![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 38975 | . 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 38976 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃))) |
9 | 7, 8 | eqtr4d 2776 | 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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 class class class wbr 5147 ‘cfv 6540 (class class class)co 7404 lecple 17200 joincjn 18260 Atomscatm 38071 HLchlt 38158 LHypclh 38793 LTrncltrn 38910 trLctrl 38967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7970 df-2nd 7971 df-map 8818 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-oposet 37984 df-ol 37986 df-oml 37987 df-covers 38074 df-ats 38075 df-atl 38106 df-cvlat 38130 df-hlat 38159 df-psubsp 38312 df-pmap 38313 df-padd 38605 df-lhyp 38797 df-laut 38798 df-ldil 38913 df-ltrn 38914 df-trl 38968 |
This theorem is referenced by: trlcoabs 39530 cdlemk1 39640 cdlemk2 39641 cdlemk1u 39668 cdlemkfid1N 39730 cdlemkid1 39731 |
Copyright terms: Public domain | W3C validator |