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Mirrors > Home > MPE Home > Th. List > Mathboxes > linepmap | Structured version Visualization version GIF version |
Description: A line described with a projective map. (Contributed by NM, 3-Feb-2012.) |
Ref | Expression |
---|---|
isline2.j | ⊢ ∨ = (join‘𝐾) |
isline2.a | ⊢ 𝐴 = (Atoms‘𝐾) |
isline2.n | ⊢ 𝑁 = (Lines‘𝐾) |
isline2.m | ⊢ 𝑀 = (pmap‘𝐾) |
Ref | Expression |
---|---|
linepmap | ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑀‘(𝑃 ∨ 𝑄)) ∈ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1227 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐾 ∈ Lat) | |
2 | simpl2 1229 | . . . . 5 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ 𝐴) | |
3 | eqid 2771 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | isline2.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 3, 4 | atbase 35097 | . . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
6 | 2, 5 | syl 17 | . . . 4 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ (Base‘𝐾)) |
7 | simpl3 1231 | . . . . 5 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ 𝐴) | |
8 | 3, 4 | atbase 35097 | . . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ (Base‘𝐾)) |
10 | isline2.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
11 | 3, 10 | latjcl 17258 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
12 | 1, 6, 9, 11 | syl3anc 1476 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
13 | eqid 2771 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
14 | isline2.m | . . . 4 ⊢ 𝑀 = (pmap‘𝐾) | |
15 | 3, 13, 4, 14 | pmapval 35565 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (𝑀‘(𝑃 ∨ 𝑄)) = {𝑟 ∈ 𝐴 ∣ 𝑟(le‘𝐾)(𝑃 ∨ 𝑄)}) |
16 | 1, 12, 15 | syl2anc 573 | . 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑀‘(𝑃 ∨ 𝑄)) = {𝑟 ∈ 𝐴 ∣ 𝑟(le‘𝐾)(𝑃 ∨ 𝑄)}) |
17 | eqid 2771 | . . 3 ⊢ {𝑟 ∈ 𝐴 ∣ 𝑟(le‘𝐾)(𝑃 ∨ 𝑄)} = {𝑟 ∈ 𝐴 ∣ 𝑟(le‘𝐾)(𝑃 ∨ 𝑄)} | |
18 | isline2.n | . . . 4 ⊢ 𝑁 = (Lines‘𝐾) | |
19 | 13, 10, 4, 18 | islinei 35548 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ {𝑟 ∈ 𝐴 ∣ 𝑟(le‘𝐾)(𝑃 ∨ 𝑄)} = {𝑟 ∈ 𝐴 ∣ 𝑟(le‘𝐾)(𝑃 ∨ 𝑄)})) → {𝑟 ∈ 𝐴 ∣ 𝑟(le‘𝐾)(𝑃 ∨ 𝑄)} ∈ 𝑁) |
20 | 17, 19 | mpanr2 684 | . 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → {𝑟 ∈ 𝐴 ∣ 𝑟(le‘𝐾)(𝑃 ∨ 𝑄)} ∈ 𝑁) |
21 | 16, 20 | eqeltrd 2850 | 1 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑀‘(𝑃 ∨ 𝑄)) ∈ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 {crab 3065 class class class wbr 4787 ‘cfv 6030 (class class class)co 6795 Basecbs 16063 lecple 16155 joincjn 17151 Latclat 17252 Atomscatm 35071 Linesclines 35302 pmapcpmap 35305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7099 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6756 df-ov 6798 df-oprab 6799 df-lub 17181 df-glb 17182 df-join 17183 df-meet 17184 df-lat 17253 df-ats 35075 df-lines 35309 df-pmap 35312 |
This theorem is referenced by: cdleme3h 36044 cdleme7ga 36057 |
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