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Mirrors > Home > HSE Home > Th. List > chrelati | Structured version Visualization version GIF version |
Description: The Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chpssat.1 | ⊢ 𝐴 ∈ Cℋ |
chpssat.2 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
chrelati | ⊢ (𝐴 ⊊ 𝐵 → ∃𝑥 ∈ HAtoms (𝐴 ⊊ (𝐴 ∨ℋ 𝑥) ∧ (𝐴 ∨ℋ 𝑥) ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chpssat.1 | . . 3 ⊢ 𝐴 ∈ Cℋ | |
2 | chpssat.2 | . . 3 ⊢ 𝐵 ∈ Cℋ | |
3 | 1, 2 | chpssati 32125 | . 2 ⊢ (𝐴 ⊊ 𝐵 → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ 𝐴)) |
4 | ancom 460 | . . . 4 ⊢ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ 𝐴) ↔ (¬ 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)) | |
5 | pssss 4090 | . . . . 5 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) | |
6 | atelch 32106 | . . . . 5 ⊢ (𝑥 ∈ HAtoms → 𝑥 ∈ Cℋ ) | |
7 | chnle 31276 | . . . . . . . 8 ⊢ ((𝐴 ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) → (¬ 𝑥 ⊆ 𝐴 ↔ 𝐴 ⊊ (𝐴 ∨ℋ 𝑥))) | |
8 | 1, 7 | mpan 687 | . . . . . . 7 ⊢ (𝑥 ∈ Cℋ → (¬ 𝑥 ⊆ 𝐴 ↔ 𝐴 ⊊ (𝐴 ∨ℋ 𝑥))) |
9 | 8 | adantl 481 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ Cℋ ) → (¬ 𝑥 ⊆ 𝐴 ↔ 𝐴 ⊊ (𝐴 ∨ℋ 𝑥))) |
10 | ibar 528 | . . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ⊆ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐵))) | |
11 | chlub 31271 | . . . . . . . 8 ⊢ ((𝐴 ∈ Cℋ ∧ 𝑥 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((𝐴 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐵) ↔ (𝐴 ∨ℋ 𝑥) ⊆ 𝐵)) | |
12 | 1, 2, 11 | mp3an13 1448 | . . . . . . 7 ⊢ (𝑥 ∈ Cℋ → ((𝐴 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐵) ↔ (𝐴 ∨ℋ 𝑥) ⊆ 𝐵)) |
13 | 10, 12 | sylan9bb 509 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ Cℋ ) → (𝑥 ⊆ 𝐵 ↔ (𝐴 ∨ℋ 𝑥) ⊆ 𝐵)) |
14 | 9, 13 | anbi12d 630 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ Cℋ ) → ((¬ 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ (𝐴 ⊊ (𝐴 ∨ℋ 𝑥) ∧ (𝐴 ∨ℋ 𝑥) ⊆ 𝐵))) |
15 | 5, 6, 14 | syl2an 595 | . . . 4 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝑥 ∈ HAtoms) → ((¬ 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ (𝐴 ⊊ (𝐴 ∨ℋ 𝑥) ∧ (𝐴 ∨ℋ 𝑥) ⊆ 𝐵))) |
16 | 4, 15 | bitrid 283 | . . 3 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝑥 ∈ HAtoms) → ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ 𝐴) ↔ (𝐴 ⊊ (𝐴 ∨ℋ 𝑥) ∧ (𝐴 ∨ℋ 𝑥) ⊆ 𝐵))) |
17 | 16 | rexbidva 3170 | . 2 ⊢ (𝐴 ⊊ 𝐵 → (∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ 𝐴) ↔ ∃𝑥 ∈ HAtoms (𝐴 ⊊ (𝐴 ∨ℋ 𝑥) ∧ (𝐴 ∨ℋ 𝑥) ⊆ 𝐵))) |
18 | 3, 17 | mpbid 231 | 1 ⊢ (𝐴 ⊊ 𝐵 → ∃𝑥 ∈ HAtoms (𝐴 ⊊ (𝐴 ∨ℋ 𝑥) ∧ (𝐴 ∨ℋ 𝑥) ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2098 ∃wrex 3064 ⊆ wss 3943 ⊊ wpss 3944 (class class class)co 7405 Cℋ cch 30691 ∨ℋ chj 30695 HAtomscat 30727 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cc 10432 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 ax-hilex 30761 ax-hfvadd 30762 ax-hvcom 30763 ax-hvass 30764 ax-hv0cl 30765 ax-hvaddid 30766 ax-hfvmul 30767 ax-hvmulid 30768 ax-hvmulass 30769 ax-hvdistr1 30770 ax-hvdistr2 30771 ax-hvmul0 30772 ax-hfi 30841 ax-his1 30844 ax-his2 30845 ax-his3 30846 ax-his4 30847 ax-hcompl 30964 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-omul 8472 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-acn 9939 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-rlim 15439 df-sum 15639 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-cn 23086 df-cnp 23087 df-lm 23088 df-haus 23174 df-tx 23421 df-hmeo 23614 df-fil 23705 df-fm 23797 df-flim 23798 df-flf 23799 df-xms 24181 df-ms 24182 df-tms 24183 df-cfil 25138 df-cau 25139 df-cmet 25140 df-grpo 30255 df-gid 30256 df-ginv 30257 df-gdiv 30258 df-ablo 30307 df-vc 30321 df-nv 30354 df-va 30357 df-ba 30358 df-sm 30359 df-0v 30360 df-vs 30361 df-nmcv 30362 df-ims 30363 df-dip 30463 df-ssp 30484 df-ph 30575 df-cbn 30625 df-hnorm 30730 df-hba 30731 df-hvsub 30733 df-hlim 30734 df-hcau 30735 df-sh 30969 df-ch 30983 df-oc 31014 df-ch0 31015 df-shs 31070 df-span 31071 df-chj 31072 df-chsup 31073 df-cv 32041 df-at 32100 |
This theorem is referenced by: chrelat2i 32127 cvati 32128 |
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