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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrnlaut | Structured version Visualization version GIF version |
Description: A lattice translation is a lattice automorphism. (Contributed by NM, 20-May-2012.) |
Ref | Expression |
---|---|
ltrnlaut.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ltrnlaut.i | ⊢ 𝐼 = (LAut‘𝐾) |
ltrnlaut.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
ltrnlaut | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrnlaut.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | eqid 2610 | . . 3 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊) | |
3 | ltrnlaut.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | ltrnldil 34426 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊)) |
5 | ltrnlaut.i | . . 3 ⊢ 𝐼 = (LAut‘𝐾) | |
6 | 1, 5, 2 | ldillaut 34415 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ ((LDil‘𝐾)‘𝑊)) → 𝐹 ∈ 𝐼) |
7 | 4, 6 | syldan 486 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 LHypclh 34288 LAutclaut 34289 LDilcldil 34404 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-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-pr 4833 |
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-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-ov 6552 df-ldil 34408 df-ltrn 34409 |
This theorem is referenced by: ltrn1o 34428 ltrncl 34429 ltrn11 34430 ltrnle 34433 ltrncnvleN 34434 ltrnm 34435 ltrnj 34436 ltrncvr 34437 ltrnid 34439 ltrneq2 34452 |
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