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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrnat | Structured version Visualization version GIF version |
Description: The lattice translation of an atom is also an atom. TODO: See if this can shorten some ltrnel 38948 uses. (Contributed by NM, 25-May-2012.) |
Ref | Expression |
---|---|
ltrnel.l | ⊢ ≤ = (le‘𝐾) |
ltrnel.a | ⊢ 𝐴 = (Atoms‘𝐾) |
ltrnel.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ltrnel.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
ltrnat | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1139 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ 𝐴) | |
2 | eqid 2733 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | ltrnel.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 2, 3 | atbase 38097 | . . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
5 | ltrnel.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | ltrnel.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
7 | 2, 3, 5, 6 | ltrnatb 38946 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑃 ∈ 𝐴 ↔ (𝐹‘𝑃) ∈ 𝐴)) |
8 | 4, 7 | syl3an3 1166 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝑃 ∈ 𝐴 ↔ (𝐹‘𝑃) ∈ 𝐴)) |
9 | 1, 8 | mpbid 231 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ‘cfv 6540 Basecbs 17140 lecple 17200 Atomscatm 38071 HLchlt 38158 LHypclh 38793 LTrncltrn 38910 |
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-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-map 8818 df-plt 18279 df-glb 18296 df-p0 18374 df-oposet 37984 df-ol 37986 df-oml 37987 df-covers 38074 df-ats 38075 df-hlat 38159 df-lhyp 38797 df-laut 38798 df-ldil 38913 df-ltrn 38914 |
This theorem is referenced by: ltrncoat 38953 trlcnv 38974 trljat2 38976 trlat 38978 trlval3 38996 trlval4 38997 cdlemc3 39002 cdlemc5 39004 cdlemg2kq 39411 cdlemg9a 39441 cdlemg9 39443 cdlemg10bALTN 39445 cdlemg10c 39448 cdlemg10a 39449 cdlemg10 39450 cdlemg12a 39452 cdlemg12c 39454 cdlemg13a 39460 cdlemg17a 39470 cdlemg17g 39476 cdlemg18a 39487 cdlemg18b 39488 cdlemg18c 39489 trlcoabs2N 39531 trlcolem 39535 cdlemg42 39538 cdlemi 39629 cdlemk3 39642 cdlemk4 39643 cdlemk6 39646 cdlemk9 39648 cdlemk9bN 39649 cdlemk10 39652 cdlemksat 39655 cdlemk7 39657 cdlemk12 39659 cdlemkole 39662 cdlemk14 39663 cdlemk15 39664 cdlemk17 39667 cdlemk5u 39670 cdlemk6u 39671 cdlemkuat 39675 cdlemk7u 39679 cdlemk12u 39681 cdlemk37 39723 cdlemk39 39725 cdlemkfid1N 39730 cdlemk47 39758 cdlemk48 39759 cdlemk50 39761 cdlemk51 39762 cdlemk52 39763 cdlemm10N 39927 |
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