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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 2775 | . . 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 2775 | . . 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 903 | . 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 2846 | . . . . . 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 1874 | . . . . . . . . . 10 ⊢ Ⅎ𝑥 𝐴 ∈ Cℋ | |
6 | nfra1 3162 | . . . . . . . . . 10 ⊢ Ⅎ𝑥∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥 | |
7 | 5, 6 | nfan 1863 | . . . . . . . . 9 ⊢ Ⅎ𝑥(𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥) |
8 | nfcv 2925 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐴 | |
9 | nfcv 2925 | . . . . . . . . 9 ⊢ Ⅎ𝑥0ℋ | |
10 | 7, 8, 9 | nfif 4373 | . . . . . . . 8 ⊢ Ⅎ𝑥if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) |
11 | 10 | nfeq2 2940 | . . . . . . 7 ⊢ Ⅎ𝑥 𝐴 = if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) |
12 | breq1 4928 | . . . . . . 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 3171 | . . . . . 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 622 | . . . . 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 2846 | . . . . . 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 2940 | . . . . . . 7 ⊢ Ⅎ𝑥0ℋ = if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) |
17 | breq1 4928 | . . . . . . 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 3171 | . . . . . 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 622 | . . . . 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 28826 | . . . . . 6 ⊢ 0ℋ ∈ Cℋ | |
21 | cm0 29182 | . . . . . . 7 ⊢ (𝑥 ∈ Cℋ → 0ℋ 𝐶ℋ 𝑥) | |
22 | 21 | rgen 3091 | . . . . . 6 ⊢ ∀𝑥 ∈ Cℋ 0ℋ 𝐶ℋ 𝑥 |
23 | 20, 22 | pm3.2i 463 | . . . . 5 ⊢ (0ℋ ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 0ℋ 𝐶ℋ 𝑥) |
24 | 14, 19, 23 | elimhyp 4407 | . . . 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 476 | . . 3 ⊢ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) ∈ Cℋ |
26 | 24 | simpri 478 | . . . 4 ⊢ ∀𝑥 ∈ Cℋ if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) 𝐶ℋ 𝑥 |
27 | nfcv 2925 | . . . . . 6 ⊢ Ⅎ𝑥 𝐶ℋ | |
28 | nfcv 2925 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
29 | 10, 27, 28 | nfbr 4972 | . . . . 5 ⊢ Ⅎ𝑥if((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥), 𝐴, 0ℋ) 𝐶ℋ 𝑦 |
30 | breq2 4929 | . . . . 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 3522 | . . . 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 29967 | . 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 4400 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ ∀𝑥 ∈ Cℋ 𝐴 𝐶ℋ 𝑥) → (𝐴 = 0ℋ ∨ 𝐴 = ℋ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∨ wo 834 = wceq 1508 ∈ wcel 2051 ∀wral 3081 ifcif 4344 class class class wbr 4925 ℋchba 28490 Cℋ cch 28500 0ℋc0h 28506 𝐶ℋ ccm 28507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-inf2 8896 ax-cc 9653 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 ax-addf 10412 ax-mulf 10413 ax-hilex 28570 ax-hfvadd 28571 ax-hvcom 28572 ax-hvass 28573 ax-hv0cl 28574 ax-hvaddid 28575 ax-hfvmul 28576 ax-hvmulid 28577 ax-hvmulass 28578 ax-hvdistr1 28579 ax-hvdistr2 28580 ax-hvmul0 28581 ax-hfi 28650 ax-his1 28653 ax-his2 28654 ax-his3 28655 ax-his4 28656 ax-hcompl 28773 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-2o 7904 df-oadd 7907 df-omul 7908 df-er 8087 df-map 8206 df-pm 8207 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-fi 8668 df-sup 8699 df-inf 8700 df-oi 8767 df-card 9160 df-acn 9163 df-cda 9386 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-q 12161 df-rp 12203 df-xneg 12322 df-xadd 12323 df-xmul 12324 df-ioo 12556 df-ico 12558 df-icc 12559 df-fz 12707 df-fzo 12848 df-fl 12975 df-seq 13183 df-exp 13243 df-hash 13504 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-clim 14704 df-rlim 14705 df-sum 14902 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-starv 16434 df-sca 16435 df-vsca 16436 df-ip 16437 df-tset 16438 df-ple 16439 df-ds 16441 df-unif 16442 df-hom 16443 df-cco 16444 df-rest 16550 df-topn 16551 df-0g 16569 df-gsum 16570 df-topgen 16571 df-pt 16572 df-prds 16575 df-xrs 16629 df-qtop 16634 df-imas 16635 df-xps 16637 df-mre 16727 df-mrc 16728 df-acs 16730 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-submnd 17816 df-mulg 18024 df-cntz 18230 df-cmn 18680 df-psmet 20254 df-xmet 20255 df-met 20256 df-bl 20257 df-mopn 20258 df-fbas 20259 df-fg 20260 df-cnfld 20263 df-top 21221 df-topon 21238 df-topsp 21260 df-bases 21273 df-cld 21346 df-ntr 21347 df-cls 21348 df-nei 21425 df-cn 21554 df-cnp 21555 df-lm 21556 df-haus 21642 df-tx 21889 df-hmeo 22082 df-fil 22173 df-fm 22265 df-flim 22266 df-flf 22267 df-xms 22648 df-ms 22649 df-tms 22650 df-cfil 23576 df-cau 23577 df-cmet 23578 df-grpo 28062 df-gid 28063 df-ginv 28064 df-gdiv 28065 df-ablo 28114 df-vc 28128 df-nv 28161 df-va 28164 df-ba 28165 df-sm 28166 df-0v 28167 df-vs 28168 df-nmcv 28169 df-ims 28170 df-dip 28270 df-ssp 28291 df-ph 28382 df-cbn 28433 df-hnorm 28539 df-hba 28540 df-hvsub 28542 df-hlim 28543 df-hcau 28544 df-sh 28778 df-ch 28792 df-oc 28823 df-ch0 28824 df-shs 28881 df-span 28882 df-chj 28883 df-chsup 28884 df-pjh 28968 df-cm 29156 df-cv 29852 df-at 29911 |
This theorem is referenced by: (None) |
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