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Theorem pmapojoinN 39142
Description: For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 39026 where only one direction holds. (Contributed by NM, 11-Apr-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmapojoin.b 𝐡 = (Baseβ€˜πΎ)
pmapojoin.l ≀ = (leβ€˜πΎ)
pmapojoin.j ∨ = (joinβ€˜πΎ)
pmapojoin.m 𝑀 = (pmapβ€˜πΎ)
pmapojoin.o βŠ₯ = (ocβ€˜πΎ)
pmapojoin.p + = (+π‘ƒβ€˜πΎ)
Assertion
Ref Expression
pmapojoinN (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ ( βŠ₯ β€˜π‘Œ)) β†’ (π‘€β€˜(𝑋 ∨ π‘Œ)) = ((π‘€β€˜π‘‹) + (π‘€β€˜π‘Œ)))

Proof of Theorem pmapojoinN
StepHypRef Expression
1 pmapojoin.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 pmapojoin.j . . . 4 ∨ = (joinβ€˜πΎ)
3 pmapojoin.m . . . 4 𝑀 = (pmapβ€˜πΎ)
4 pmapojoin.p . . . 4 + = (+π‘ƒβ€˜πΎ)
5 eqid 2730 . . . 4 (βŠ₯π‘ƒβ€˜πΎ) = (βŠ₯π‘ƒβ€˜πΎ)
61, 2, 3, 4, 5pmapj2N 39103 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘€β€˜(𝑋 ∨ π‘Œ)) = ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜((π‘€β€˜π‘‹) + (π‘€β€˜π‘Œ)))))
76adantr 479 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ ( βŠ₯ β€˜π‘Œ)) β†’ (π‘€β€˜(𝑋 ∨ π‘Œ)) = ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜((π‘€β€˜π‘‹) + (π‘€β€˜π‘Œ)))))
8 simpl1 1189 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ ( βŠ₯ β€˜π‘Œ)) β†’ 𝐾 ∈ HL)
9 simpl2 1190 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ ( βŠ₯ β€˜π‘Œ)) β†’ 𝑋 ∈ 𝐡)
10 eqid 2730 . . . . . 6 (PSubClβ€˜πΎ) = (PSubClβ€˜πΎ)
111, 3, 10pmapsubclN 39120 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ (π‘€β€˜π‘‹) ∈ (PSubClβ€˜πΎ))
128, 9, 11syl2anc 582 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ ( βŠ₯ β€˜π‘Œ)) β†’ (π‘€β€˜π‘‹) ∈ (PSubClβ€˜πΎ))
13 simpl3 1191 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ ( βŠ₯ β€˜π‘Œ)) β†’ π‘Œ ∈ 𝐡)
141, 3, 10pmapsubclN 39120 . . . . 5 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐡) β†’ (π‘€β€˜π‘Œ) ∈ (PSubClβ€˜πΎ))
158, 13, 14syl2anc 582 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ ( βŠ₯ β€˜π‘Œ)) β†’ (π‘€β€˜π‘Œ) ∈ (PSubClβ€˜πΎ))
16 hlop 38535 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
17163ad2ant1 1131 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ OP)
18 simp3 1136 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ ∈ 𝐡)
19 pmapojoin.o . . . . . . . . 9 βŠ₯ = (ocβ€˜πΎ)
201, 19opoccl 38367 . . . . . . . 8 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
2117, 18, 20syl2anc 582 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
22 pmapojoin.l . . . . . . . 8 ≀ = (leβ€˜πΎ)
231, 22, 3pmaple 38935 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡) β†’ (𝑋 ≀ ( βŠ₯ β€˜π‘Œ) ↔ (π‘€β€˜π‘‹) βŠ† (π‘€β€˜( βŠ₯ β€˜π‘Œ))))
2421, 23syld3an3 1407 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ ( βŠ₯ β€˜π‘Œ) ↔ (π‘€β€˜π‘‹) βŠ† (π‘€β€˜( βŠ₯ β€˜π‘Œ))))
2524biimpa 475 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ ( βŠ₯ β€˜π‘Œ)) β†’ (π‘€β€˜π‘‹) βŠ† (π‘€β€˜( βŠ₯ β€˜π‘Œ)))
261, 19, 3, 5polpmapN 39086 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐡) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜(π‘€β€˜π‘Œ)) = (π‘€β€˜( βŠ₯ β€˜π‘Œ)))
278, 13, 26syl2anc 582 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ ( βŠ₯ β€˜π‘Œ)) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜(π‘€β€˜π‘Œ)) = (π‘€β€˜( βŠ₯ β€˜π‘Œ)))
2825, 27sseqtrrd 4022 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ ( βŠ₯ β€˜π‘Œ)) β†’ (π‘€β€˜π‘‹) βŠ† ((βŠ₯π‘ƒβ€˜πΎ)β€˜(π‘€β€˜π‘Œ)))
294, 5, 10osumclN 39141 . . . 4 (((𝐾 ∈ HL ∧ (π‘€β€˜π‘‹) ∈ (PSubClβ€˜πΎ) ∧ (π‘€β€˜π‘Œ) ∈ (PSubClβ€˜πΎ)) ∧ (π‘€β€˜π‘‹) βŠ† ((βŠ₯π‘ƒβ€˜πΎ)β€˜(π‘€β€˜π‘Œ))) β†’ ((π‘€β€˜π‘‹) + (π‘€β€˜π‘Œ)) ∈ (PSubClβ€˜πΎ))
308, 12, 15, 28, 29syl31anc 1371 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ ( βŠ₯ β€˜π‘Œ)) β†’ ((π‘€β€˜π‘‹) + (π‘€β€˜π‘Œ)) ∈ (PSubClβ€˜πΎ))
315, 10psubcli2N 39113 . . 3 ((𝐾 ∈ HL ∧ ((π‘€β€˜π‘‹) + (π‘€β€˜π‘Œ)) ∈ (PSubClβ€˜πΎ)) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜((π‘€β€˜π‘‹) + (π‘€β€˜π‘Œ)))) = ((π‘€β€˜π‘‹) + (π‘€β€˜π‘Œ)))
328, 30, 31syl2anc 582 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ ( βŠ₯ β€˜π‘Œ)) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜((π‘€β€˜π‘‹) + (π‘€β€˜π‘Œ)))) = ((π‘€β€˜π‘‹) + (π‘€β€˜π‘Œ)))
337, 32eqtrd 2770 1 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ ( βŠ₯ β€˜π‘Œ)) β†’ (π‘€β€˜(𝑋 ∨ π‘Œ)) = ((π‘€β€˜π‘‹) + (π‘€β€˜π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   βŠ† wss 3947   class class class wbr 5147  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  lecple 17208  occoc 17209  joincjn 18268  OPcops 38345  HLchlt 38523  pmapcpmap 38671  +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-pss 3966  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:  pl42lem1N  39153
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