![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrnval1 | Structured version Visualization version GIF version |
Description: Value of a lattice translation under its co-atom. (Contributed by NM, 20-May-2012.) |
Ref | Expression |
---|---|
ltrnval1.b | ⊢ 𝐵 = (Base‘𝐾) |
ltrnval1.l | ⊢ ≤ = (le‘𝐾) |
ltrnval1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ltrnval1.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
ltrnval1 | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrnval1.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | eqid 2726 | . . . 4 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊) | |
3 | ltrnval1.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | ltrnldil 39821 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊)) |
5 | 4 | 3adant3 1129 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊)) |
6 | ltrnval1.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
7 | ltrnval1.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
8 | 6, 7, 1, 2 | ldilval 39812 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋) |
9 | 5, 8 | syld3an2 1408 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 class class class wbr 5153 ‘cfv 6554 Basecbs 17213 lecple 17273 LHypclh 39683 LDilcldil 39799 LTrncltrn 39800 |
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 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-ldil 39803 df-ltrn 39804 |
This theorem is referenced by: ltrnid 39834 ltrnatb 39836 ltrnel 39838 ltrncnvel 39841 ltrneq 39848 cdlemc2 39891 cdlemd2 39898 cdlemg7N 40325 |
Copyright terms: Public domain | W3C validator |