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Mirrors > Home > HSE Home > Th. List > chirred | Structured version Visualization version GIF version |
Description: The Hilbert lattice is irreducible: any element that commutes with all elements must be zero or one. Theorem 14.8.4 of [BeltramettiCassinelli] p. 166. (Contributed by NM, 16-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chirred | ⊢ ((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥) → (𝐴 = 0ℋ ∨ 𝐴 = ℋ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2730 | . . 3 ⊢ (𝐴 = if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) → (𝐴 = 0ℋ ↔ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) = 0ℋ)) | |
2 | eqeq1 2730 | . . 3 ⊢ (𝐴 = if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) → (𝐴 = ℋ ↔ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) = ℋ)) | |
3 | 1, 2 | orbi12d 916 | . 2 ⊢ (𝐴 = if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) → ((𝐴 = 0ℋ ∨ 𝐴 = ℋ) ↔ (if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) = 0ℋ ∨ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) = ℋ))) |
4 | eleq1 2814 | . . . . . 6 ⊢ (𝐴 = if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) → (𝐴 ∈ Cℋ ↔ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) ∈ Cℋ )) | |
5 | nfv 1910 | . . . . . . . . . 10 ⊢ Ⅎ𝑥 𝐴 ∈ Cℋ | |
6 | nfra1 3272 | . . . . . . . . . 10 ⊢ Ⅎ𝑥∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥 | |
7 | 5, 6 | nfan 1895 | . . . . . . . . 9 ⊢ Ⅎ𝑥(𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥) |
8 | nfcv 2892 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐴 | |
9 | nfcv 2892 | . . . . . . . . 9 ⊢ Ⅎ𝑥0ℋ | |
10 | 7, 8, 9 | nfif 4555 | . . . . . . . 8 ⊢ Ⅎ𝑥if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) |
11 | 10 | nfeq2 2910 | . . . . . . 7 ⊢ Ⅎ𝑥 𝐴 = if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) |
12 | breq1 5148 | . . . . . . 7 ⊢ (𝐴 = if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) → (𝐴 𝐶ℋ 𝑥 ↔ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) 𝐶ℋ 𝑥)) | |
13 | 11, 12 | ralbid 3261 | . . . . . 6 ⊢ (𝐴 = if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) → (∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥 ↔ ∀𝑥 ∈ Cℋ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) 𝐶ℋ 𝑥)) |
14 | 4, 13 | anbi12d 630 | . . . . 5 ⊢ (𝐴 = if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) → ((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥) ↔ (if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) 𝐶ℋ 𝑥))) |
15 | eleq1 2814 | . . . . . 6 ⊢ (0ℋ = if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) → (0ℋ ∈ Cℋ ↔ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) ∈ Cℋ )) | |
16 | 10 | nfeq2 2910 | . . . . . . 7 ⊢ Ⅎ𝑥0ℋ = if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) |
17 | breq1 5148 | . . . . . . 7 ⊢ (0ℋ = if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) → (0ℋ 𝐶ℋ 𝑥 ↔ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) 𝐶ℋ 𝑥)) | |
18 | 16, 17 | ralbid 3261 | . . . . . 6 ⊢ (0ℋ = if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) → (∀𝑥 ∈ Cℋ 0ℋ 𝐶ℋ 𝑥 ↔ ∀𝑥 ∈ Cℋ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) 𝐶ℋ 𝑥)) |
19 | 15, 18 | anbi12d 630 | . . . . 5 ⊢ (0ℋ = if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) → ((0ℋ ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 0ℋ 𝐶ℋ 𝑥) ↔ (if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) 𝐶ℋ 𝑥))) |
20 | h0elch 31184 | . . . . . 6 ⊢ 0ℋ ∈ Cℋ | |
21 | cm0 31538 | . . . . . . 7 ⊢ (𝑥 ∈ Cℋ → 0ℋ 𝐶ℋ 𝑥) | |
22 | 21 | rgen 3053 | . . . . . 6 ⊢ ∀𝑥 ∈ Cℋ 0ℋ 𝐶ℋ 𝑥 |
23 | 20, 22 | pm3.2i 469 | . . . . 5 ⊢ (0ℋ ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 0ℋ 𝐶ℋ 𝑥) |
24 | 14, 19, 23 | elimhyp 4590 | . . . 4 ⊢ (if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) 𝐶ℋ 𝑥) |
25 | 24 | simpli 482 | . . 3 ⊢ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) ∈ Cℋ |
26 | 24 | simpri 484 | . . . 4 ⊢ ∀𝑥 ∈ Cℋ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) 𝐶ℋ 𝑥 |
27 | nfcv 2892 | . . . . . 6 ⊢ Ⅎ𝑥 𝐶ℋ | |
28 | nfcv 2892 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
29 | 10, 27, 28 | nfbr 5192 | . . . . 5 ⊢ Ⅎ𝑥if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) 𝐶ℋ 𝑦 |
30 | breq2 5149 | . . . . 5 ⊢ (𝑥 = 𝑦 → (if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) 𝐶ℋ 𝑥 ↔ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) 𝐶ℋ 𝑦)) | |
31 | 29, 30 | rspc 3597 | . . . 4 ⊢ (𝑦 ∈ Cℋ → (∀𝑥 ∈ Cℋ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) 𝐶ℋ 𝑥 → if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) 𝐶ℋ 𝑦)) |
32 | 26, 31 | mpi 20 | . . 3 ⊢ (𝑦 ∈ Cℋ → if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) 𝐶ℋ 𝑦) |
33 | 25, 32 | chirredi 32323 | . 2 ⊢ (if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) = 0ℋ ∨ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) = ℋ) |
34 | 3, 33 | dedth 4583 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥) → (𝐴 = 0ℋ ∨ 𝐴 = ℋ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ifcif 4525 class class class wbr 5145 ℋchba 30848 Cℋ cch 30858 0ℋc0h 30864 𝐶ℋ ccm 30865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7737 ax-inf2 9676 ax-cc 10468 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 ax-addf 11227 ax-mulf 11228 ax-hilex 30928 ax-hfvadd 30929 ax-hvcom 30930 ax-hvass 30931 ax-hv0cl 30932 ax-hvaddid 30933 ax-hfvmul 30934 ax-hvmulid 30935 ax-hvmulass 30936 ax-hvdistr1 30937 ax-hvdistr2 30938 ax-hvmul0 30939 ax-hfi 31008 ax-his1 31011 ax-his2 31012 ax-his3 31013 ax-his4 31014 ax-hcompl 31131 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3466 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3968 df-nul 4325 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4908 df-int 4949 df-iun 4997 df-iin 4998 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6497 df-fun 6547 df-fn 6548 df-f 6549 df-f1 6550 df-fo 6551 df-f1o 6552 df-fv 6553 df-isom 6554 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-omul 8492 df-er 8725 df-map 8848 df-pm 8849 df-ixp 8918 df-en 8966 df-dom 8967 df-sdom 8968 df-fin 8969 df-fsupp 9398 df-fi 9446 df-sup 9477 df-inf 9478 df-oi 9545 df-card 9974 df-acn 9977 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12258 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12518 df-z 12604 df-dec 12723 df-uz 12868 df-q 12978 df-rp 13022 df-xneg 13139 df-xadd 13140 df-xmul 13141 df-ioo 13375 df-ico 13377 df-icc 13378 df-fz 13532 df-fzo 13675 df-fl 13805 df-seq 14015 df-exp 14075 df-hash 14342 df-cj 15098 df-re 15099 df-im 15100 df-sqrt 15234 df-abs 15235 df-clim 15484 df-rlim 15485 df-sum 15685 df-struct 17143 df-sets 17160 df-slot 17178 df-ndx 17190 df-base 17208 df-ress 17237 df-plusg 17273 df-mulr 17274 df-starv 17275 df-sca 17276 df-vsca 17277 df-ip 17278 df-tset 17279 df-ple 17280 df-ds 17282 df-unif 17283 df-hom 17284 df-cco 17285 df-rest 17431 df-topn 17432 df-0g 17450 df-gsum 17451 df-topgen 17452 df-pt 17453 df-prds 17456 df-xrs 17511 df-qtop 17516 df-imas 17517 df-xps 17519 df-mre 17593 df-mrc 17594 df-acs 17596 df-mgm 18627 df-sgrp 18706 df-mnd 18722 df-submnd 18768 df-mulg 19057 df-cntz 19306 df-cmn 19775 df-psmet 21330 df-xmet 21331 df-met 21332 df-bl 21333 df-mopn 21334 df-fbas 21335 df-fg 21336 df-cnfld 21339 df-top 22883 df-topon 22900 df-topsp 22922 df-bases 22936 df-cld 23010 df-ntr 23011 df-cls 23012 df-nei 23089 df-cn 23218 df-cnp 23219 df-lm 23220 df-haus 23306 df-tx 23553 df-hmeo 23746 df-fil 23837 df-fm 23929 df-flim 23930 df-flf 23931 df-xms 24313 df-ms 24314 df-tms 24315 df-cfil 25270 df-cau 25271 df-cmet 25272 df-grpo 30422 df-gid 30423 df-ginv 30424 df-gdiv 30425 df-ablo 30474 df-vc 30488 df-nv 30521 df-va 30524 df-ba 30525 df-sm 30526 df-0v 30527 df-vs 30528 df-nmcv 30529 df-ims 30530 df-dip 30630 df-ssp 30651 df-ph 30742 df-cbn 30792 df-hnorm 30897 df-hba 30898 df-hvsub 30900 df-hlim 30901 df-hcau 30902 df-sh 31136 df-ch 31150 df-oc 31181 df-ch0 31182 df-shs 31237 df-span 31238 df-chj 31239 df-chsup 31240 df-pjh 31324 df-cm 31512 df-cv 32208 df-at 32267 |
This theorem is referenced by: (None) |
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