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Theorem osumcllem3N 39132
Description: Lemma for osumclN 39141. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
osumcllem.l ≀ = (leβ€˜πΎ)
osumcllem.j ∨ = (joinβ€˜πΎ)
osumcllem.a 𝐴 = (Atomsβ€˜πΎ)
osumcllem.p + = (+π‘ƒβ€˜πΎ)
osumcllem.o βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
osumcllem.c 𝐢 = (PSubClβ€˜πΎ)
osumcllem.m 𝑀 = (𝑋 + {𝑝})
osumcllem.u π‘ˆ = ( βŠ₯ β€˜( βŠ₯ β€˜(𝑋 + π‘Œ)))
Assertion
Ref Expression
osumcllem3N ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ (( βŠ₯ β€˜π‘‹) ∩ π‘ˆ) = π‘Œ)

Proof of Theorem osumcllem3N
StepHypRef Expression
1 incom 4200 . 2 (( βŠ₯ β€˜π‘‹) ∩ π‘ˆ) = (π‘ˆ ∩ ( βŠ₯ β€˜π‘‹))
2 osumcllem.u . . . . 5 π‘ˆ = ( βŠ₯ β€˜( βŠ₯ β€˜(𝑋 + π‘Œ)))
3 simp1 1134 . . . . . . . 8 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ 𝐾 ∈ HL)
4 simp3 1136 . . . . . . . . 9 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ))
5 osumcllem.a . . . . . . . . . . . 12 𝐴 = (Atomsβ€˜πΎ)
6 osumcllem.c . . . . . . . . . . . 12 𝐢 = (PSubClβ€˜πΎ)
75, 6psubclssatN 39115 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢) β†’ π‘Œ βŠ† 𝐴)
873adant3 1130 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ π‘Œ βŠ† 𝐴)
9 osumcllem.o . . . . . . . . . . 11 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
105, 9polssatN 39082 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘Œ) βŠ† 𝐴)
113, 8, 10syl2anc 582 . . . . . . . . 9 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ ( βŠ₯ β€˜π‘Œ) βŠ† 𝐴)
124, 11sstrd 3991 . . . . . . . 8 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ 𝑋 βŠ† 𝐴)
13 osumcllem.p . . . . . . . . 9 + = (+π‘ƒβ€˜πΎ)
145, 13, 9poldmj1N 39102 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ ( βŠ₯ β€˜(𝑋 + π‘Œ)) = (( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ)))
153, 12, 8, 14syl3anc 1369 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ ( βŠ₯ β€˜(𝑋 + π‘Œ)) = (( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ)))
16 incom 4200 . . . . . . 7 (( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ)) = (( βŠ₯ β€˜π‘Œ) ∩ ( βŠ₯ β€˜π‘‹))
1715, 16eqtrdi 2786 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ ( βŠ₯ β€˜(𝑋 + π‘Œ)) = (( βŠ₯ β€˜π‘Œ) ∩ ( βŠ₯ β€˜π‘‹)))
1817fveq2d 6894 . . . . 5 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜(𝑋 + π‘Œ))) = ( βŠ₯ β€˜(( βŠ₯ β€˜π‘Œ) ∩ ( βŠ₯ β€˜π‘‹))))
192, 18eqtrid 2782 . . . 4 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ π‘ˆ = ( βŠ₯ β€˜(( βŠ₯ β€˜π‘Œ) ∩ ( βŠ₯ β€˜π‘‹))))
2019ineq1d 4210 . . 3 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ (π‘ˆ ∩ ( βŠ₯ β€˜π‘‹)) = (( βŠ₯ β€˜(( βŠ₯ β€˜π‘Œ) ∩ ( βŠ₯ β€˜π‘‹))) ∩ ( βŠ₯ β€˜π‘‹)))
215, 9polcon2N 39093 . . . . 5 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ π‘Œ βŠ† ( βŠ₯ β€˜π‘‹))
228, 21syld3an2 1409 . . . 4 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ π‘Œ βŠ† ( βŠ₯ β€˜π‘‹))
235, 9poml5N 39128 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† ( βŠ₯ β€˜π‘‹)) β†’ (( βŠ₯ β€˜(( βŠ₯ β€˜π‘Œ) ∩ ( βŠ₯ β€˜π‘‹))) ∩ ( βŠ₯ β€˜π‘‹)) = ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)))
243, 12, 22, 23syl3anc 1369 . . 3 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ (( βŠ₯ β€˜(( βŠ₯ β€˜π‘Œ) ∩ ( βŠ₯ β€˜π‘‹))) ∩ ( βŠ₯ β€˜π‘‹)) = ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)))
259, 6psubcli2N 39113 . . . 4 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)) = π‘Œ)
26253adant3 1130 . . 3 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)) = π‘Œ)
2720, 24, 263eqtrd 2774 . 2 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ (π‘ˆ ∩ ( βŠ₯ β€˜π‘‹)) = π‘Œ)
281, 27eqtrid 2782 1 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ (( βŠ₯ β€˜π‘‹) ∩ π‘ˆ) = π‘Œ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   ∩ cin 3946   βŠ† wss 3947  {csn 4627  β€˜cfv 6542  (class class class)co 7411  lecple 17208  joincjn 18268  Atomscatm 38436  HLchlt 38523  +𝑃cpadd 38969  βŠ₯𝑃cpolN 39076  PSubClcpscN 39108
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 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-proset 18252  df-poset 18270  df-plt 18287  df-lub 18303  df-glb 18304  df-join 18305  df-meet 18306  df-p0 18382  df-p1 18383  df-lat 18389  df-clat 18456  df-oposet 38349  df-ol 38351  df-oml 38352  df-covers 38439  df-ats 38440  df-atl 38471  df-cvlat 38495  df-hlat 38524  df-psubsp 38677  df-pmap 38678  df-padd 38970  df-polarityN 39077  df-psubclN 39109
This theorem is referenced by:  osumcllem9N  39138
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