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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 1196 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐾 ∈ HL) | |
2 | 1 | hllatd 37374 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐾 ∈ Lat) |
3 | ltrnm.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | eqid 2740 | . . . 4 ⊢ (LAut‘𝐾) = (LAut‘𝐾) | |
5 | ltrnm.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
6 | 3, 4, 5 | ltrnlaut 38133 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (LAut‘𝐾)) |
7 | 6 | 3adant3 1131 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐹 ∈ (LAut‘𝐾)) |
8 | simp3l 1200 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
9 | simp3r 1201 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
10 | ltrnm.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
11 | ltrnm.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
12 | 10, 11, 4 | lautm 38104 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝐹 ∈ (LAut‘𝐾) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘(𝑋 ∧ 𝑌)) = ((𝐹‘𝑋) ∧ (𝐹‘𝑌))) |
13 | 2, 7, 8, 9, 12 | syl13anc 1371 | 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 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ‘cfv 6432 (class class class)co 7271 Basecbs 16910 meetcmee 18028 Latclat 18147 HLchlt 37360 LHypclh 37994 LAutclaut 37995 LTrncltrn 38111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-map 8600 df-proset 18011 df-poset 18029 df-lub 18062 df-glb 18063 df-join 18064 df-meet 18065 df-lat 18148 df-atl 37308 df-cvlat 37332 df-hlat 37361 df-laut 37999 df-ldil 38114 df-ltrn 38115 |
This theorem is referenced by: cdlemd2 38209 cdlemg17 38687 |
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