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Mirrors > Home > MPE Home > Th. List > Mathboxes > llnexch2N | Structured version Visualization version GIF version |
Description: Line exchange property (compare cvlatexch2 34942 for atoms). (Contributed by NM, 18-Nov-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
llnexch.l | ⊢ ≤ = (le‘𝐾) |
llnexch.j | ⊢ ∨ = (join‘𝐾) |
llnexch.m | ⊢ ∧ = (meet‘𝐾) |
llnexch.a | ⊢ 𝐴 = (Atoms‘𝐾) |
llnexch.n | ⊢ 𝑁 = (LLines‘𝐾) |
Ref | Expression |
---|---|
llnexch2N | ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑍 ∈ 𝑁) ∧ ((𝑋 ∧ 𝑌) ∈ 𝐴 ∧ 𝑋 ≠ 𝑍)) → ((𝑋 ∧ 𝑌) ≤ 𝑍 → (𝑋 ∧ 𝑍) ≤ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | llnexch.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
2 | llnexch.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
3 | llnexch.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
4 | llnexch.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | llnexch.n | . . 3 ⊢ 𝑁 = (LLines‘𝐾) | |
6 | 1, 2, 3, 4, 5 | llnexchb2 35473 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑍 ∈ 𝑁) ∧ ((𝑋 ∧ 𝑌) ∈ 𝐴 ∧ 𝑋 ≠ 𝑍)) → ((𝑋 ∧ 𝑌) ≤ 𝑍 ↔ (𝑋 ∧ 𝑌) = (𝑋 ∧ 𝑍))) |
7 | hllat 34968 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
8 | 7 | 3ad2ant1 1102 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑍 ∈ 𝑁) ∧ ((𝑋 ∧ 𝑌) ∈ 𝐴 ∧ 𝑋 ≠ 𝑍)) → 𝐾 ∈ Lat) |
9 | simp21 1114 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑍 ∈ 𝑁) ∧ ((𝑋 ∧ 𝑌) ∈ 𝐴 ∧ 𝑋 ≠ 𝑍)) → 𝑋 ∈ 𝑁) | |
10 | eqid 2651 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
11 | 10, 5 | llnbase 35113 | . . . . 5 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ (Base‘𝐾)) |
12 | 9, 11 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑍 ∈ 𝑁) ∧ ((𝑋 ∧ 𝑌) ∈ 𝐴 ∧ 𝑋 ≠ 𝑍)) → 𝑋 ∈ (Base‘𝐾)) |
13 | simp22 1115 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑍 ∈ 𝑁) ∧ ((𝑋 ∧ 𝑌) ∈ 𝐴 ∧ 𝑋 ≠ 𝑍)) → 𝑌 ∈ 𝑁) | |
14 | 10, 5 | llnbase 35113 | . . . . 5 ⊢ (𝑌 ∈ 𝑁 → 𝑌 ∈ (Base‘𝐾)) |
15 | 13, 14 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑍 ∈ 𝑁) ∧ ((𝑋 ∧ 𝑌) ∈ 𝐴 ∧ 𝑋 ≠ 𝑍)) → 𝑌 ∈ (Base‘𝐾)) |
16 | 10, 1, 3 | latmle2 17124 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑌) ≤ 𝑌) |
17 | 8, 12, 15, 16 | syl3anc 1366 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑍 ∈ 𝑁) ∧ ((𝑋 ∧ 𝑌) ∈ 𝐴 ∧ 𝑋 ≠ 𝑍)) → (𝑋 ∧ 𝑌) ≤ 𝑌) |
18 | breq1 4688 | . . 3 ⊢ ((𝑋 ∧ 𝑌) = (𝑋 ∧ 𝑍) → ((𝑋 ∧ 𝑌) ≤ 𝑌 ↔ (𝑋 ∧ 𝑍) ≤ 𝑌)) | |
19 | 17, 18 | syl5ibcom 235 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑍 ∈ 𝑁) ∧ ((𝑋 ∧ 𝑌) ∈ 𝐴 ∧ 𝑋 ≠ 𝑍)) → ((𝑋 ∧ 𝑌) = (𝑋 ∧ 𝑍) → (𝑋 ∧ 𝑍) ≤ 𝑌)) |
20 | 6, 19 | sylbid 230 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑍 ∈ 𝑁) ∧ ((𝑋 ∧ 𝑌) ∈ 𝐴 ∧ 𝑋 ≠ 𝑍)) → ((𝑋 ∧ 𝑌) ≤ 𝑍 → (𝑋 ∧ 𝑍) ≤ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 lecple 15995 joincjn 16991 meetcmee 16992 Latclat 17092 Atomscatm 34868 HLchlt 34955 LLinesclln 35095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-preset 16975 df-poset 16993 df-plt 17005 df-lub 17021 df-glb 17022 df-join 17023 df-meet 17024 df-p0 17086 df-lat 17093 df-clat 17155 df-oposet 34781 df-ol 34783 df-oml 34784 df-covers 34871 df-ats 34872 df-atl 34903 df-cvlat 34927 df-hlat 34956 df-llines 35102 df-psubsp 35107 df-pmap 35108 df-padd 35400 |
This theorem is referenced by: (None) |
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