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Theorem poldmj1N 38885
Description: De Morgan's law for polarity of projective sum. (oldmj1 38177 analog.) (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
paddun.a 𝐴 = (Atomsβ€˜πΎ)
paddun.p + = (+π‘ƒβ€˜πΎ)
paddun.o βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
Assertion
Ref Expression
poldmj1N ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ( βŠ₯ β€˜(𝑆 + 𝑇)) = (( βŠ₯ β€˜π‘†) ∩ ( βŠ₯ β€˜π‘‡)))
Proof of Theorem poldmj1N
StepHypRef Expression
1 paddun.a . . 3 𝐴 = (Atomsβ€˜πΎ)
2 paddun.p . . 3 + = (+π‘ƒβ€˜πΎ)
3 paddun.o . . 3 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
41, 2, 3paddunN 38884 . 2 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ( βŠ₯ β€˜(𝑆 + 𝑇)) = ( βŠ₯ β€˜(𝑆 βˆͺ 𝑇)))
5 simp1 1136 . . 3 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ 𝐾 ∈ HL)
6 unss 4184 . . . . 5 ((𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) ↔ (𝑆 βˆͺ 𝑇) βŠ† 𝐴)
76biimpi 215 . . . 4 ((𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ (𝑆 βˆͺ 𝑇) βŠ† 𝐴)
873adant1 1130 . . 3 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ (𝑆 βˆͺ 𝑇) βŠ† 𝐴)
9 eqid 2732 . . . 4 (lubβ€˜πΎ) = (lubβ€˜πΎ)
10 eqid 2732 . . . 4 (ocβ€˜πΎ) = (ocβ€˜πΎ)
11 eqid 2732 . . . 4 (pmapβ€˜πΎ) = (pmapβ€˜πΎ)
129, 10, 1, 11, 3polval2N 38863 . . 3 ((𝐾 ∈ HL ∧ (𝑆 βˆͺ 𝑇) βŠ† 𝐴) β†’ ( βŠ₯ β€˜(𝑆 βˆͺ 𝑇)) = ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜(𝑆 βˆͺ 𝑇)))))
135, 8, 12syl2anc 584 . 2 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ( βŠ₯ β€˜(𝑆 βˆͺ 𝑇)) = ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜(𝑆 βˆͺ 𝑇)))))
14 hlop 38318 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
15143ad2ant1 1133 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ 𝐾 ∈ OP)
16 hlclat 38314 . . . . . . 7 (𝐾 ∈ HL β†’ 𝐾 ∈ CLat)
17163ad2ant1 1133 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ 𝐾 ∈ CLat)
18 simp2 1137 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ 𝑆 βŠ† 𝐴)
19 eqid 2732 . . . . . . . 8 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2019, 1atssbase 38246 . . . . . . 7 𝐴 βŠ† (Baseβ€˜πΎ)
2118, 20sstrdi 3994 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ 𝑆 βŠ† (Baseβ€˜πΎ))
2219, 9clatlubcl 18458 . . . . . 6 ((𝐾 ∈ CLat ∧ 𝑆 βŠ† (Baseβ€˜πΎ)) β†’ ((lubβ€˜πΎ)β€˜π‘†) ∈ (Baseβ€˜πΎ))
2317, 21, 22syl2anc 584 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ((lubβ€˜πΎ)β€˜π‘†) ∈ (Baseβ€˜πΎ))
2419, 10opoccl 38150 . . . . 5 ((𝐾 ∈ OP ∧ ((lubβ€˜πΎ)β€˜π‘†) ∈ (Baseβ€˜πΎ)) β†’ ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘†)) ∈ (Baseβ€˜πΎ))
2515, 23, 24syl2anc 584 . . . 4 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘†)) ∈ (Baseβ€˜πΎ))
26 simp3 1138 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ 𝑇 βŠ† 𝐴)
2726, 20sstrdi 3994 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ 𝑇 βŠ† (Baseβ€˜πΎ))
2819, 9clatlubcl 18458 . . . . . 6 ((𝐾 ∈ CLat ∧ 𝑇 βŠ† (Baseβ€˜πΎ)) β†’ ((lubβ€˜πΎ)β€˜π‘‡) ∈ (Baseβ€˜πΎ))
2917, 27, 28syl2anc 584 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ((lubβ€˜πΎ)β€˜π‘‡) ∈ (Baseβ€˜πΎ))
3019, 10opoccl 38150 . . . . 5 ((𝐾 ∈ OP ∧ ((lubβ€˜πΎ)β€˜π‘‡) ∈ (Baseβ€˜πΎ)) β†’ ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‡)) ∈ (Baseβ€˜πΎ))
3115, 29, 30syl2anc 584 . . . 4 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‡)) ∈ (Baseβ€˜πΎ))
32 eqid 2732 . . . . 5 (meetβ€˜πΎ) = (meetβ€˜πΎ)
3319, 32, 1, 11pmapmeet 38730 . . . 4 ((𝐾 ∈ HL ∧ ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘†)) ∈ (Baseβ€˜πΎ) ∧ ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‡)) ∈ (Baseβ€˜πΎ)) β†’ ((pmapβ€˜πΎ)β€˜(((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘†))(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‡)))) = (((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘†))) ∩ ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‡)))))
345, 25, 31, 33syl3anc 1371 . . 3 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ((pmapβ€˜πΎ)β€˜(((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘†))(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‡)))) = (((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘†))) ∩ ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‡)))))
35 eqid 2732 . . . . . . . 8 (joinβ€˜πΎ) = (joinβ€˜πΎ)
3619, 35, 9lubun 18470 . . . . . . 7 ((𝐾 ∈ CLat ∧ 𝑆 βŠ† (Baseβ€˜πΎ) ∧ 𝑇 βŠ† (Baseβ€˜πΎ)) β†’ ((lubβ€˜πΎ)β€˜(𝑆 βˆͺ 𝑇)) = (((lubβ€˜πΎ)β€˜π‘†)(joinβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘‡)))
3717, 21, 27, 36syl3anc 1371 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ((lubβ€˜πΎ)β€˜(𝑆 βˆͺ 𝑇)) = (((lubβ€˜πΎ)β€˜π‘†)(joinβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘‡)))
3837fveq2d 6895 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜(𝑆 βˆͺ 𝑇))) = ((ocβ€˜πΎ)β€˜(((lubβ€˜πΎ)β€˜π‘†)(joinβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘‡))))
39 hlol 38317 . . . . . . 7 (𝐾 ∈ HL β†’ 𝐾 ∈ OL)
40393ad2ant1 1133 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ 𝐾 ∈ OL)
4119, 35, 32, 10oldmj1 38177 . . . . . 6 ((𝐾 ∈ OL ∧ ((lubβ€˜πΎ)β€˜π‘†) ∈ (Baseβ€˜πΎ) ∧ ((lubβ€˜πΎ)β€˜π‘‡) ∈ (Baseβ€˜πΎ)) β†’ ((ocβ€˜πΎ)β€˜(((lubβ€˜πΎ)β€˜π‘†)(joinβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘‡))) = (((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘†))(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‡))))
4240, 23, 29, 41syl3anc 1371 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ((ocβ€˜πΎ)β€˜(((lubβ€˜πΎ)β€˜π‘†)(joinβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘‡))) = (((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘†))(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‡))))
4338, 42eqtrd 2772 . . . 4 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜(𝑆 βˆͺ 𝑇))) = (((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘†))(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‡))))
4443fveq2d 6895 . . 3 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜(𝑆 βˆͺ 𝑇)))) = ((pmapβ€˜πΎ)β€˜(((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘†))(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‡)))))
459, 10, 1, 11, 3polval2N 38863 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘†) = ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘†))))
46453adant3 1132 . . . 4 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘†) = ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘†))))
479, 10, 1, 11, 3polval2N 38863 . . . . 5 ((𝐾 ∈ HL ∧ 𝑇 βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘‡) = ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‡))))
48473adant2 1131 . . . 4 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘‡) = ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‡))))
4946, 48ineq12d 4213 . . 3 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ (( βŠ₯ β€˜π‘†) ∩ ( βŠ₯ β€˜π‘‡)) = (((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘†))) ∩ ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‡)))))
5034, 44, 493eqtr4d 2782 . 2 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜(𝑆 βˆͺ 𝑇)))) = (( βŠ₯ β€˜π‘†) ∩ ( βŠ₯ β€˜π‘‡)))
514, 13, 503eqtrd 2776 1 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ( βŠ₯ β€˜(𝑆 + 𝑇)) = (( βŠ₯ β€˜π‘†) ∩ ( βŠ₯ β€˜π‘‡)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  occoc 17207  lubclub 18264  joincjn 18266  meetcmee 18267  CLatccla 18453  OPcops 38128  OLcol 38130  Atomscatm 38219  HLchlt 38306  pmapcpmap 38454  +𝑃cpadd 38752  βŠ₯𝑃cpolN 38859
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-proset 18250  df-poset 18268  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-p1 18381  df-lat 18387  df-clat 18454  df-oposet 38132  df-ol 38134  df-oml 38135  df-covers 38222  df-ats 38223  df-atl 38254  df-cvlat 38278  df-hlat 38307  df-psubsp 38460  df-pmap 38461  df-padd 38753  df-polarityN 38860
This theorem is referenced by:  pmapj2N  38886  osumcllem3N  38915  pexmidN  38926
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