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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 28176 | . . . . . . . . . . . 12 ⊢ (𝐴 +ℎ 𝐵) ∈ ℋ |
| 4 | spansnid 28723 | . . . . . . . . . . . 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 2830 | . . . . . . . . . 10 ⊢ (𝐴 +ℎ 𝐵) ∈ 𝐻 |
| 8 | nonbool.3 | . . . . . . . . . . . . 13 ⊢ 𝐹 = (span‘{𝐴}) | |
| 9 | 1 | spansnchi 28722 | . . . . . . . . . . . . . 14 ⊢ (span‘{𝐴}) ∈ Cℋ |
| 10 | 9 | chshii 28385 | . . . . . . . . . . . . 13 ⊢ (span‘{𝐴}) ∈ Sℋ |
| 11 | 8, 10 | eqeltri 2827 | . . . . . . . . . . . 12 ⊢ 𝐹 ∈ Sℋ |
| 12 | nonbool.4 | . . . . . . . . . . . . 13 ⊢ 𝐺 = (span‘{𝐵}) | |
| 13 | 2 | spansnchi 28722 | . . . . . . . . . . . . . 14 ⊢ (span‘{𝐵}) ∈ Cℋ |
| 14 | 13 | chshii 28385 | . . . . . . . . . . . . 13 ⊢ (span‘{𝐵}) ∈ Sℋ |
| 15 | 12, 14 | eqeltri 2827 | . . . . . . . . . . . 12 ⊢ 𝐺 ∈ Sℋ |
| 16 | 11, 15 | shsleji 28530 | . . . . . . . . . . 11 ⊢ (𝐹 +ℋ 𝐺) ⊆ (𝐹 ∨ℋ 𝐺) |
| 17 | spansnid 28723 | . . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℋ → 𝐴 ∈ (span‘{𝐴})) | |
| 18 | 1, 17 | ax-mp 5 | . . . . . . . . . . . . 13 ⊢ 𝐴 ∈ (span‘{𝐴}) |
| 19 | 18, 8 | eleqtrri 2830 | . . . . . . . . . . . 12 ⊢ 𝐴 ∈ 𝐹 |
| 20 | spansnid 28723 | . . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℋ → 𝐵 ∈ (span‘{𝐵})) | |
| 21 | 2, 20 | ax-mp 5 | . . . . . . . . . . . . 13 ⊢ 𝐵 ∈ (span‘{𝐵}) |
| 22 | 21, 12 | eleqtrri 2830 | . . . . . . . . . . . 12 ⊢ 𝐵 ∈ 𝐺 |
| 23 | 11, 15 | shsvai 28524 | . . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺) → (𝐴 +ℎ 𝐵) ∈ (𝐹 +ℋ 𝐺)) |
| 24 | 19, 22, 23 | mp2an 710 | . . . . . . . . . . 11 ⊢ (𝐴 +ℎ 𝐵) ∈ (𝐹 +ℋ 𝐺) |
| 25 | 16, 24 | sselii 3733 | . . . . . . . . . 10 ⊢ (𝐴 +ℎ 𝐵) ∈ (𝐹 ∨ℋ 𝐺) |
| 26 | elin 3931 | . . . . . . . . . 10 ⊢ ((𝐴 +ℎ 𝐵) ∈ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ↔ ((𝐴 +ℎ 𝐵) ∈ 𝐻 ∧ (𝐴 +ℎ 𝐵) ∈ (𝐹 ∨ℋ 𝐺))) | |
| 27 | 7, 25, 26 | mpbir2an 993 | . . . . . . . . 9 ⊢ (𝐴 +ℎ 𝐵) ∈ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) |
| 28 | eleq2 2820 | . . . . . . . . 9 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → ((𝐴 +ℎ 𝐵) ∈ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ↔ (𝐴 +ℎ 𝐵) ∈ 0ℋ)) | |
| 29 | 27, 28 | mpbii 223 | . . . . . . . 8 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → (𝐴 +ℎ 𝐵) ∈ 0ℋ) |
| 30 | elch0 28412 | . . . . . . . 8 ⊢ ((𝐴 +ℎ 𝐵) ∈ 0ℋ ↔ (𝐴 +ℎ 𝐵) = 0ℎ) | |
| 31 | 29, 30 | sylib 208 | . . . . . . 7 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → (𝐴 +ℎ 𝐵) = 0ℎ) |
| 32 | ch0 28386 | . . . . . . . 8 ⊢ ((span‘{𝐴}) ∈ Cℋ → 0ℎ ∈ (span‘{𝐴})) | |
| 33 | 9, 32 | ax-mp 5 | . . . . . . 7 ⊢ 0ℎ ∈ (span‘{𝐴}) |
| 34 | 31, 33 | syl6eqel 2839 | . . . . . 6 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → (𝐴 +ℎ 𝐵) ∈ (span‘{𝐴})) |
| 35 | 8 | eleq2i 2823 | . . . . . . 7 ⊢ (𝐵 ∈ 𝐹 ↔ 𝐵 ∈ (span‘{𝐴})) |
| 36 | sumspansn 28809 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 +ℎ 𝐵) ∈ (span‘{𝐴}) ↔ 𝐵 ∈ (span‘{𝐴}))) | |
| 37 | 1, 2, 36 | mp2an 710 | . . . . . . 7 ⊢ ((𝐴 +ℎ 𝐵) ∈ (span‘{𝐴}) ↔ 𝐵 ∈ (span‘{𝐴})) |
| 38 | 35, 37 | bitr4i 267 | . . . . . 6 ⊢ (𝐵 ∈ 𝐹 ↔ (𝐴 +ℎ 𝐵) ∈ (span‘{𝐴})) |
| 39 | 34, 38 | sylibr 224 | . . . . 5 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → 𝐵 ∈ 𝐹) |
| 40 | 39 | con3i 150 | . . . 4 ⊢ (¬ 𝐵 ∈ 𝐹 → ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ) |
| 41 | 40 | adantl 473 | . . 3 ⊢ ((¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹) → ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ) |
| 42 | 6, 8 | ineq12i 3947 | . . . . . 6 ⊢ (𝐻 ∩ 𝐹) = ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐴})) |
| 43 | 3, 1 | spansnm0i 28810 | . . . . . . 7 ⊢ (¬ (𝐴 +ℎ 𝐵) ∈ (span‘{𝐴}) → ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐴})) = 0ℋ) |
| 44 | 38, 43 | sylnbi 319 | . . . . . 6 ⊢ (¬ 𝐵 ∈ 𝐹 → ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐴})) = 0ℋ) |
| 45 | 42, 44 | syl5eq 2798 | . . . . 5 ⊢ (¬ 𝐵 ∈ 𝐹 → (𝐻 ∩ 𝐹) = 0ℋ) |
| 46 | 6, 12 | ineq12i 3947 | . . . . . 6 ⊢ (𝐻 ∩ 𝐺) = ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐵})) |
| 47 | sumspansn 28809 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 +ℎ 𝐴) ∈ (span‘{𝐵}) ↔ 𝐴 ∈ (span‘{𝐵}))) | |
| 48 | 2, 1, 47 | mp2an 710 | . . . . . . . 8 ⊢ ((𝐵 +ℎ 𝐴) ∈ (span‘{𝐵}) ↔ 𝐴 ∈ (span‘{𝐵})) |
| 49 | 1, 2 | hvcomi 28177 | . . . . . . . . 9 ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴) |
| 50 | 49 | eleq1i 2822 | . . . . . . . 8 ⊢ ((𝐴 +ℎ 𝐵) ∈ (span‘{𝐵}) ↔ (𝐵 +ℎ 𝐴) ∈ (span‘{𝐵})) |
| 51 | 12 | eleq2i 2823 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝐺 ↔ 𝐴 ∈ (span‘{𝐵})) |
| 52 | 48, 50, 51 | 3bitr4ri 293 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐺 ↔ (𝐴 +ℎ 𝐵) ∈ (span‘{𝐵})) |
| 53 | 3, 2 | spansnm0i 28810 | . . . . . . 7 ⊢ (¬ (𝐴 +ℎ 𝐵) ∈ (span‘{𝐵}) → ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐵})) = 0ℋ) |
| 54 | 52, 53 | sylnbi 319 | . . . . . 6 ⊢ (¬ 𝐴 ∈ 𝐺 → ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐵})) = 0ℋ) |
| 55 | 46, 54 | syl5eq 2798 | . . . . 5 ⊢ (¬ 𝐴 ∈ 𝐺 → (𝐻 ∩ 𝐺) = 0ℋ) |
| 56 | 45, 55 | oveqan12rd 6825 | . . . 4 ⊢ ((¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹) → ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = (0ℋ ∨ℋ 0ℋ)) |
| 57 | h0elch 28413 | . . . . 5 ⊢ 0ℋ ∈ Cℋ | |
| 58 | 57 | chj0i 28615 | . . . 4 ⊢ (0ℋ ∨ℋ 0ℋ) = 0ℋ |
| 59 | 56, 58 | syl6eq 2802 | . . 3 ⊢ ((¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹) → ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = 0ℋ) |
| 60 | eqeq2 2763 | . . . . 5 ⊢ (((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = 0ℋ → ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) ↔ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ)) | |
| 61 | 60 | notbid 307 | . . . 4 ⊢ (((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = 0ℋ → (¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) ↔ ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ)) |
| 62 | 61 | biimparc 505 | . . 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 696 | . 2 ⊢ ((¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹) → ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺))) |
| 64 | ioran 512 | . 2 ⊢ (¬ (𝐴 ∈ 𝐺 ∨ 𝐵 ∈ 𝐹) ↔ (¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹)) | |
| 65 | df-ne 2925 | . 2 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ≠ ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) ↔ ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺))) | |
| 66 | 63, 64, 65 | 3imtr4i 281 | 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 196 ∨ wo 382 ∧ wa 383 = wceq 1624 ∈ wcel 2131 ≠ wne 2924 ∩ cin 3706 {csn 4313 ‘cfv 6041 (class class class)co 6805 ℋchil 28077 +ℎ cva 28078 0ℎc0v 28082 Sℋ csh 28086 Cℋ cch 28087 +ℋ cph 28089 spancspn 28090 ∨ℋ chj 28091 0ℋc0h 28093 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-rep 4915 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-inf2 8703 ax-cc 9441 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 ax-pre-sup 10198 ax-addf 10199 ax-mulf 10200 ax-hilex 28157 ax-hfvadd 28158 ax-hvcom 28159 ax-hvass 28160 ax-hv0cl 28161 ax-hvaddid 28162 ax-hfvmul 28163 ax-hvmulid 28164 ax-hvmulass 28165 ax-hvdistr1 28166 ax-hvdistr2 28167 ax-hvmul0 28168 ax-hfi 28237 ax-his1 28240 ax-his2 28241 ax-his3 28242 ax-his4 28243 ax-hcompl 28360 |
| This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-fal 1630 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-int 4620 df-iun 4666 df-iin 4667 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-se 5218 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-isom 6050 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-of 7054 df-om 7223 df-1st 7325 df-2nd 7326 df-supp 7456 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-1o 7721 df-2o 7722 df-oadd 7725 df-omul 7726 df-er 7903 df-map 8017 df-pm 8018 df-ixp 8067 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-fsupp 8433 df-fi 8474 df-sup 8505 df-inf 8506 df-oi 8572 df-card 8947 df-acn 8950 df-cda 9174 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-div 10869 df-nn 11205 df-2 11263 df-3 11264 df-4 11265 df-5 11266 df-6 11267 df-7 11268 df-8 11269 df-9 11270 df-n0 11477 df-z 11562 df-dec 11678 df-uz 11872 df-q 11974 df-rp 12018 df-xneg 12131 df-xadd 12132 df-xmul 12133 df-ioo 12364 df-ico 12366 df-icc 12367 df-fz 12512 df-fzo 12652 df-fl 12779 df-seq 12988 df-exp 13047 df-hash 13304 df-cj 14030 df-re 14031 df-im 14032 df-sqrt 14166 df-abs 14167 df-clim 14410 df-rlim 14411 df-sum 14608 df-struct 16053 df-ndx 16054 df-slot 16055 df-base 16057 df-sets 16058 df-ress 16059 df-plusg 16148 df-mulr 16149 df-starv 16150 df-sca 16151 df-vsca 16152 df-ip 16153 df-tset 16154 df-ple 16155 df-ds 16158 df-unif 16159 df-hom 16160 df-cco 16161 df-rest 16277 df-topn 16278 df-0g 16296 df-gsum 16297 df-topgen 16298 df-pt 16299 df-prds 16302 df-xrs 16356 df-qtop 16361 df-imas 16362 df-xps 16364 df-mre 16440 df-mrc 16441 df-acs 16443 df-mgm 17435 df-sgrp 17477 df-mnd 17488 df-submnd 17529 df-mulg 17734 df-cntz 17942 df-cmn 18387 df-psmet 19932 df-xmet 19933 df-met 19934 df-bl 19935 df-mopn 19936 df-fbas 19937 df-fg 19938 df-cnfld 19941 df-top 20893 df-topon 20910 df-topsp 20931 df-bases 20944 df-cld 21017 df-ntr 21018 df-cls 21019 df-nei 21096 df-cn 21225 df-cnp 21226 df-lm 21227 df-haus 21313 df-tx 21559 df-hmeo 21752 df-fil 21843 df-fm 21935 df-flim 21936 df-flf 21937 df-xms 22318 df-ms 22319 df-tms 22320 df-cfil 23245 df-cau 23246 df-cmet 23247 df-grpo 27648 df-gid 27649 df-ginv 27650 df-gdiv 27651 df-ablo 27700 df-vc 27715 df-nv 27748 df-va 27751 df-ba 27752 df-sm 27753 df-0v 27754 df-vs 27755 df-nmcv 27756 df-ims 27757 df-dip 27857 df-ssp 27878 df-ph 27969 df-cbn 28020 df-hnorm 28126 df-hba 28127 df-hvsub 28129 df-hlim 28130 df-hcau 28131 df-sh 28365 df-ch 28379 df-oc 28410 df-ch0 28411 df-shs 28468 df-span 28469 df-chj 28470 |
| This theorem is referenced by: (None) |
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