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Mathbox for Norm Megill |
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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 2824 | . . . 4 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊) | |
3 | ltrnval1.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | ltrnldil 36196 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊)) |
5 | 4 | 3adant3 1168 | . 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 36187 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋) |
9 | 5, 8 | syld3an2 1537 | 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 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 class class class wbr 4872 ‘cfv 6122 Basecbs 16221 lecple 16311 LHypclh 36058 LDilcldil 36174 LTrncltrn 36175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pr 5126 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rex 3122 df-reu 3123 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-ov 6907 df-ldil 36178 df-ltrn 36179 |
This theorem is referenced by: ltrnid 36209 ltrnatb 36211 ltrnel 36213 ltrncnvel 36216 ltrneq 36223 cdlemc2 36266 cdlemd2 36273 cdlemg7N 36700 |
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