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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 31167 | . . . . . . . . . . . 12 ⊢ (𝐴 +ℎ 𝐵) ∈ ℋ |
| 4 | spansnid 31712 | . . . . . . . . . . . 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 2860 | . . . . . . . . . 10 ⊢ (𝐴 +ℎ 𝐵) ∈ 𝐻 |
| 8 | nonbool.3 | . . . . . . . . . . . . 13 ⊢ 𝐹 = (span‘{𝐴}) | |
| 9 | 1 | spansnchi 31711 | . . . . . . . . . . . . . 14 ⊢ (span‘{𝐴}) ∈ Cℋ |
| 10 | 9 | chshii 31376 | . . . . . . . . . . . . 13 ⊢ (span‘{𝐴}) ∈ Sℋ |
| 11 | 8, 10 | eqeltri 2857 | . . . . . . . . . . . 12 ⊢ 𝐹 ∈ Sℋ |
| 12 | nonbool.4 | . . . . . . . . . . . . 13 ⊢ 𝐺 = (span‘{𝐵}) | |
| 13 | 2 | spansnchi 31711 | . . . . . . . . . . . . . 14 ⊢ (span‘{𝐵}) ∈ Cℋ |
| 14 | 13 | chshii 31376 | . . . . . . . . . . . . 13 ⊢ (span‘{𝐵}) ∈ Sℋ |
| 15 | 12, 14 | eqeltri 2857 | . . . . . . . . . . . 12 ⊢ 𝐺 ∈ Sℋ |
| 16 | 11, 15 | shsleji 31519 | . . . . . . . . . . 11 ⊢ (𝐹 +ℋ 𝐺) ⊆ (𝐹 ∨ℋ 𝐺) |
| 17 | spansnid 31712 | . . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℋ → 𝐴 ∈ (span‘{𝐴})) | |
| 18 | 1, 17 | ax-mp 5 | . . . . . . . . . . . . 13 ⊢ 𝐴 ∈ (span‘{𝐴}) |
| 19 | 18, 8 | eleqtrri 2860 | . . . . . . . . . . . 12 ⊢ 𝐴 ∈ 𝐹 |
| 20 | spansnid 31712 | . . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℋ → 𝐵 ∈ (span‘{𝐵})) | |
| 21 | 2, 20 | ax-mp 5 | . . . . . . . . . . . . 13 ⊢ 𝐵 ∈ (span‘{𝐵}) |
| 22 | 21, 12 | eleqtrri 2860 | . . . . . . . . . . . 12 ⊢ 𝐵 ∈ 𝐺 |
| 23 | 11, 15 | shsvai 31513 | . . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺) → (𝐴 +ℎ 𝐵) ∈ (𝐹 +ℋ 𝐺)) |
| 24 | 19, 22, 23 | mp2an 702 | . . . . . . . . . . 11 ⊢ (𝐴 +ℎ 𝐵) ∈ (𝐹 +ℋ 𝐺) |
| 25 | 16, 24 | sselii 3933 | . . . . . . . . . 10 ⊢ (𝐴 +ℎ 𝐵) ∈ (𝐹 ∨ℋ 𝐺) |
| 26 | elin 3920 | . . . . . . . . . 10 ⊢ ((𝐴 +ℎ 𝐵) ∈ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ↔ ((𝐴 +ℎ 𝐵) ∈ 𝐻 ∧ (𝐴 +ℎ 𝐵) ∈ (𝐹 ∨ℋ 𝐺))) | |
| 27 | 7, 25, 26 | mpbir2an 721 | . . . . . . . . 9 ⊢ (𝐴 +ℎ 𝐵) ∈ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) |
| 28 | eleq2 2850 | . . . . . . . . 9 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → ((𝐴 +ℎ 𝐵) ∈ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ↔ (𝐴 +ℎ 𝐵) ∈ 0ℋ)) | |
| 29 | 27, 28 | mpbii 235 | . . . . . . . 8 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → (𝐴 +ℎ 𝐵) ∈ 0ℋ) |
| 30 | elch0 31403 | . . . . . . . 8 ⊢ ((𝐴 +ℎ 𝐵) ∈ 0ℋ ↔ (𝐴 +ℎ 𝐵) = 0ℎ) | |
| 31 | 29, 30 | sylib 220 | . . . . . . 7 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → (𝐴 +ℎ 𝐵) = 0ℎ) |
| 32 | ch0 31377 | . . . . . . . 8 ⊢ ((span‘{𝐴}) ∈ Cℋ → 0ℎ ∈ (span‘{𝐴})) | |
| 33 | 9, 32 | ax-mp 5 | . . . . . . 7 ⊢ 0ℎ ∈ (span‘{𝐴}) |
| 34 | 31, 33 | eqeltrdi 2869 | . . . . . 6 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → (𝐴 +ℎ 𝐵) ∈ (span‘{𝐴})) |
| 35 | 8 | eleq2i 2853 | . . . . . . 7 ⊢ (𝐵 ∈ 𝐹 ↔ 𝐵 ∈ (span‘{𝐴})) |
| 36 | sumspansn 31798 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 +ℎ 𝐵) ∈ (span‘{𝐴}) ↔ 𝐵 ∈ (span‘{𝐴}))) | |
| 37 | 1, 2, 36 | mp2an 702 | . . . . . . 7 ⊢ ((𝐴 +ℎ 𝐵) ∈ (span‘{𝐴}) ↔ 𝐵 ∈ (span‘{𝐴})) |
| 38 | 35, 37 | bitr4i 280 | . . . . . 6 ⊢ (𝐵 ∈ 𝐹 ↔ (𝐴 +ℎ 𝐵) ∈ (span‘{𝐴})) |
| 39 | 34, 38 | sylibr 236 | . . . . 5 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ → 𝐵 ∈ 𝐹) |
| 40 | 39 | con3i 154 | . . . 4 ⊢ (¬ 𝐵 ∈ 𝐹 → ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ) |
| 41 | 40 | adantl 485 | . . 3 ⊢ ((¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹) → ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ) |
| 42 | 6, 8 | ineq12i 4170 | . . . . . 6 ⊢ (𝐻 ∩ 𝐹) = ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐴})) |
| 43 | 3, 1 | spansnm0i 31799 | . . . . . . 7 ⊢ (¬ (𝐴 +ℎ 𝐵) ∈ (span‘{𝐴}) → ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐴})) = 0ℋ) |
| 44 | 38, 43 | sylnbi 332 | . . . . . 6 ⊢ (¬ 𝐵 ∈ 𝐹 → ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐴})) = 0ℋ) |
| 45 | 42, 44 | eqtrid 2808 | . . . . 5 ⊢ (¬ 𝐵 ∈ 𝐹 → (𝐻 ∩ 𝐹) = 0ℋ) |
| 46 | 6, 12 | ineq12i 4170 | . . . . . 6 ⊢ (𝐻 ∩ 𝐺) = ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐵})) |
| 47 | sumspansn 31798 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 +ℎ 𝐴) ∈ (span‘{𝐵}) ↔ 𝐴 ∈ (span‘{𝐵}))) | |
| 48 | 2, 1, 47 | mp2an 702 | . . . . . . . 8 ⊢ ((𝐵 +ℎ 𝐴) ∈ (span‘{𝐵}) ↔ 𝐴 ∈ (span‘{𝐵})) |
| 49 | 1, 2 | hvcomi 31168 | . . . . . . . . 9 ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴) |
| 50 | 49 | eleq1i 2852 | . . . . . . . 8 ⊢ ((𝐴 +ℎ 𝐵) ∈ (span‘{𝐵}) ↔ (𝐵 +ℎ 𝐴) ∈ (span‘{𝐵})) |
| 51 | 12 | eleq2i 2853 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝐺 ↔ 𝐴 ∈ (span‘{𝐵})) |
| 52 | 48, 50, 51 | 3bitr4ri 306 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐺 ↔ (𝐴 +ℎ 𝐵) ∈ (span‘{𝐵})) |
| 53 | 3, 2 | spansnm0i 31799 | . . . . . . 7 ⊢ (¬ (𝐴 +ℎ 𝐵) ∈ (span‘{𝐵}) → ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐵})) = 0ℋ) |
| 54 | 52, 53 | sylnbi 332 | . . . . . 6 ⊢ (¬ 𝐴 ∈ 𝐺 → ((span‘{(𝐴 +ℎ 𝐵)}) ∩ (span‘{𝐵})) = 0ℋ) |
| 55 | 46, 54 | eqtrid 2808 | . . . . 5 ⊢ (¬ 𝐴 ∈ 𝐺 → (𝐻 ∩ 𝐺) = 0ℋ) |
| 56 | 45, 55 | oveqan12rd 7412 | . . . 4 ⊢ ((¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹) → ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = (0ℋ ∨ℋ 0ℋ)) |
| 57 | h0elch 31404 | . . . . 5 ⊢ 0ℋ ∈ Cℋ | |
| 58 | 57 | chj0i 31604 | . . . 4 ⊢ (0ℋ ∨ℋ 0ℋ) = 0ℋ |
| 59 | 56, 58 | eqtrdi 2812 | . . 3 ⊢ ((¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹) → ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = 0ℋ) |
| 60 | eqeq2 2773 | . . . . 5 ⊢ (((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = 0ℋ → ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) ↔ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ)) | |
| 61 | 60 | notbid 320 | . . . 4 ⊢ (((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) = 0ℋ → (¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) ↔ ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = 0ℋ)) |
| 62 | 61 | biimparc 483 | . . 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 593 | . 2 ⊢ ((¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹) → ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺))) |
| 64 | ioran 996 | . 2 ⊢ (¬ (𝐴 ∈ 𝐺 ∨ 𝐵 ∈ 𝐹) ↔ (¬ 𝐴 ∈ 𝐺 ∧ ¬ 𝐵 ∈ 𝐹)) | |
| 65 | df-ne 2957 | . 2 ⊢ ((𝐻 ∩ (𝐹 ∨ℋ 𝐺)) ≠ ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺)) ↔ ¬ (𝐻 ∩ (𝐹 ∨ℋ 𝐺)) = ((𝐻 ∩ 𝐹) ∨ℋ (𝐻 ∩ 𝐺))) | |
| 66 | 63, 64, 65 | 3imtr4i 294 | 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 208 ∧ wa 399 ∨ wo 858 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∩ cin 3903 {csn 4581 ‘cfv 6517 (class class class)co 7392 ℋchba 31068 +ℎ cva 31069 0ℎc0v 31073 Sℋ csh 31077 Cℋ cch 31078 +ℋ cph 31080 spancspn 31081 ∨ℋ chj 31082 0ℋc0h 31084 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-inf2 9593 ax-cc 10389 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148 ax-addf 11149 ax-mulf 11150 ax-hilex 31148 ax-hfvadd 31149 ax-hvcom 31150 ax-hvass 31151 ax-hv0cl 31152 ax-hvaddid 31153 ax-hfvmul 31154 ax-hvmulid 31155 ax-hvmulass 31156 ax-hvdistr1 31157 ax-hvdistr2 31158 ax-hvmul0 31159 ax-hfi 31228 ax-his1 31231 ax-his2 31232 ax-his3 31233 ax-his4 31234 ax-hcompl 31351 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-om 7843 df-1st 7966 df-2nd 7967 df-supp 8136 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-oadd 8436 df-omul 8437 df-er 8673 df-map 8805 df-pm 8806 df-ixp 8876 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-fsupp 9305 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9455 df-card 9894 df-acn 9897 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-z 12566 df-dec 12686 df-uz 12837 df-q 12947 df-rp 12991 df-xneg 13111 df-xadd 13112 df-xmul 13113 df-ioo 13350 df-ico 13352 df-icc 13353 df-fz 13510 df-fzo 13657 df-fl 13799 df-seq 14012 df-exp 14072 df-hash 14341 df-cj 15109 df-re 15110 df-im 15111 df-sqrt 15245 df-abs 15246 df-clim 15498 df-rlim 15499 df-sum 15697 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17515 df-qtop 17520 df-imas 17521 df-xps 17523 df-mre 17597 df-mrc 17598 df-acs 17600 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-submnd 18801 df-mulg 19093 df-cntz 19340 df-cmn 19805 df-psmet 21396 df-xmet 21397 df-met 21398 df-bl 21399 df-mopn 21400 df-fbas 21401 df-fg 21402 df-cnfld 21405 df-top 22934 df-topon 22951 df-topsp 22973 df-bases 22986 df-cld 23059 df-ntr 23060 df-cls 23061 df-nei 23138 df-cn 23267 df-cnp 23268 df-lm 23269 df-haus 23355 df-tx 23602 df-hmeo 23795 df-fil 23886 df-fm 23978 df-flim 23979 df-flf 23980 df-xms 24360 df-ms 24361 df-tms 24362 df-cfil 25297 df-cau 25298 df-cmet 25299 df-grpo 30642 df-gid 30643 df-ginv 30644 df-gdiv 30645 df-ablo 30694 df-vc 30708 df-nv 30741 df-va 30744 df-ba 30745 df-sm 30746 df-0v 30747 df-vs 30748 df-nmcv 30749 df-ims 30750 df-dip 30850 df-ssp 30871 df-ph 30962 df-cbn 31012 df-hnorm 31117 df-hba 31118 df-hvsub 31120 df-hlim 31121 df-hcau 31122 df-sh 31356 df-ch 31370 df-oc 31401 df-ch0 31402 df-shs 31457 df-span 31458 df-chj 31459 |
| This theorem is referenced by: (None) |
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