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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihmeetlem5 | Structured version Visualization version GIF version |
Description: Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014.) |
Ref | Expression |
---|---|
dihmeetlem5.b | ⊢ 𝐵 = (Base‘𝐾) |
dihmeetlem5.l | ⊢ ≤ = (le‘𝐾) |
dihmeetlem5.j | ⊢ ∨ = (join‘𝐾) |
dihmeetlem5.m | ⊢ ∧ = (meet‘𝐾) |
dihmeetlem5.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
dihmeetlem5 | ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → (𝑋 ∧ (𝑌 ∨ 𝑄)) = ((𝑋 ∧ 𝑌) ∨ 𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1193 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → 𝐾 ∈ HL) | |
2 | simprl 771 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → 𝑄 ∈ 𝐴) | |
3 | simpl2 1194 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → 𝑋 ∈ 𝐵) | |
4 | simpl3 1195 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → 𝑌 ∈ 𝐵) | |
5 | simprr 773 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → 𝑄 ≤ 𝑋) | |
6 | dihmeetlem5.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
7 | dihmeetlem5.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
8 | dihmeetlem5.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
9 | dihmeetlem5.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
10 | dihmeetlem5.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
11 | 6, 7, 8, 9, 10 | atmod2i1 37569 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑄 ≤ 𝑋) → ((𝑋 ∧ 𝑌) ∨ 𝑄) = (𝑋 ∧ (𝑌 ∨ 𝑄))) |
12 | 1, 2, 3, 4, 5, 11 | syl131anc 1385 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → ((𝑋 ∧ 𝑌) ∨ 𝑄) = (𝑋 ∧ (𝑌 ∨ 𝑄))) |
13 | 12 | eqcomd 2740 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → (𝑋 ∧ (𝑌 ∨ 𝑄)) = ((𝑋 ∧ 𝑌) ∨ 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 class class class wbr 5043 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 lecple 16774 joincjn 17790 meetcmee 17791 Atomscatm 36971 HLchlt 37058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-1st 7750 df-2nd 7751 df-proset 17774 df-poset 17792 df-plt 17808 df-lub 17824 df-glb 17825 df-join 17826 df-meet 17827 df-p0 17903 df-lat 17910 df-clat 17977 df-oposet 36884 df-ol 36886 df-oml 36887 df-covers 36974 df-ats 36975 df-atl 37006 df-cvlat 37030 df-hlat 37059 df-psubsp 37211 df-pmap 37212 df-padd 37504 |
This theorem is referenced by: dihmeetlem6 39017 dihjatc1 39019 dihmeetlem10N 39024 |
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