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Theorem poldmj1N 39312
Description: De Morgan's law for polarity of projective sum. (oldmj1 38604 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 39311 . 2 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ( βŠ₯ β€˜(𝑆 + 𝑇)) = ( βŠ₯ β€˜(𝑆 βˆͺ 𝑇)))
5 simp1 1133 . . 3 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ 𝐾 ∈ HL)
6 unss 4179 . . . . 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 1127 . . 3 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ (𝑆 βˆͺ 𝑇) βŠ† 𝐴)
9 eqid 2726 . . . 4 (lubβ€˜πΎ) = (lubβ€˜πΎ)
10 eqid 2726 . . . 4 (ocβ€˜πΎ) = (ocβ€˜πΎ)
11 eqid 2726 . . . 4 (pmapβ€˜πΎ) = (pmapβ€˜πΎ)
129, 10, 1, 11, 3polval2N 39290 . . 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 583 . 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 38745 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
15143ad2ant1 1130 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ 𝐾 ∈ OP)
16 hlclat 38741 . . . . . . 7 (𝐾 ∈ HL β†’ 𝐾 ∈ CLat)
17163ad2ant1 1130 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ 𝐾 ∈ CLat)
18 simp2 1134 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ 𝑆 βŠ† 𝐴)
19 eqid 2726 . . . . . . . 8 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2019, 1atssbase 38673 . . . . . . 7 𝐴 βŠ† (Baseβ€˜πΎ)
2118, 20sstrdi 3989 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ 𝑆 βŠ† (Baseβ€˜πΎ))
2219, 9clatlubcl 18468 . . . . . 6 ((𝐾 ∈ CLat ∧ 𝑆 βŠ† (Baseβ€˜πΎ)) β†’ ((lubβ€˜πΎ)β€˜π‘†) ∈ (Baseβ€˜πΎ))
2317, 21, 22syl2anc 583 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ((lubβ€˜πΎ)β€˜π‘†) ∈ (Baseβ€˜πΎ))
2419, 10opoccl 38577 . . . . 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 583 . . . 4 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘†)) ∈ (Baseβ€˜πΎ))
26 simp3 1135 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ 𝑇 βŠ† 𝐴)
2726, 20sstrdi 3989 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ 𝑇 βŠ† (Baseβ€˜πΎ))
2819, 9clatlubcl 18468 . . . . . 6 ((𝐾 ∈ CLat ∧ 𝑇 βŠ† (Baseβ€˜πΎ)) β†’ ((lubβ€˜πΎ)β€˜π‘‡) ∈ (Baseβ€˜πΎ))
2917, 27, 28syl2anc 583 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ((lubβ€˜πΎ)β€˜π‘‡) ∈ (Baseβ€˜πΎ))
3019, 10opoccl 38577 . . . . 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 583 . . . 4 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‡)) ∈ (Baseβ€˜πΎ))
32 eqid 2726 . . . . 5 (meetβ€˜πΎ) = (meetβ€˜πΎ)
3319, 32, 1, 11pmapmeet 39157 . . . 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 1368 . . 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 2726 . . . . . . . 8 (joinβ€˜πΎ) = (joinβ€˜πΎ)
3619, 35, 9lubun 18480 . . . . . . 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 1368 . . . . . 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 6889 . . . . 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 38744 . . . . . . 7 (𝐾 ∈ HL β†’ 𝐾 ∈ OL)
40393ad2ant1 1130 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴 ∧ 𝑇 βŠ† 𝐴) β†’ 𝐾 ∈ OL)
4119, 35, 32, 10oldmj1 38604 . . . . . 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 1368 . . . . 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 2766 . . . 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 6889 . . 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 39290 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘†) = ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘†))))
46453adant3 1129 . . . 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 39290 . . . . 5 ((𝐾 ∈ HL ∧ 𝑇 βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘‡) = ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‡))))
48473adant2 1128 . . . 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 4208 . . 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 2776 . 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 2770 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βˆͺ cun 3941   ∩ cin 3942   βŠ† wss 3943  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  occoc 17214  lubclub 18274  joincjn 18276  meetcmee 18277  CLatccla 18463  OPcops 38555  OLcol 38557  Atomscatm 38646  HLchlt 38733  pmapcpmap 38881  +𝑃cpadd 39179  βŠ₯𝑃cpolN 39286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-p1 18391  df-lat 18397  df-clat 18464  df-oposet 38559  df-ol 38561  df-oml 38562  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734  df-psubsp 38887  df-pmap 38888  df-padd 39180  df-polarityN 39287
This theorem is referenced by:  pmapj2N  39313  osumcllem3N  39342  pexmidN  39353
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