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| Mirrors > Home > HSE Home > Th. List > nonbooli | Structured version Visualization version GIF version | ||
| Description: A Hilbert lattice with two or more dimensions fails the distributive law and therefore cannot be a Boolean algebra. This counterexample demonstrates a condition where ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = 0ℋ but (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ≠ 0ℋ. The antecedent specifies that the vectors 𝐴 and 𝐵 are nonzero and non-colinear. The last three hypotheses assign one-dimensional subspaces to 𝐹, 𝐺, and 𝐻. (Contributed by NM, 1-Nov-2005.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nonbool.1 | ⊢ 𝐴 ∈ ℋ |
| nonbool.2 | ⊢ 𝐵 ∈ ℋ |
| nonbool.3 | ⊢ 𝐹 = (span‘{𝐴}) |
| nonbool.4 | ⊢ 𝐺 = (span‘{𝐵}) |
| nonbool.5 | ⊢ 𝐻 = (span‘{(𝐴 +ℎ 𝐵)}) |
| Ref | Expression |
|---|---|
| nonbooli | ⊢ (¬ (𝐴 ∈ 𝐺 ∨ 𝐵 ∈ 𝐹) → (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ≠ ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nonbool.1 | . . . . . . . . . . . . 13 ⊢ 𝐴 ∈ ℋ | |
| 2 | nonbool.2 | . . . . . . . . . . . . 13 ⊢ 𝐵 ∈ ℋ | |
| 3 | 1, 2 | hvaddcli 30999 | . . . . . . . . . . . 12 ⊢ (𝐴 +ℎ 𝐵) ∈ ℋ |
| 4 | spansnid 31544 | . . . . . . . . . . . 12 ⊢ ((𝐴 +ℎ 𝐵) ∈ ℋ → (𝐴 +ℎ 𝐵) ∈ (span‘{(𝐴 +ℎ 𝐵)})) | |
| 5 | 3, 4 | ax-mp 5 | . . . . . . . . . . 11 ⊢ (𝐴 +ℎ 𝐵) ∈ (span‘{(𝐴 +ℎ 𝐵)}) |
| 6 | nonbool.5 | . . . . . . . . . . 11 ⊢ 𝐻 = (span‘{(𝐴 +ℎ 𝐵)}) | |
| 7 | 5, 6 | eleqtrri 2833 | . . . . . . . . . 10 ⊢ (𝐴 +ℎ 𝐵) ∈ 𝐻 |
| 8 | nonbool.3 | . . . . . . . . . . . . 13 ⊢ 𝐹 = (span‘{𝐴}) | |
| 9 | 1 | spansnchi 31543 | . . . . . . . . . . . . . 14 ⊢ (span‘{𝐴}) ∈ Cℋ |
| 10 | 9 | chshii 31208 | . . . . . . . . . . . . 13 ⊢ (span‘{𝐴}) ∈ Sℋ |
| 11 | 8, 10 | eqeltri 2830 | . . . . . . . . . . . 12 ⊢ 𝐹 ∈ Sℋ |
| 12 | nonbool.4 | . . . . . . . . . . . . 13 ⊢ 𝐺 = (span‘{𝐵}) | |
| 13 | 2 | spansnchi 31543 | . . . . . . . . . . . . . 14 ⊢ (span‘{𝐵}) ∈ Cℋ |
| 14 | 13 | chshii 31208 | . . . . . . . . . . . . 13 ⊢ (span‘{𝐵}) ∈ Sℋ |
| 15 | 12, 14 | eqeltri 2830 | . . . . . . . . . . . 12 ⊢ 𝐺 ∈ Sℋ |
| 16 | 11, 15 | shsleji 31351 | . . . . . . . . . . 11 ⊢ (𝐹 +ℋ 𝐺) ⊆ (𝐹 ∨ℋ 𝐺) |
| 17 | spansnid 31544 | . . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℋ → 𝐴 ∈ (span‘{𝐴})) | |
| 18 | 1, 17 | ax-mp 5 | . . . . . . . . . . . . 13 ⊢ 𝐴 ∈ (span‘{𝐴}) |
| 19 | 18, 8 | eleqtrri 2833 | . . . . . . . . . . . 12 ⊢ 𝐴 ∈ 𝐹 |
| 20 | spansnid 31544 | . . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℋ → 𝐵 ∈ (span‘{𝐵})) | |
| 21 | 2, 20 | ax-mp 5 | . . . . . . . . . . . . 13 ⊢ 𝐵 ∈ (span‘{𝐵}) |
| 22 | 21, 12 | eleqtrri 2833 | . . . . . . . . . . . 12 ⊢ 𝐵 ∈ 𝐺 |
| 23 | 11, 15 | shsvai 31345 | . . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺) → (𝐴 +ℎ 𝐵) ∈ (𝐹 +ℋ 𝐺)) |
| 24 | 19, 22, 23 | mp2an 692 | . . . . . . . . . . 11 ⊢ (𝐴 +ℎ 𝐵) ∈ (𝐹 +ℋ 𝐺) |
| 25 | 16, 24 | sselii 3955 | . . . . . . . . . 10 ⊢ (𝐴 +ℎ 𝐵) ∈ (𝐹 ∨ℋ 𝐺) |
| 26 | elin 3942 | . . . . . . . . . 10 ⊢ ((𝐴 +ℎ 𝐵) ∈ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ↔ ((𝐴 +ℎ 𝐵) ∈ 𝐻 ∧ (𝐴 +ℎ 𝐵) ∈ (𝐹 ∨ℋ 𝐺))) | |
| 27 | 7, 25, 26 | mpbir2an 711 | . . . . . . . . 9 ⊢ (𝐴 +ℎ 𝐵) ∈ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) |
| 28 | eleq2 2823 | . . . . . . . . 9 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → ((𝐴 +ℎ 𝐵) ∈ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ↔ (𝐴 +ℎ 𝐵) ∈ 0ℋ)) | |
| 29 | 27, 28 | mpbii 233 | . . . . . . . 8 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → (𝐴 +ℎ 𝐵) ∈ 0ℋ) |
| 30 | elch0 31235 | . . . . . . . 8 ⊢ ((𝐴 +ℎ 𝐵) ∈ 0ℋ ↔ (𝐴 +ℎ 𝐵) = 0ℎ) | |
| 31 | 29, 30 | sylib 218 | . . . . . . 7 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → (𝐴 +ℎ 𝐵) = 0ℎ) |
| 32 | ch0 31209 | . . . . . . . 8 ⊢ ((span‘{𝐴}) ∈ Cℋ → 0ℎ ∈ (span‘{𝐴})) | |
| 33 | 9, 32 | ax-mp 5 | . . . . . . 7 ⊢ 0ℎ ∈ (span‘{𝐴}) |
| 34 | 31, 33 | eqeltrdi 2842 | . . . . . 6 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → (𝐴 +ℎ 𝐵) ∈ (span‘{𝐴})) |
| 35 | 8 | eleq2i 2826 | . . . . . . 7 ⊢ (𝐵 ∈ 𝐹 ↔ 𝐵 ∈ (span‘{𝐴})) |
| 36 | sumspansn 31630 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 +ℎ 𝐵) ∈ (span‘{𝐴}) ↔ 𝐵 ∈ (span‘{𝐴}))) | |
| 37 | 1, 2, 36 | mp2an 692 | . . . . . . 7 ⊢ ((𝐴 +ℎ 𝐵) ∈ (span‘{𝐴}) ↔ 𝐵 ∈ (span‘{𝐴})) |
| 38 | 35, 37 | bitr4i 278 | . . . . . 6 ⊢ (𝐵 ∈ 𝐹 ↔ (𝐴 +ℎ 𝐵) ∈ (span‘{𝐴})) |
| 39 | 34, 38 | sylibr 234 | . . . . 5 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → 𝐵 ∈ 𝐹) |
| 40 | 39 | con3i 154 | . . . 4 ⊢ (¬ 𝐵 ∈ 𝐹 → ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ) |
| 41 | 40 | adantl 481 | . . 3 ⊢ ((¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹) → ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ) |
| 42 | 6, 8 | ineq12i 4193 | . . . . . 6 ⊢ (𝐻 ∩ 𝐹) = ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐴})) |
| 43 | 3, 1 | spansnm0i 31631 | . . . . . . 7 ⊢ (¬ (𝐴 +ℎ 𝐵) ∈ (span‘{𝐴}) → ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐴})) = 0ℋ) |
| 44 | 38, 43 | sylnbi 330 | . . . . . 6 ⊢ (¬ 𝐵 ∈ 𝐹 → ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐴})) = 0ℋ) |
| 45 | 42, 44 | eqtrid 2782 | . . . . 5 ⊢ (¬ 𝐵 ∈ 𝐹 → (𝐻 ∩ 𝐹) = 0ℋ) |
| 46 | 6, 12 | ineq12i 4193 | . . . . . 6 ⊢ (𝐻 ∩ 𝐺) = ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐵})) |
| 47 | sumspansn 31630 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 +ℎ 𝐴) ∈ (span‘{𝐵}) ↔ 𝐴 ∈ (span‘{𝐵}))) | |
| 48 | 2, 1, 47 | mp2an 692 | . . . . . . . 8 ⊢ ((𝐵 +ℎ 𝐴) ∈ (span‘{𝐵}) ↔ 𝐴 ∈ (span‘{𝐵})) |
| 49 | 1, 2 | hvcomi 31000 | . . . . . . . . 9 ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴) |
| 50 | 49 | eleq1i 2825 | . . . . . . . 8 ⊢ ((𝐴 +ℎ 𝐵) ∈ (span‘{𝐵}) ↔ (𝐵 +ℎ 𝐴) ∈ (span‘{𝐵})) |
| 51 | 12 | eleq2i 2826 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝐺 ↔ 𝐴 ∈ (span‘{𝐵})) |
| 52 | 48, 50, 51 | 3bitr4ri 304 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐺 ↔ (𝐴 +ℎ 𝐵) ∈ (span‘{𝐵})) |
| 53 | 3, 2 | spansnm0i 31631 | . . . . . . 7 ⊢ (¬ (𝐴 +ℎ 𝐵) ∈ (span‘{𝐵}) → ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐵})) = 0ℋ) |
| 54 | 52, 53 | sylnbi 330 | . . . . . 6 ⊢ (¬ 𝐴 ∈ 𝐺 → ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐵})) = 0ℋ) |
| 55 | 46, 54 | eqtrid 2782 | . . . . 5 ⊢ (¬ 𝐴 ∈ 𝐺 → (𝐻 ∩ 𝐺) = 0ℋ) |
| 56 | 45, 55 | oveqan12rd 7425 | . . . 4 ⊢ ((¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹) → ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = (0ℋ ∨ℋ 0ℋ)) |
| 57 | h0elch 31236 | . . . . 5 ⊢ 0ℋ ∈ Cℋ | |
| 58 | 57 | chj0i 31436 | . . . 4 ⊢ (0ℋ ∨ℋ 0ℋ) = 0ℋ |
| 59 | 56, 58 | eqtrdi 2786 | . . 3 ⊢ ((¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹) → ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = 0ℋ) |
| 60 | eqeq2 2747 | . . . . 5 ⊢ (((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = 0ℋ → ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) ↔ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ)) | |
| 61 | 60 | notbid 318 | . . . 4 ⊢ (((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = 0ℋ → (¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) ↔ ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ)) |
| 62 | 61 | biimparc 479 | . . 3 ⊢ ((¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ ∧ ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = 0ℋ) → ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺))) |
| 63 | 41, 59, 62 | syl2anc 584 | . 2 ⊢ ((¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹) → ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺))) |
| 64 | ioran 985 | . 2 ⊢ (¬ (𝐴 ∈ 𝐺 ∨ 𝐵 ∈ 𝐹) ↔ (¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹)) | |
| 65 | df-ne 2933 | . 2 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ≠ ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) ↔ ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺))) | |
| 66 | 63, 64, 65 | 3imtr4i 292 | 1 ⊢ (¬ (𝐴 ∈ 𝐺 ∨ 𝐵 ∈ 𝐹) → (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ≠ ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∩ cin 3925 {csn 4601 ‘cfv 6531 (class class class)co 7405 ℋchba 30900 +ℎ cva 30901 0ℎc0v 30905 Sℋ csh 30909 Cℋ cch 30910 +ℋ cph 30912 spancspn 30913 ∨ℋ chj 30914 0ℋc0h 30916 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cc 10449 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208 ax-mulf 11209 ax-hilex 30980 ax-hfvadd 30981 ax-hvcom 30982 ax-hvass 30983 ax-hv0cl 30984 ax-hvaddid 30985 ax-hfvmul 30986 ax-hvmulid 30987 ax-hvmulass 30988 ax-hvdistr1 30989 ax-hvdistr2 30990 ax-hvmul0 30991 ax-hfi 31060 ax-his1 31063 ax-his2 31064 ax-his3 31065 ax-his4 31066 ax-hcompl 31183 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-omul 8485 df-er 8719 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-card 9953 df-acn 9956 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13366 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-fl 13809 df-seq 14020 df-exp 14080 df-hash 14349 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-clim 15504 df-rlim 15505 df-sum 15703 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-rest 17436 df-topn 17437 df-0g 17455 df-gsum 17456 df-topgen 17457 df-pt 17458 df-prds 17461 df-xrs 17516 df-qtop 17521 df-imas 17522 df-xps 17524 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-mulg 19051 df-cntz 19300 df-cmn 19763 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-fbas 21312 df-fg 21313 df-cnfld 21316 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cld 22957 df-ntr 22958 df-cls 22959 df-nei 23036 df-cn 23165 df-cnp 23166 df-lm 23167 df-haus 23253 df-tx 23500 df-hmeo 23693 df-fil 23784 df-fm 23876 df-flim 23877 df-flf 23878 df-xms 24259 df-ms 24260 df-tms 24261 df-cfil 25207 df-cau 25208 df-cmet 25209 df-grpo 30474 df-gid 30475 df-ginv 30476 df-gdiv 30477 df-ablo 30526 df-vc 30540 df-nv 30573 df-va 30576 df-ba 30577 df-sm 30578 df-0v 30579 df-vs 30580 df-nmcv 30581 df-ims 30582 df-dip 30682 df-ssp 30703 df-ph 30794 df-cbn 30844 df-hnorm 30949 df-hba 30950 df-hvsub 30952 df-hlim 30953 df-hcau 30954 df-sh 31188 df-ch 31202 df-oc 31233 df-ch0 31234 df-shs 31289 df-span 31290 df-chj 31291 |
| This theorem is referenced by: (None) |
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