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Mirrors > Home > HSE Home > Th. List > fh3i | 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 |
---|---|
fh3i | ⊢ (𝐴 ∨ℋ (𝐵 ∩ 𝐶)) = ((𝐴 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fh1.1 | . . . . . 6 ⊢ 𝐴 ∈ Cℋ | |
2 | 1 | choccli 28499 | . . . . 5 ⊢ (⊥‘𝐴) ∈ Cℋ |
3 | fh1.2 | . . . . . 6 ⊢ 𝐵 ∈ Cℋ | |
4 | 3 | choccli 28499 | . . . . 5 ⊢ (⊥‘𝐵) ∈ Cℋ |
5 | fh1.3 | . . . . . 6 ⊢ 𝐶 ∈ Cℋ | |
6 | 5 | choccli 28499 | . . . . 5 ⊢ (⊥‘𝐶) ∈ Cℋ |
7 | fh1.4 | . . . . . . 7 ⊢ 𝐴 𝐶ℋ 𝐵 | |
8 | 1, 3, 7 | cmcm3ii 28791 | . . . . . 6 ⊢ (⊥‘𝐴) 𝐶ℋ 𝐵 |
9 | 2, 3, 8 | cmcm2ii 28790 | . . . . 5 ⊢ (⊥‘𝐴) 𝐶ℋ (⊥‘𝐵) |
10 | fh1.5 | . . . . . . 7 ⊢ 𝐴 𝐶ℋ 𝐶 | |
11 | 1, 5, 10 | cmcm3ii 28791 | . . . . . 6 ⊢ (⊥‘𝐴) 𝐶ℋ 𝐶 |
12 | 2, 5, 11 | cmcm2ii 28790 | . . . . 5 ⊢ (⊥‘𝐴) 𝐶ℋ (⊥‘𝐶) |
13 | 2, 4, 6, 9, 12 | fh1i 28813 | . . . 4 ⊢ ((⊥‘𝐴) ∩ ((⊥‘𝐵) ∨ℋ (⊥‘𝐶))) = (((⊥‘𝐴) ∩ (⊥‘𝐵)) ∨ℋ ((⊥‘𝐴) ∩ (⊥‘𝐶))) |
14 | 3, 5 | chdmm1i 28669 | . . . . 5 ⊢ (⊥‘(𝐵 ∩ 𝐶)) = ((⊥‘𝐵) ∨ℋ (⊥‘𝐶)) |
15 | 14 | ineq2i 3962 | . . . 4 ⊢ ((⊥‘𝐴) ∩ (⊥‘(𝐵 ∩ 𝐶))) = ((⊥‘𝐴) ∩ ((⊥‘𝐵) ∨ℋ (⊥‘𝐶))) |
16 | 1, 3 | chdmj1i 28673 | . . . . 5 ⊢ (⊥‘(𝐴 ∨ℋ 𝐵)) = ((⊥‘𝐴) ∩ (⊥‘𝐵)) |
17 | 1, 5 | chdmj1i 28673 | . . . . 5 ⊢ (⊥‘(𝐴 ∨ℋ 𝐶)) = ((⊥‘𝐴) ∩ (⊥‘𝐶)) |
18 | 16, 17 | oveq12i 6803 | . . . 4 ⊢ ((⊥‘(𝐴 ∨ℋ 𝐵)) ∨ℋ (⊥‘(𝐴 ∨ℋ 𝐶))) = (((⊥‘𝐴) ∩ (⊥‘𝐵)) ∨ℋ ((⊥‘𝐴) ∩ (⊥‘𝐶))) |
19 | 13, 15, 18 | 3eqtr4ri 2804 | . . 3 ⊢ ((⊥‘(𝐴 ∨ℋ 𝐵)) ∨ℋ (⊥‘(𝐴 ∨ℋ 𝐶))) = ((⊥‘𝐴) ∩ (⊥‘(𝐵 ∩ 𝐶))) |
20 | 1, 3 | chjcli 28649 | . . . 4 ⊢ (𝐴 ∨ℋ 𝐵) ∈ Cℋ |
21 | 1, 5 | chjcli 28649 | . . . 4 ⊢ (𝐴 ∨ℋ 𝐶) ∈ Cℋ |
22 | 20, 21 | chdmm1i 28669 | . . 3 ⊢ (⊥‘((𝐴 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐶))) = ((⊥‘(𝐴 ∨ℋ 𝐵)) ∨ℋ (⊥‘(𝐴 ∨ℋ 𝐶))) |
23 | 3, 5 | chincli 28652 | . . . 4 ⊢ (𝐵 ∩ 𝐶) ∈ Cℋ |
24 | 1, 23 | chdmj1i 28673 | . . 3 ⊢ (⊥‘(𝐴 ∨ℋ (𝐵 ∩ 𝐶))) = ((⊥‘𝐴) ∩ (⊥‘(𝐵 ∩ 𝐶))) |
25 | 19, 22, 24 | 3eqtr4i 2803 | . 2 ⊢ (⊥‘((𝐴 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐶))) = (⊥‘(𝐴 ∨ℋ (𝐵 ∩ 𝐶))) |
26 | 1, 23 | chjcli 28649 | . . 3 ⊢ (𝐴 ∨ℋ (𝐵 ∩ 𝐶)) ∈ Cℋ |
27 | 20, 21 | chincli 28652 | . . 3 ⊢ ((𝐴 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐶)) ∈ Cℋ |
28 | 26, 27 | chcon3i 28658 | . 2 ⊢ ((𝐴 ∨ℋ (𝐵 ∩ 𝐶)) = ((𝐴 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐶)) ↔ (⊥‘((𝐴 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐶))) = (⊥‘(𝐴 ∨ℋ (𝐵 ∩ 𝐶)))) |
29 | 25, 28 | mpbir 221 | 1 ⊢ (𝐴 ∨ℋ (𝐵 ∩ 𝐶)) = ((𝐴 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 ∈ wcel 2145 ∩ cin 3722 class class class wbr 4786 ‘cfv 6029 (class class class)co 6791 Cℋ cch 28119 ⊥cort 28120 ∨ℋ chj 28123 𝐶ℋ ccm 28126 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-inf2 8700 ax-cc 9457 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 ax-pre-sup 10214 ax-addf 10215 ax-mulf 10216 ax-hilex 28189 ax-hfvadd 28190 ax-hvcom 28191 ax-hvass 28192 ax-hv0cl 28193 ax-hvaddid 28194 ax-hfvmul 28195 ax-hvmulid 28196 ax-hvmulass 28197 ax-hvdistr1 28198 ax-hvdistr2 28199 ax-hvmul0 28200 ax-hfi 28269 ax-his1 28272 ax-his2 28273 ax-his3 28274 ax-his4 28275 ax-hcompl 28392 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-isom 6038 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-of 7042 df-om 7211 df-1st 7313 df-2nd 7314 df-supp 7445 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-2o 7712 df-oadd 7715 df-omul 7716 df-er 7894 df-map 8009 df-pm 8010 df-ixp 8061 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-fsupp 8430 df-fi 8471 df-sup 8502 df-inf 8503 df-oi 8569 df-card 8963 df-acn 8966 df-cda 9190 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-div 10885 df-nn 11221 df-2 11279 df-3 11280 df-4 11281 df-5 11282 df-6 11283 df-7 11284 df-8 11285 df-9 11286 df-n0 11493 df-z 11578 df-dec 11694 df-uz 11887 df-q 11990 df-rp 12029 df-xneg 12144 df-xadd 12145 df-xmul 12146 df-ioo 12377 df-ico 12379 df-icc 12380 df-fz 12527 df-fzo 12667 df-fl 12794 df-seq 13002 df-exp 13061 df-hash 13315 df-cj 14040 df-re 14041 df-im 14042 df-sqrt 14176 df-abs 14177 df-clim 14420 df-rlim 14421 df-sum 14618 df-struct 16059 df-ndx 16060 df-slot 16061 df-base 16063 df-sets 16064 df-ress 16065 df-plusg 16155 df-mulr 16156 df-starv 16157 df-sca 16158 df-vsca 16159 df-ip 16160 df-tset 16161 df-ple 16162 df-ds 16165 df-unif 16166 df-hom 16167 df-cco 16168 df-rest 16284 df-topn 16285 df-0g 16303 df-gsum 16304 df-topgen 16305 df-pt 16306 df-prds 16309 df-xrs 16363 df-qtop 16368 df-imas 16369 df-xps 16371 df-mre 16447 df-mrc 16448 df-acs 16450 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-submnd 17537 df-mulg 17742 df-cntz 17950 df-cmn 18395 df-psmet 19946 df-xmet 19947 df-met 19948 df-bl 19949 df-mopn 19950 df-fbas 19951 df-fg 19952 df-cnfld 19955 df-top 20912 df-topon 20929 df-topsp 20951 df-bases 20964 df-cld 21037 df-ntr 21038 df-cls 21039 df-nei 21116 df-cn 21245 df-cnp 21246 df-lm 21247 df-haus 21333 df-tx 21579 df-hmeo 21772 df-fil 21863 df-fm 21955 df-flim 21956 df-flf 21957 df-xms 22338 df-ms 22339 df-tms 22340 df-cfil 23265 df-cau 23266 df-cmet 23267 df-grpo 27680 df-gid 27681 df-ginv 27682 df-gdiv 27683 df-ablo 27732 df-vc 27747 df-nv 27780 df-va 27783 df-ba 27784 df-sm 27785 df-0v 27786 df-vs 27787 df-nmcv 27788 df-ims 27789 df-dip 27889 df-ssp 27910 df-ph 28001 df-cbn 28052 df-hnorm 28158 df-hba 28159 df-hvsub 28161 df-hlim 28162 df-hcau 28163 df-sh 28397 df-ch 28411 df-oc 28442 df-ch0 28443 df-shs 28500 df-chj 28502 df-cm 28775 |
This theorem is referenced by: mayetes3i 28921 |
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