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Mirrors > Home > HSE Home > Th. List > dmdcompli | Structured version Visualization version GIF version |
Description: A condition equivalent to the dual modular pair property. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mdcompl.1 | ⊢ 𝐴 ∈ Cℋ |
mdcompl.2 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
dmdcompli | ⊢ (𝐴 𝑀ℋ* 𝐵 ↔ (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdcompl.1 | . . . 4 ⊢ 𝐴 ∈ Cℋ | |
2 | mdcompl.2 | . . . 4 ⊢ 𝐵 ∈ Cℋ | |
3 | 1, 2 | chincli 28652 | . . 3 ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ |
4 | 3 | mdoc1i 29617 | . 2 ⊢ (𝐴 ∩ 𝐵) 𝑀ℋ (⊥‘(𝐴 ∩ 𝐵)) |
5 | 3 | dmdoc2i 29620 | . 2 ⊢ (⊥‘(𝐴 ∩ 𝐵)) 𝑀ℋ* (𝐴 ∩ 𝐵) |
6 | ssid 3773 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ (𝐴 ∩ 𝐵) | |
7 | 1, 2 | chjcli 28649 | . . . 4 ⊢ (𝐴 ∨ℋ 𝐵) ∈ Cℋ |
8 | 7 | chssii 28421 | . . 3 ⊢ (𝐴 ∨ℋ 𝐵) ⊆ ℋ |
9 | 3 | chjoi 28680 | . . 3 ⊢ ((𝐴 ∩ 𝐵) ∨ℋ (⊥‘(𝐴 ∩ 𝐵))) = ℋ |
10 | 8, 9 | sseqtr4i 3787 | . 2 ⊢ (𝐴 ∨ℋ 𝐵) ⊆ ((𝐴 ∩ 𝐵) ∨ℋ (⊥‘(𝐴 ∩ 𝐵))) |
11 | 3 | choccli 28499 | . . 3 ⊢ (⊥‘(𝐴 ∩ 𝐵)) ∈ Cℋ |
12 | 3, 11, 1, 2 | mdsldmd1i 29523 | . 2 ⊢ ((((𝐴 ∩ 𝐵) 𝑀ℋ (⊥‘(𝐴 ∩ 𝐵)) ∧ (⊥‘(𝐴 ∩ 𝐵)) 𝑀ℋ* (𝐴 ∩ 𝐵)) ∧ ((𝐴 ∩ 𝐵) ⊆ (𝐴 ∩ 𝐵) ∧ (𝐴 ∨ℋ 𝐵) ⊆ ((𝐴 ∩ 𝐵) ∨ℋ (⊥‘(𝐴 ∩ 𝐵))))) → (𝐴 𝑀ℋ* 𝐵 ↔ (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))))) |
13 | 4, 5, 6, 10, 12 | mp4an 673 | 1 ⊢ (𝐴 𝑀ℋ* 𝐵 ↔ (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∈ wcel 2145 ∩ cin 3722 ⊆ wss 3723 class class class wbr 4786 ‘cfv 6029 (class class class)co 6791 ℋchil 28109 Cℋ cch 28119 ⊥cort 28120 ∨ℋ chj 28123 𝑀ℋ cmd 28156 𝑀ℋ* cdmd 28157 |
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-pjh 28587 df-cm 28775 df-md 29472 df-dmd 29473 |
This theorem is referenced by: mddmdin0i 29623 |
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