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Theorem polatN 37987
Description: The polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
polat.o βŠ₯ = (ocβ€˜πΎ)
polat.a 𝐴 = (Atomsβ€˜πΎ)
polat.m 𝑀 = (pmapβ€˜πΎ)
polat.p 𝑃 = (βŠ₯π‘ƒβ€˜πΎ)
Assertion
Ref Expression
polatN ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) β†’ (π‘ƒβ€˜{𝑄}) = (π‘€β€˜( βŠ₯ β€˜π‘„)))

Proof of Theorem polatN
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 snssi 4747 . . 3 (𝑄 ∈ 𝐴 β†’ {𝑄} βŠ† 𝐴)
2 polat.o . . . 4 βŠ₯ = (ocβ€˜πΎ)
3 polat.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
4 polat.m . . . 4 𝑀 = (pmapβ€˜πΎ)
5 polat.p . . . 4 𝑃 = (βŠ₯π‘ƒβ€˜πΎ)
62, 3, 4, 5polvalN 37961 . . 3 ((𝐾 ∈ OL ∧ {𝑄} βŠ† 𝐴) β†’ (π‘ƒβ€˜{𝑄}) = (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (π‘€β€˜( βŠ₯ β€˜π‘))))
71, 6sylan2 594 . 2 ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) β†’ (π‘ƒβ€˜{𝑄}) = (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (π‘€β€˜( βŠ₯ β€˜π‘))))
8 2fveq3 6809 . . . . 5 (𝑝 = 𝑄 β†’ (π‘€β€˜( βŠ₯ β€˜π‘)) = (π‘€β€˜( βŠ₯ β€˜π‘„)))
98iinxsng 5024 . . . 4 (𝑄 ∈ 𝐴 β†’ ∩ 𝑝 ∈ {𝑄} (π‘€β€˜( βŠ₯ β€˜π‘)) = (π‘€β€˜( βŠ₯ β€˜π‘„)))
109adantl 483 . . 3 ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) β†’ ∩ 𝑝 ∈ {𝑄} (π‘€β€˜( βŠ₯ β€˜π‘)) = (π‘€β€˜( βŠ₯ β€˜π‘„)))
1110ineq2d 4152 . 2 ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) β†’ (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (π‘€β€˜( βŠ₯ β€˜π‘))) = (𝐴 ∩ (π‘€β€˜( βŠ₯ β€˜π‘„))))
12 olop 37270 . . . . 5 (𝐾 ∈ OL β†’ 𝐾 ∈ OP)
13 eqid 2736 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
1413, 3atbase 37345 . . . . 5 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
1513, 2opoccl 37250 . . . . 5 ((𝐾 ∈ OP ∧ 𝑄 ∈ (Baseβ€˜πΎ)) β†’ ( βŠ₯ β€˜π‘„) ∈ (Baseβ€˜πΎ))
1612, 14, 15syl2an 597 . . . 4 ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) β†’ ( βŠ₯ β€˜π‘„) ∈ (Baseβ€˜πΎ))
1713, 3, 4pmapssat 37815 . . . 4 ((𝐾 ∈ OL ∧ ( βŠ₯ β€˜π‘„) ∈ (Baseβ€˜πΎ)) β†’ (π‘€β€˜( βŠ₯ β€˜π‘„)) βŠ† 𝐴)
1816, 17syldan 592 . . 3 ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) β†’ (π‘€β€˜( βŠ₯ β€˜π‘„)) βŠ† 𝐴)
19 sseqin2 4155 . . 3 ((π‘€β€˜( βŠ₯ β€˜π‘„)) βŠ† 𝐴 ↔ (𝐴 ∩ (π‘€β€˜( βŠ₯ β€˜π‘„))) = (π‘€β€˜( βŠ₯ β€˜π‘„)))
2018, 19sylib 217 . 2 ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) β†’ (𝐴 ∩ (π‘€β€˜( βŠ₯ β€˜π‘„))) = (π‘€β€˜( βŠ₯ β€˜π‘„)))
217, 11, 203eqtrd 2780 1 ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) β†’ (π‘ƒβ€˜{𝑄}) = (π‘€β€˜( βŠ₯ β€˜π‘„)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1539   ∈ wcel 2104   ∩ cin 3891   βŠ† wss 3892  {csn 4565  βˆ© ciin 4932  β€˜cfv 6458  Basecbs 16957  occoc 17015  OPcops 37228  OLcol 37230  Atomscatm 37319  pmapcpmap 37553  βŠ₯𝑃cpolN 37958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  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 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-iin 4934  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oposet 37232  df-ol 37234  df-ats 37323  df-pmap 37560  df-polarityN 37959
This theorem is referenced by:  2polatN  37988
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