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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrncvr | Structured version Visualization version GIF version |
Description: Covering property of a lattice translation. (Contributed by NM, 20-May-2012.) |
Ref | Expression |
---|---|
ltrncvr.b | ⊢ 𝐵 = (Base‘𝐾) |
ltrncvr.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
ltrncvr.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ltrncvr.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
ltrncvr | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐶𝑌 ↔ (𝐹‘𝑋)𝐶(𝐹‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1l 1195 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐾 ∈ 𝑉) | |
2 | ltrncvr.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | eqid 2728 | . . . 4 ⊢ (LAut‘𝐾) = (LAut‘𝐾) | |
4 | ltrncvr.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | 2, 3, 4 | ltrnlaut 39596 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (LAut‘𝐾)) |
6 | 5 | 3adant3 1130 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐹 ∈ (LAut‘𝐾)) |
7 | simp3l 1199 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
8 | simp3r 1200 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
9 | ltrncvr.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
10 | ltrncvr.c | . . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
11 | 9, 10, 3 | lautcvr 39565 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ (𝐹 ∈ (LAut‘𝐾) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐶𝑌 ↔ (𝐹‘𝑋)𝐶(𝐹‘𝑌))) |
12 | 1, 6, 7, 8, 11 | syl13anc 1370 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐶𝑌 ↔ (𝐹‘𝑋)𝐶(𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 ‘cfv 6548 Basecbs 17180 ⋖ ccvr 38734 LHypclh 39457 LAutclaut 39458 LTrncltrn 39574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8847 df-plt 18322 df-covers 38738 df-laut 39462 df-ldil 39577 df-ltrn 39578 |
This theorem is referenced by: ltrnatb 39610 |
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