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Mirrors > Home > MPE Home > Th. List > Mathboxes > trlval5 | Structured version Visualization version GIF version |
Description: The value of the trace of a lattice translation in terms of itself. (Contributed by NM, 19-Jul-2013.) |
Ref | Expression |
---|---|
trlval3.l | ⊢ ≤ = (le‘𝐾) |
trlval3.j | ⊢ ∨ = (join‘𝐾) |
trlval3.m | ⊢ ∧ = (meet‘𝐾) |
trlval3.a | ⊢ 𝐴 = (Atoms‘𝐾) |
trlval3.h | ⊢ 𝐻 = (LHyp‘𝐾) |
trlval3.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
trlval3.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
trlval5 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝑅‘𝐹)) ∧ 𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlval3.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
2 | trlval3.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
3 | trlval3.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
4 | trlval3.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | trlval3.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | trlval3.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
7 | trlval3.r | . . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | trlval2 37863 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) |
9 | 1, 2, 4, 5, 6, 7 | trljat1 37866 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃))) |
10 | 9 | oveq1d 7206 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑃 ∨ (𝑅‘𝐹)) ∧ 𝑊) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) |
11 | 8, 10 | eqtr4d 2774 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝑅‘𝐹)) ∧ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 lecple 16756 joincjn 17772 meetcmee 17773 Atomscatm 36963 HLchlt 37050 LHypclh 37684 LTrncltrn 37801 trLctrl 37858 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-map 8488 df-proset 17756 df-poset 17774 df-plt 17790 df-lub 17806 df-glb 17807 df-join 17808 df-meet 17809 df-p0 17885 df-p1 17886 df-lat 17892 df-clat 17959 df-oposet 36876 df-ol 36878 df-oml 36879 df-covers 36966 df-ats 36967 df-atl 36998 df-cvlat 37022 df-hlat 37051 df-psubsp 37203 df-pmap 37204 df-padd 37496 df-lhyp 37688 df-laut 37689 df-ldil 37804 df-ltrn 37805 df-trl 37859 |
This theorem is referenced by: cdlemk39 38616 |
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