊥‘𝐴)) ∨ℋ ((⊥‘𝐵) ∩ (⊥‘𝐶))) |
19 | 13, 15, 18 | 3eqtr4ri 2779 | . . 3 ⊢ ((⊥‘(𝐵 ∨ℋ 𝐴)) ∨ℋ (⊥‘(𝐵 ∨ℋ 𝐶))) = ((⊥‘𝐵) ∩ (⊥‘(𝐴 ∩ 𝐶))) |
20 | 3, 1 | chjcli 29813 | . . . 4 ⊢ (𝐵 ∨ℋ 𝐴) ∈ Cℋ |
21 | 3, 5 | chjcli 29813 | . . . 4 ⊢ (𝐵 ∨ℋ 𝐶) ∈ Cℋ |
22 | 20, 21 | chdmm1i 29833 | . . 3 ⊢ (⊥‘((𝐵 ∨ℋ 𝐴) ∩ (𝐵 ∨ℋ 𝐶))) = ((⊥‘(𝐵 ∨ℋ 𝐴)) ∨ℋ (⊥‘(𝐵 ∨ℋ 𝐶))) |
23 | 1, 5 | chincli 29816 | . . . 4 ⊢ (𝐴 ∩ 𝐶) ∈ Cℋ |
24 | 3, 23 | chdmj1i 29837 | . . 3 ⊢ (⊥‘(𝐵 ∨ℋ (𝐴 ∩ 𝐶))) = ((⊥‘𝐵) ∩ (⊥‘(𝐴 ∩ 𝐶))) |
25 | 19, 22, 24 | 3eqtr4i 2778 | . 2 ⊢ (⊥‘((𝐵 ∨ℋ 𝐴) ∩ (𝐵 ∨ℋ 𝐶))) = (⊥‘(𝐵 ∨ℋ (𝐴 ∩ 𝐶))) |
26 | 3, 23 | chjcli 29813 | . . 3 ⊢ (𝐵 ∨ℋ (𝐴 ∩ 𝐶)) ∈ Cℋ |
27 | 20, 21 | chincli 29816 | . . 3 ⊢ ((𝐵 ∨ℋ 𝐴) ∩ (𝐵 ∨ℋ 𝐶)) ∈ Cℋ |
28 | 26, 27 | chcon3i 29822 | . 2 ⊢ ((𝐵 ∨ℋ (𝐴 ∩ 𝐶)) = ((𝐵 ∨ℋ 𝐴) ∩ (𝐵 ∨ℋ 𝐶)) ↔ (⊥‘((𝐵 ∨ℋ 𝐴) ∩ (𝐵 ∨ℋ 𝐶))) = (⊥‘(𝐵 ∨ℋ (𝐴 ∩ 𝐶)))) |
29 | 25, 28 | mpbir 230 | 1 ⊢ (𝐵 ∨ℋ (𝐴 ∩ 𝐶)) = ((𝐵 ∨ℋ 𝐴) ∩ (𝐵 ∨ℋ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2110 ∩ cin 3891 class class class wbr 5079 ‘cfv 6431 (class class class)co 7269 Cℋ cch 29285 ⊥cort 29286 ∨ℋ chj 29289 𝐶ℋ ccm 29292 |
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 df-cm 29939 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |