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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrnm | Structured version Visualization version GIF version |
Description: Lattice translation of a meet. (Contributed by NM, 20-May-2012.) |
Ref | Expression |
---|---|
ltrnm.b | ⊢ 𝐵 = (Base‘𝐾) |
ltrnm.m | ⊢ ∧ = (meet‘𝐾) |
ltrnm.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ltrnm.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
ltrnm | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘(𝑋 ∧ 𝑌)) = ((𝐹‘𝑋) ∧ (𝐹‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1l 1240 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐾 ∈ HL) | |
2 | hllat 35151 | . . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐾 ∈ Lat) |
4 | ltrnm.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | eqid 2758 | . . . 4 ⊢ (LAut‘𝐾) = (LAut‘𝐾) | |
6 | ltrnm.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
7 | 4, 5, 6 | ltrnlaut 35910 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (LAut‘𝐾)) |
8 | 7 | 3adant3 1127 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐹 ∈ (LAut‘𝐾)) |
9 | simp3l 1244 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
10 | simp3r 1245 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
11 | ltrnm.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
12 | ltrnm.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
13 | 11, 12, 5 | lautm 35881 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝐹 ∈ (LAut‘𝐾) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘(𝑋 ∧ 𝑌)) = ((𝐹‘𝑋) ∧ (𝐹‘𝑌))) |
14 | 3, 8, 9, 10, 13 | syl13anc 1479 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘(𝑋 ∧ 𝑌)) = ((𝐹‘𝑋) ∧ (𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1630 ∈ wcel 2137 ‘cfv 6047 (class class class)co 6811 Basecbs 16057 meetcmee 17144 Latclat 17244 HLchlt 35138 LHypclh 35771 LAutclaut 35772 LTrncltrn 35888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-8 2139 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-rep 4921 ax-sep 4931 ax-nul 4939 ax-pow 4990 ax-pr 5053 ax-un 7112 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-mo 2610 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-ne 2931 df-ral 3053 df-rex 3054 df-reu 3055 df-rab 3057 df-v 3340 df-sbc 3575 df-csb 3673 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-nul 4057 df-if 4229 df-pw 4302 df-sn 4320 df-pr 4322 df-op 4326 df-uni 4587 df-iun 4672 df-br 4803 df-opab 4863 df-mpt 4880 df-id 5172 df-xp 5270 df-rel 5271 df-cnv 5272 df-co 5273 df-dm 5274 df-rn 5275 df-res 5276 df-ima 5277 df-iota 6010 df-fun 6049 df-fn 6050 df-f 6051 df-f1 6052 df-fo 6053 df-f1o 6054 df-fv 6055 df-riota 6772 df-ov 6814 df-oprab 6815 df-mpt2 6816 df-map 8023 df-preset 17127 df-poset 17145 df-lub 17173 df-glb 17174 df-join 17175 df-meet 17176 df-lat 17245 df-atl 35086 df-cvlat 35110 df-hlat 35139 df-laut 35776 df-ldil 35891 df-ltrn 35892 |
This theorem is referenced by: ltrnmwOLD 35939 cdlemd2 35987 cdlemg17 36465 |
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