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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1dimN | Structured version Visualization version GIF version |
Description: An atom is covered by a height-2 element (1-dimensional line). (Contributed by NM, 3-May-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2dim.j | ⊢ ∨ = (join‘𝐾) |
2dim.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
2dim.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
1dimN | ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → ∃𝑞 ∈ 𝐴 𝑃𝐶(𝑃 ∨ 𝑞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2dim.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
2 | 2dim.c | . . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
3 | 2dim.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 1, 2, 3 | 2dim 35482 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑃𝐶(𝑃 ∨ 𝑞) ∧ (𝑃 ∨ 𝑞)𝐶((𝑃 ∨ 𝑞) ∨ 𝑟))) |
5 | r19.42v 3271 | . . . 4 ⊢ (∃𝑟 ∈ 𝐴 (𝑃𝐶(𝑃 ∨ 𝑞) ∧ (𝑃 ∨ 𝑞)𝐶((𝑃 ∨ 𝑞) ∨ 𝑟)) ↔ (𝑃𝐶(𝑃 ∨ 𝑞) ∧ ∃𝑟 ∈ 𝐴 (𝑃 ∨ 𝑞)𝐶((𝑃 ∨ 𝑞) ∨ 𝑟))) | |
6 | 5 | simplbi 492 | . . 3 ⊢ (∃𝑟 ∈ 𝐴 (𝑃𝐶(𝑃 ∨ 𝑞) ∧ (𝑃 ∨ 𝑞)𝐶((𝑃 ∨ 𝑞) ∨ 𝑟)) → 𝑃𝐶(𝑃 ∨ 𝑞)) |
7 | 6 | reximi 3189 | . 2 ⊢ (∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑃𝐶(𝑃 ∨ 𝑞) ∧ (𝑃 ∨ 𝑞)𝐶((𝑃 ∨ 𝑞) ∨ 𝑟)) → ∃𝑞 ∈ 𝐴 𝑃𝐶(𝑃 ∨ 𝑞)) |
8 | 4, 7 | syl 17 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → ∃𝑞 ∈ 𝐴 𝑃𝐶(𝑃 ∨ 𝑞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∃wrex 3088 class class class wbr 4841 ‘cfv 6099 (class class class)co 6876 joincjn 17255 ⋖ ccvr 35274 Atomscatm 35275 HLchlt 35362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-proset 17239 df-poset 17257 df-plt 17269 df-lub 17285 df-glb 17286 df-join 17287 df-meet 17288 df-p0 17350 df-p1 17351 df-lat 17357 df-clat 17419 df-oposet 35188 df-ol 35190 df-oml 35191 df-covers 35278 df-ats 35279 df-atl 35310 df-cvlat 35334 df-hlat 35363 |
This theorem is referenced by: (None) |
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