![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > polatN | Structured version Visualization version GIF version |
Description: The polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
polat.o | ⊢ ⊥ = (oc‘𝐾) |
polat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
polat.m | ⊢ 𝑀 = (pmap‘𝐾) |
polat.p | ⊢ 𝑃 = (⊥𝑃‘𝐾) |
Ref | Expression |
---|---|
polatN | ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = (𝑀‘( ⊥ ‘𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 4701 | . . 3 ⊢ (𝑄 ∈ 𝐴 → {𝑄} ⊆ 𝐴) | |
2 | polat.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
3 | polat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | polat.m | . . . 4 ⊢ 𝑀 = (pmap‘𝐾) | |
5 | polat.p | . . . 4 ⊢ 𝑃 = (⊥𝑃‘𝐾) | |
6 | 2, 3, 4, 5 | polvalN 37201 | . . 3 ⊢ ((𝐾 ∈ OL ∧ {𝑄} ⊆ 𝐴) → (𝑃‘{𝑄}) = (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝)))) |
7 | 1, 6 | sylan2 595 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝)))) |
8 | 2fveq3 6650 | . . . . 5 ⊢ (𝑝 = 𝑄 → (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄))) | |
9 | 8 | iinxsng 4973 | . . . 4 ⊢ (𝑄 ∈ 𝐴 → ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄))) |
10 | 9 | adantl 485 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄))) |
11 | 10 | ineq2d 4139 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝))) = (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄)))) |
12 | olop 36510 | . . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
13 | eqid 2798 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
14 | 13, 3 | atbase 36585 | . . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
15 | 13, 2 | opoccl 36490 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑄 ∈ (Base‘𝐾)) → ( ⊥ ‘𝑄) ∈ (Base‘𝐾)) |
16 | 12, 14, 15 | syl2an 598 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → ( ⊥ ‘𝑄) ∈ (Base‘𝐾)) |
17 | 13, 3, 4 | pmapssat 37055 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ ( ⊥ ‘𝑄) ∈ (Base‘𝐾)) → (𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴) |
18 | 16, 17 | syldan 594 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴) |
19 | sseqin2 4142 | . . 3 ⊢ ((𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄))) = (𝑀‘( ⊥ ‘𝑄))) | |
20 | 18, 19 | sylib 221 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄))) = (𝑀‘( ⊥ ‘𝑄))) |
21 | 7, 11, 20 | 3eqtrd 2837 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = (𝑀‘( ⊥ ‘𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ⊆ wss 3881 {csn 4525 ∩ ciin 4882 ‘cfv 6324 Basecbs 16475 occoc 16565 OPcops 36468 OLcol 36470 Atomscatm 36559 pmapcpmap 36793 ⊥𝑃cpolN 37198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oposet 36472 df-ol 36474 df-ats 36563 df-pmap 36800 df-polarityN 37199 |
This theorem is referenced by: 2polatN 37228 |
Copyright terms: Public domain | W3C validator |