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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 37308, ltrnat 37309 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 1131 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ∈ 𝐴) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | simp2l 1194 | . 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 37309 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐺‘𝑃) ∈ 𝐴) |
8 | 7 | 3adant2l 1173 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ∈ 𝐴) → (𝐺‘𝑃) ∈ 𝐴) |
9 | 3, 4, 5, 6 | ltrnat 37309 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝐺‘𝑃) ∈ 𝐴) → (𝐹‘(𝐺‘𝑃)) ∈ 𝐴) |
10 | 1, 2, 8, 9 | syl3anc 1366 | 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 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 lecple 16567 Atomscatm 36432 HLchlt 36519 LHypclh 37153 LTrncltrn 37270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-map 8401 df-plt 17563 df-glb 17580 df-p0 17644 df-oposet 36345 df-ol 36347 df-oml 36348 df-covers 36435 df-ats 36436 df-hlat 36520 df-lhyp 37157 df-laut 37158 df-ldil 37273 df-ltrn 37274 |
This theorem is referenced by: cdlemg9a 37801 cdlemg9 37803 cdlemg11aq 37807 cdlemg12a 37812 cdlemg12c 37814 cdlemg12f 37817 cdlemg12g 37818 cdlemg12 37819 cdlemg13a 37820 cdlemg13 37821 cdlemg17f 37835 cdlemg17g 37836 cdlemg17 37846 cdlemg19a 37852 cdlemg19 37853 |
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