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Theorem 2pmaplubN 39310
Description: Double projective map of an LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
sspmaplub.u π‘ˆ = (lubβ€˜πΎ)
sspmaplub.a 𝐴 = (Atomsβ€˜πΎ)
sspmaplub.m 𝑀 = (pmapβ€˜πΎ)
Assertion
Ref Expression
2pmaplubN ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴) β†’ (π‘€β€˜(π‘ˆβ€˜(π‘€β€˜(π‘ˆβ€˜π‘†)))) = (π‘€β€˜(π‘ˆβ€˜π‘†)))

Proof of Theorem 2pmaplubN
StepHypRef Expression
1 sspmaplub.u . . . . . . 7 π‘ˆ = (lubβ€˜πΎ)
2 sspmaplub.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
3 sspmaplub.m . . . . . . 7 𝑀 = (pmapβ€˜πΎ)
4 eqid 2726 . . . . . . 7 (βŠ₯π‘ƒβ€˜πΎ) = (βŠ₯π‘ƒβ€˜πΎ)
51, 2, 3, 42polvalN 39298 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘†)) = (π‘€β€˜(π‘ˆβ€˜π‘†)))
65fveq2d 6889 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘†))) = ((βŠ₯π‘ƒβ€˜πΎ)β€˜(π‘€β€˜(π‘ˆβ€˜π‘†))))
76fveq2d 6889 . . . 4 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘†)))) = ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜(π‘€β€˜(π‘ˆβ€˜π‘†)))))
82, 4polssatN 39292 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘†) βŠ† 𝐴)
92, 43polN 39300 . . . . 5 ((𝐾 ∈ HL ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘†) βŠ† 𝐴) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘†)))) = ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘†)))
108, 9syldan 590 . . . 4 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘†)))) = ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘†)))
117, 10eqtr3d 2768 . . 3 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜(π‘€β€˜(π‘ˆβ€˜π‘†)))) = ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘†)))
12 hlclat 38741 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ CLat)
13 eqid 2726 . . . . . . . 8 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
1413, 2atssbase 38673 . . . . . . 7 𝐴 βŠ† (Baseβ€˜πΎ)
15 sstr 3985 . . . . . . 7 ((𝑆 βŠ† 𝐴 ∧ 𝐴 βŠ† (Baseβ€˜πΎ)) β†’ 𝑆 βŠ† (Baseβ€˜πΎ))
1614, 15mpan2 688 . . . . . 6 (𝑆 βŠ† 𝐴 β†’ 𝑆 βŠ† (Baseβ€˜πΎ))
1713, 1clatlubcl 18468 . . . . . 6 ((𝐾 ∈ CLat ∧ 𝑆 βŠ† (Baseβ€˜πΎ)) β†’ (π‘ˆβ€˜π‘†) ∈ (Baseβ€˜πΎ))
1812, 16, 17syl2an 595 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘†) ∈ (Baseβ€˜πΎ))
1913, 2, 3pmapssat 39143 . . . . 5 ((𝐾 ∈ HL ∧ (π‘ˆβ€˜π‘†) ∈ (Baseβ€˜πΎ)) β†’ (π‘€β€˜(π‘ˆβ€˜π‘†)) βŠ† 𝐴)
2018, 19syldan 590 . . . 4 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴) β†’ (π‘€β€˜(π‘ˆβ€˜π‘†)) βŠ† 𝐴)
211, 2, 3, 42polvalN 39298 . . . 4 ((𝐾 ∈ HL ∧ (π‘€β€˜(π‘ˆβ€˜π‘†)) βŠ† 𝐴) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜(π‘€β€˜(π‘ˆβ€˜π‘†)))) = (π‘€β€˜(π‘ˆβ€˜(π‘€β€˜(π‘ˆβ€˜π‘†)))))
2220, 21syldan 590 . . 3 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜(π‘€β€˜(π‘ˆβ€˜π‘†)))) = (π‘€β€˜(π‘ˆβ€˜(π‘€β€˜(π‘ˆβ€˜π‘†)))))
2311, 22eqtr3d 2768 . 2 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘†)) = (π‘€β€˜(π‘ˆβ€˜(π‘€β€˜(π‘ˆβ€˜π‘†)))))
2423, 5eqtr3d 2768 1 ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴) β†’ (π‘€β€˜(π‘ˆβ€˜(π‘€β€˜(π‘ˆβ€˜π‘†)))) = (π‘€β€˜(π‘ˆβ€˜π‘†)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943  β€˜cfv 6537  Basecbs 17153  lubclub 18274  CLatccla 18463  Atomscatm 38646  HLchlt 38733  pmapcpmap 38881  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-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-polarityN 39287
This theorem is referenced by:  paddunN  39311
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