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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > trljat3 | Structured version Visualization version GIF version |
Description: The value of a translation of an atom 𝑃 not under the fiducial co-atom 𝑊, joined with trace. Equation above Lemma C in [Crawley] p. 112. (Contributed by NM, 22-May-2012.) |
Ref | Expression |
---|---|
trljat.l | ⊢ ≤ = (le‘𝐾) |
trljat.j | ⊢ ∨ = (join‘𝐾) |
trljat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
trljat.h | ⊢ 𝐻 = (LHyp‘𝐾) |
trljat.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
trljat.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
trljat3 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝐹)) = ((𝐹‘𝑃) ∨ (𝑅‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trljat.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
2 | trljat.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
3 | trljat.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | trljat.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | trljat.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
6 | trljat.r | . . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
7 | 1, 2, 3, 4, 5, 6 | trljat1 40148 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃))) |
8 | 1, 2, 3, 4, 5, 6 | trljat2 40149 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃))) |
9 | 7, 8 | eqtr4d 2777 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝐹)) = ((𝐹‘𝑃) ∨ (𝑅‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 lecple 17304 joincjn 18368 Atomscatm 39244 HLchlt 39331 LHypclh 39966 LTrncltrn 40083 trLctrl 40140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-map 8866 df-proset 18351 df-poset 18370 df-plt 18387 df-lub 18403 df-glb 18404 df-join 18405 df-meet 18406 df-p0 18482 df-p1 18483 df-lat 18489 df-clat 18556 df-oposet 39157 df-ol 39159 df-oml 39160 df-covers 39247 df-ats 39248 df-atl 39279 df-cvlat 39303 df-hlat 39332 df-psubsp 39485 df-pmap 39486 df-padd 39778 df-lhyp 39970 df-laut 39971 df-ldil 40086 df-ltrn 40087 df-trl 40141 |
This theorem is referenced by: trlcoabs 40703 cdlemk1 40813 cdlemk2 40814 cdlemk1u 40841 cdlemkfid1N 40903 cdlemkid1 40904 |
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