![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > pmapjat2 | Structured version Visualization version GIF version |
Description: The projective map of the join of an atom with a lattice element. (Contributed by NM, 12-May-2012.) |
Ref | Expression |
---|---|
pmapjat.b | ⊢ 𝐵 = (Base‘𝐾) |
pmapjat.j | ⊢ ∨ = (join‘𝐾) |
pmapjat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
pmapjat.m | ⊢ 𝑀 = (pmap‘𝐾) |
pmapjat.p | ⊢ + = (+𝑃‘𝐾) |
Ref | Expression |
---|---|
pmapjat2 | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘(𝑄 ∨ 𝑋)) = ((𝑀‘𝑄) + (𝑀‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmapjat.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | pmapjat.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
3 | pmapjat.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | pmapjat.m | . . 3 ⊢ 𝑀 = (pmap‘𝐾) | |
5 | pmapjat.p | . . 3 ⊢ + = (+𝑃‘𝐾) | |
6 | 1, 2, 3, 4, 5 | pmapjat1 35923 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘(𝑋 ∨ 𝑄)) = ((𝑀‘𝑋) + (𝑀‘𝑄))) |
7 | hllat 35433 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
8 | 7 | 3ad2ant1 1167 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ Lat) |
9 | 1, 3 | atbase 35359 | . . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵) |
10 | 9 | 3ad2ant3 1169 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → 𝑄 ∈ 𝐵) |
11 | simp2 1171 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
12 | 1, 2 | latjcom 17419 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄 ∨ 𝑋) = (𝑋 ∨ 𝑄)) |
13 | 8, 10, 11, 12 | syl3anc 1494 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑋) = (𝑋 ∨ 𝑄)) |
14 | 13 | fveq2d 6441 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘(𝑄 ∨ 𝑋)) = (𝑀‘(𝑋 ∨ 𝑄))) |
15 | simp1 1170 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ HL) | |
16 | 1, 3, 4 | pmapssat 35829 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐵) → (𝑀‘𝑄) ⊆ 𝐴) |
17 | 15, 10, 16 | syl2anc 579 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘𝑄) ⊆ 𝐴) |
18 | 1, 3, 4 | pmapssat 35829 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ 𝐴) |
19 | 18 | 3adant3 1166 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘𝑋) ⊆ 𝐴) |
20 | 3, 5 | paddcom 35883 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑀‘𝑄) ⊆ 𝐴 ∧ (𝑀‘𝑋) ⊆ 𝐴) → ((𝑀‘𝑄) + (𝑀‘𝑋)) = ((𝑀‘𝑋) + (𝑀‘𝑄))) |
21 | 8, 17, 19, 20 | syl3anc 1494 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → ((𝑀‘𝑄) + (𝑀‘𝑋)) = ((𝑀‘𝑋) + (𝑀‘𝑄))) |
22 | 6, 14, 21 | 3eqtr4d 2871 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘(𝑄 ∨ 𝑋)) = ((𝑀‘𝑄) + (𝑀‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 joincjn 17304 Latclat 17405 Atomscatm 35333 HLchlt 35420 pmapcpmap 35567 +𝑃cpadd 35865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-proset 17288 df-poset 17306 df-plt 17318 df-lub 17334 df-glb 17335 df-join 17336 df-meet 17337 df-p0 17399 df-lat 17406 df-clat 17468 df-oposet 35246 df-ol 35248 df-oml 35249 df-covers 35336 df-ats 35337 df-atl 35368 df-cvlat 35392 df-hlat 35421 df-pmap 35574 df-padd 35866 |
This theorem is referenced by: atmod1i1 35927 |
Copyright terms: Public domain | W3C validator |