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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 35743 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 1083 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ 𝐴) | |
2 | eqid 2651 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | ltrnel.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 2, 3 | atbase 34894 | . . 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 35741 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑃 ∈ 𝐴 ↔ (𝐹‘𝑃) ∈ 𝐴)) |
8 | 4, 7 | syl3an3 1401 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝑃 ∈ 𝐴 ↔ (𝐹‘𝑃) ∈ 𝐴)) |
9 | 1, 8 | mpbid 222 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 Basecbs 15904 lecple 15995 Atomscatm 34868 HLchlt 34955 LHypclh 35588 LTrncltrn 35705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-map 7901 df-plt 17005 df-glb 17022 df-p0 17086 df-oposet 34781 df-ol 34783 df-oml 34784 df-covers 34871 df-ats 34872 df-hlat 34956 df-lhyp 35592 df-laut 35593 df-ldil 35708 df-ltrn 35709 |
This theorem is referenced by: ltrncoat 35748 trlcnv 35770 trljat2 35772 trlat 35774 trlval3 35792 trlval4 35793 cdlemc3 35798 cdlemc5 35800 cdlemg2kq 36207 cdlemg9a 36237 cdlemg9 36239 cdlemg10bALTN 36241 cdlemg10c 36244 cdlemg10a 36245 cdlemg10 36246 cdlemg12a 36248 cdlemg12c 36250 cdlemg13a 36256 cdlemg17a 36266 cdlemg17g 36272 cdlemg18a 36283 cdlemg18b 36284 cdlemg18c 36285 trlcoabs2N 36327 trlcolem 36331 cdlemg42 36334 cdlemi 36425 cdlemk3 36438 cdlemk4 36439 cdlemk6 36442 cdlemk9 36444 cdlemk9bN 36445 cdlemk10 36448 cdlemksat 36451 cdlemk7 36453 cdlemk12 36455 cdlemkole 36458 cdlemk14 36459 cdlemk15 36460 cdlemk17 36463 cdlemk5u 36466 cdlemk6u 36467 cdlemkuat 36471 cdlemk7u 36475 cdlemk12u 36477 cdlemk37 36519 cdlemk39 36521 cdlemkfid1N 36526 cdlemk47 36554 cdlemk48 36555 cdlemk50 36557 cdlemk51 36558 cdlemk52 36559 cdlemm10N 36724 |
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