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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 34443 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 1056 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ 𝐴) | |
2 | eqid 2610 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | ltrnel.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 2, 3 | atbase 33594 | . . 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 34441 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑃 ∈ 𝐴 ↔ (𝐹‘𝑃) ∈ 𝐴)) |
8 | 4, 7 | syl3an3 1353 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝑃 ∈ 𝐴 ↔ (𝐹‘𝑃) ∈ 𝐴)) |
9 | 1, 8 | mpbid 221 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 Basecbs 15695 lecple 15775 Atomscatm 33568 HLchlt 33655 LHypclh 34288 LTrncltrn 34405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-plt 16781 df-glb 16798 df-p0 16862 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-hlat 33656 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 |
This theorem is referenced by: ltrncoat 34448 trlcnv 34470 trljat2 34472 trlat 34474 trlval3 34492 trlval4 34493 cdlemc3 34498 cdlemc5 34500 cdlemg2kq 34908 cdlemg9a 34938 cdlemg9 34940 cdlemg10bALTN 34942 cdlemg10c 34945 cdlemg10a 34946 cdlemg10 34947 cdlemg12a 34949 cdlemg12c 34951 cdlemg13a 34957 cdlemg17a 34967 cdlemg17g 34973 cdlemg18a 34984 cdlemg18b 34985 cdlemg18c 34986 trlcoabs2N 35028 trlcolem 35032 cdlemg42 35035 cdlemi 35126 cdlemk3 35139 cdlemk4 35140 cdlemk6 35143 cdlemk9 35145 cdlemk9bN 35146 cdlemk10 35149 cdlemksat 35152 cdlemk7 35154 cdlemk12 35156 cdlemkole 35159 cdlemk14 35160 cdlemk15 35161 cdlemk17 35164 cdlemk5u 35167 cdlemk6u 35168 cdlemkuat 35172 cdlemk7u 35176 cdlemk12u 35178 cdlemk37 35220 cdlemk39 35222 cdlemkfid1N 35227 cdlemk47 35255 cdlemk48 35256 cdlemk50 35258 cdlemk51 35259 cdlemk52 35260 cdlemm10N 35425 |
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