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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 30951 | . . . . . . . . . . . 12 ⊢ (𝐴 +ℎ 𝐵) ∈ ℋ |
| 4 | spansnid 31496 | . . . . . . . . . . . 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 2825 | . . . . . . . . . 10 ⊢ (𝐴 +ℎ 𝐵) ∈ 𝐻 |
| 8 | nonbool.3 | . . . . . . . . . . . . 13 ⊢ 𝐹 = (span‘{𝐴}) | |
| 9 | 1 | spansnchi 31495 | . . . . . . . . . . . . . 14 ⊢ (span‘{𝐴}) ∈ Cℋ |
| 10 | 9 | chshii 31160 | . . . . . . . . . . . . 13 ⊢ (span‘{𝐴}) ∈ Sℋ |
| 11 | 8, 10 | eqeltri 2822 | . . . . . . . . . . . 12 ⊢ 𝐹 ∈ Sℋ |
| 12 | nonbool.4 | . . . . . . . . . . . . 13 ⊢ 𝐺 = (span‘{𝐵}) | |
| 13 | 2 | spansnchi 31495 | . . . . . . . . . . . . . 14 ⊢ (span‘{𝐵}) ∈ Cℋ |
| 14 | 13 | chshii 31160 | . . . . . . . . . . . . 13 ⊢ (span‘{𝐵}) ∈ Sℋ |
| 15 | 12, 14 | eqeltri 2822 | . . . . . . . . . . . 12 ⊢ 𝐺 ∈ Sℋ |
| 16 | 11, 15 | shsleji 31303 | . . . . . . . . . . 11 ⊢ (𝐹 +ℋ 𝐺) ⊆ (𝐹 ∨ℋ 𝐺) |
| 17 | spansnid 31496 | . . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℋ → 𝐴 ∈ (span‘{𝐴})) | |
| 18 | 1, 17 | ax-mp 5 | . . . . . . . . . . . . 13 ⊢ 𝐴 ∈ (span‘{𝐴}) |
| 19 | 18, 8 | eleqtrri 2825 | . . . . . . . . . . . 12 ⊢ 𝐴 ∈ 𝐹 |
| 20 | spansnid 31496 | . . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℋ → 𝐵 ∈ (span‘{𝐵})) | |
| 21 | 2, 20 | ax-mp 5 | . . . . . . . . . . . . 13 ⊢ 𝐵 ∈ (span‘{𝐵}) |
| 22 | 21, 12 | eleqtrri 2825 | . . . . . . . . . . . 12 ⊢ 𝐵 ∈ 𝐺 |
| 23 | 11, 15 | shsvai 31297 | . . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺) → (𝐴 +ℎ 𝐵) ∈ (𝐹 +ℋ 𝐺)) |
| 24 | 19, 22, 23 | mp2an 690 | . . . . . . . . . . 11 ⊢ (𝐴 +ℎ 𝐵) ∈ (𝐹 +ℋ 𝐺) |
| 25 | 16, 24 | sselii 3976 | . . . . . . . . . 10 ⊢ (𝐴 +ℎ 𝐵) ∈ (𝐹 ∨ℋ 𝐺) |
| 26 | elin 3963 | . . . . . . . . . 10 ⊢ ((𝐴 +ℎ 𝐵) ∈ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ↔ ((𝐴 +ℎ 𝐵) ∈ 𝐻 ∧ (𝐴 +ℎ 𝐵) ∈ (𝐹 ∨ℋ 𝐺))) | |
| 27 | 7, 25, 26 | mpbir2an 709 | . . . . . . . . 9 ⊢ (𝐴 +ℎ 𝐵) ∈ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) |
| 28 | eleq2 2815 | . . . . . . . . 9 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → ((𝐴 +ℎ 𝐵) ∈ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ↔ (𝐴 +ℎ 𝐵) ∈ 0ℋ)) | |
| 29 | 27, 28 | mpbii 232 | . . . . . . . 8 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → (𝐴 +ℎ 𝐵) ∈ 0ℋ) |
| 30 | elch0 31187 | . . . . . . . 8 ⊢ ((𝐴 +ℎ 𝐵) ∈ 0ℋ ↔ (𝐴 +ℎ 𝐵) = 0ℎ) | |
| 31 | 29, 30 | sylib 217 | . . . . . . 7 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → (𝐴 +ℎ 𝐵) = 0ℎ) |
| 32 | ch0 31161 | . . . . . . . 8 ⊢ ((span‘{𝐴}) ∈ Cℋ → 0ℎ ∈ (span‘{𝐴})) | |
| 33 | 9, 32 | ax-mp 5 | . . . . . . 7 ⊢ 0ℎ ∈ (span‘{𝐴}) |
| 34 | 31, 33 | eqeltrdi 2834 | . . . . . 6 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → (𝐴 +ℎ 𝐵) ∈ (span‘{𝐴})) |
| 35 | 8 | eleq2i 2818 | . . . . . . 7 ⊢ (𝐵 ∈ 𝐹 ↔ 𝐵 ∈ (span‘{𝐴})) |
| 36 | sumspansn 31582 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 +ℎ 𝐵) ∈ (span‘{𝐴}) ↔ 𝐵 ∈ (span‘{𝐴}))) | |
| 37 | 1, 2, 36 | mp2an 690 | . . . . . . 7 ⊢ ((𝐴 +ℎ 𝐵) ∈ (span‘{𝐴}) ↔ 𝐵 ∈ (span‘{𝐴})) |
| 38 | 35, 37 | bitr4i 277 | . . . . . 6 ⊢ (𝐵 ∈ 𝐹 ↔ (𝐴 +ℎ 𝐵) ∈ (span‘{𝐴})) |
| 39 | 34, 38 | sylibr 233 | . . . . 5 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → 𝐵 ∈ 𝐹) |
| 40 | 39 | con3i 154 | . . . 4 ⊢ (¬ 𝐵 ∈ 𝐹 → ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ) |
| 41 | 40 | adantl 480 | . . 3 ⊢ ((¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹) → ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ) |
| 42 | 6, 8 | ineq12i 4211 | . . . . . 6 ⊢ (𝐻 ∩ 𝐹) = ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐴})) |
| 43 | 3, 1 | spansnm0i 31583 | . . . . . . 7 ⊢ (¬ (𝐴 +ℎ 𝐵) ∈ (span‘{𝐴}) → ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐴})) = 0ℋ) |
| 44 | 38, 43 | sylnbi 329 | . . . . . 6 ⊢ (¬ 𝐵 ∈ 𝐹 → ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐴})) = 0ℋ) |
| 45 | 42, 44 | eqtrid 2778 | . . . . 5 ⊢ (¬ 𝐵 ∈ 𝐹 → (𝐻 ∩ 𝐹) = 0ℋ) |
| 46 | 6, 12 | ineq12i 4211 | . . . . . 6 ⊢ (𝐻 ∩ 𝐺) = ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐵})) |
| 47 | sumspansn 31582 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 +ℎ 𝐴) ∈ (span‘{𝐵}) ↔ 𝐴 ∈ (span‘{𝐵}))) | |
| 48 | 2, 1, 47 | mp2an 690 | . . . . . . . 8 ⊢ ((𝐵 +ℎ 𝐴) ∈ (span‘{𝐵}) ↔ 𝐴 ∈ (span‘{𝐵})) |
| 49 | 1, 2 | hvcomi 30952 | . . . . . . . . 9 ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴) |
| 50 | 49 | eleq1i 2817 | . . . . . . . 8 ⊢ ((𝐴 +ℎ 𝐵) ∈ (span‘{𝐵}) ↔ (𝐵 +ℎ 𝐴) ∈ (span‘{𝐵})) |
| 51 | 12 | eleq2i 2818 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝐺 ↔ 𝐴 ∈ (span‘{𝐵})) |
| 52 | 48, 50, 51 | 3bitr4ri 303 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐺 ↔ (𝐴 +ℎ 𝐵) ∈ (span‘{𝐵})) |
| 53 | 3, 2 | spansnm0i 31583 | . . . . . . 7 ⊢ (¬ (𝐴 +ℎ 𝐵) ∈ (span‘{𝐵}) → ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐵})) = 0ℋ) |
| 54 | 52, 53 | sylnbi 329 | . . . . . 6 ⊢ (¬ 𝐴 ∈ 𝐺 → ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐵})) = 0ℋ) |
| 55 | 46, 54 | eqtrid 2778 | . . . . 5 ⊢ (¬ 𝐴 ∈ 𝐺 → (𝐻 ∩ 𝐺) = 0ℋ) |
| 56 | 45, 55 | oveqan12rd 7444 | . . . 4 ⊢ ((¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹) → ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = (0ℋ ∨ℋ 0ℋ)) |
| 57 | h0elch 31188 | . . . . 5 ⊢ 0ℋ ∈ Cℋ | |
| 58 | 57 | chj0i 31388 | . . . 4 ⊢ (0ℋ ∨ℋ 0ℋ) = 0ℋ |
| 59 | 56, 58 | eqtrdi 2782 | . . 3 ⊢ ((¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹) → ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = 0ℋ) |
| 60 | eqeq2 2738 | . . . . 5 ⊢ (((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = 0ℋ → ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) ↔ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ)) | |
| 61 | 60 | notbid 317 | . . . 4 ⊢ (((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = 0ℋ → (¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) ↔ ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ)) |
| 62 | 61 | biimparc 478 | . . 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 582 | . 2 ⊢ ((¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹) → ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺))) |
| 64 | ioran 981 | . 2 ⊢ (¬ (𝐴 ∈ 𝐺 ∨ 𝐵 ∈ 𝐹) ↔ (¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹)) | |
| 65 | df-ne 2931 | . 2 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ≠ ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) ↔ ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺))) | |
| 66 | 63, 64, 65 | 3imtr4i 291 | 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 205 ∧ wa 394 ∨ wo 845 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∩ cin 3946 {csn 4633 ‘cfv 6554 (class class class)co 7424 ℋchba 30852 +ℎ cva 30853 0ℎc0v 30857 Sℋ csh 30861 Cℋ cch 30862 +ℋ cph 30864 spancspn 30865 ∨ℋ chj 30866 0ℋc0h 30868 |
| 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 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cc 10478 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237 ax-mulf 11238 ax-hilex 30932 ax-hfvadd 30933 ax-hvcom 30934 ax-hvass 30935 ax-hv0cl 30936 ax-hvaddid 30937 ax-hfvmul 30938 ax-hvmulid 30939 ax-hvmulass 30940 ax-hvdistr1 30941 ax-hvdistr2 30942 ax-hvmul0 30943 ax-hfi 31012 ax-his1 31015 ax-his2 31016 ax-his3 31017 ax-his4 31018 ax-hcompl 31135 |
| 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 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-oadd 8500 df-omul 8501 df-er 8734 df-map 8857 df-pm 8858 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-fi 9454 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-acn 9985 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-ioo 13382 df-ico 13384 df-icc 13385 df-fz 13539 df-fzo 13682 df-fl 13812 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-rlim 15491 df-sum 15691 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-hom 17290 df-cco 17291 df-rest 17437 df-topn 17438 df-0g 17456 df-gsum 17457 df-topgen 17458 df-pt 17459 df-prds 17462 df-xrs 17517 df-qtop 17522 df-imas 17523 df-xps 17525 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-mulg 19062 df-cntz 19311 df-cmn 19780 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 df-mopn 21339 df-fbas 21340 df-fg 21341 df-cnfld 21344 df-top 22887 df-topon 22904 df-topsp 22926 df-bases 22940 df-cld 23014 df-ntr 23015 df-cls 23016 df-nei 23093 df-cn 23222 df-cnp 23223 df-lm 23224 df-haus 23310 df-tx 23557 df-hmeo 23750 df-fil 23841 df-fm 23933 df-flim 23934 df-flf 23935 df-xms 24317 df-ms 24318 df-tms 24319 df-cfil 25274 df-cau 25275 df-cmet 25276 df-grpo 30426 df-gid 30427 df-ginv 30428 df-gdiv 30429 df-ablo 30478 df-vc 30492 df-nv 30525 df-va 30528 df-ba 30529 df-sm 30530 df-0v 30531 df-vs 30532 df-nmcv 30533 df-ims 30534 df-dip 30634 df-ssp 30655 df-ph 30746 df-cbn 30796 df-hnorm 30901 df-hba 30902 df-hvsub 30904 df-hlim 30905 df-hcau 30906 df-sh 31140 df-ch 31154 df-oc 31185 df-ch0 31186 df-shs 31241 df-span 31242 df-chj 31243 |
| This theorem is referenced by: (None) |
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