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Mathbox for Norm Megill |
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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 4813 | . . 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 39888 | . . 3 ⊢ ((𝐾 ∈ OL ∧ {𝑄} ⊆ 𝐴) → (𝑃‘{𝑄}) = (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝)))) |
7 | 1, 6 | sylan2 593 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝)))) |
8 | 2fveq3 6912 | . . . . 5 ⊢ (𝑝 = 𝑄 → (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄))) | |
9 | 8 | iinxsng 5093 | . . . 4 ⊢ (𝑄 ∈ 𝐴 → ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄))) |
10 | 9 | adantl 481 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄))) |
11 | 10 | ineq2d 4228 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝))) = (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄)))) |
12 | olop 39196 | . . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
13 | eqid 2735 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
14 | 13, 3 | atbase 39271 | . . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
15 | 13, 2 | opoccl 39176 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑄 ∈ (Base‘𝐾)) → ( ⊥ ‘𝑄) ∈ (Base‘𝐾)) |
16 | 12, 14, 15 | syl2an 596 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → ( ⊥ ‘𝑄) ∈ (Base‘𝐾)) |
17 | 13, 3, 4 | pmapssat 39742 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ ( ⊥ ‘𝑄) ∈ (Base‘𝐾)) → (𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴) |
18 | 16, 17 | syldan 591 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴) |
19 | sseqin2 4231 | . . 3 ⊢ ((𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄))) = (𝑀‘( ⊥ ‘𝑄))) | |
20 | 18, 19 | sylib 218 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄))) = (𝑀‘( ⊥ ‘𝑄))) |
21 | 7, 11, 20 | 3eqtrd 2779 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = (𝑀‘( ⊥ ‘𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ⊆ wss 3963 {csn 4631 ∩ ciin 4997 ‘cfv 6563 Basecbs 17245 occoc 17306 OPcops 39154 OLcol 39156 Atomscatm 39245 pmapcpmap 39480 ⊥𝑃cpolN 39885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oposet 39158 df-ol 39160 df-ats 39249 df-pmap 39487 df-polarityN 39886 |
This theorem is referenced by: 2polatN 39915 |
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