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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 37181 | . 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 37184 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃))) |
10 | 9 | oveq1d 7160 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑃 ∨ (𝑅‘𝐹)) ∧ 𝑊) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) |
11 | 8, 10 | eqtr4d 2859 | 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 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 class class class wbr 5058 ‘cfv 6349 (class class class)co 7145 lecple 16562 joincjn 17544 meetcmee 17545 Atomscatm 36281 HLchlt 36368 LHypclh 37002 LTrncltrn 37119 trLctrl 37176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 df-iin 4915 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7680 df-2nd 7681 df-map 8398 df-proset 17528 df-poset 17546 df-plt 17558 df-lub 17574 df-glb 17575 df-join 17576 df-meet 17577 df-p0 17639 df-p1 17640 df-lat 17646 df-clat 17708 df-oposet 36194 df-ol 36196 df-oml 36197 df-covers 36284 df-ats 36285 df-atl 36316 df-cvlat 36340 df-hlat 36369 df-psubsp 36521 df-pmap 36522 df-padd 36814 df-lhyp 37006 df-laut 37007 df-ldil 37122 df-ltrn 37123 df-trl 37177 |
This theorem is referenced by: cdlemk39 37934 |
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