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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrncoat | Structured version Visualization version GIF version |
Description: Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 37435, ltrnat 37436 uses. (Contributed by NM, 1-May-2013.) |
Ref | Expression |
---|---|
ltrnel.l | ⊢ ≤ = (le‘𝐾) |
ltrnel.a | ⊢ 𝐴 = (Atoms‘𝐾) |
ltrnel.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ltrnel.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
ltrncoat | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ∈ 𝐴) → (𝐹‘(𝐺‘𝑃)) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ∈ 𝐴) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | simp2l 1196 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ∈ 𝐴) → 𝐹 ∈ 𝑇) | |
3 | ltrnel.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
4 | ltrnel.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | ltrnel.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | ltrnel.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
7 | 3, 4, 5, 6 | ltrnat 37436 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐺‘𝑃) ∈ 𝐴) |
8 | 7 | 3adant2l 1175 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ∈ 𝐴) → (𝐺‘𝑃) ∈ 𝐴) |
9 | 3, 4, 5, 6 | ltrnat 37436 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝐺‘𝑃) ∈ 𝐴) → (𝐹‘(𝐺‘𝑃)) ∈ 𝐴) |
10 | 1, 2, 8, 9 | syl3anc 1368 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ∈ 𝐴) → (𝐹‘(𝐺‘𝑃)) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 lecple 16564 Atomscatm 36559 HLchlt 36646 LHypclh 37280 LTrncltrn 37397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-plt 17560 df-glb 17577 df-p0 17641 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-hlat 36647 df-lhyp 37284 df-laut 37285 df-ldil 37400 df-ltrn 37401 |
This theorem is referenced by: cdlemg9a 37928 cdlemg9 37930 cdlemg11aq 37934 cdlemg12a 37939 cdlemg12c 37941 cdlemg12f 37944 cdlemg12g 37945 cdlemg12 37946 cdlemg13a 37947 cdlemg13 37948 cdlemg17f 37962 cdlemg17g 37963 cdlemg17 37973 cdlemg19a 37979 cdlemg19 37980 |
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