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Theorem pexmidN 38840
Description: Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 38824. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 38838. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pexmid.a 𝐴 = (Atomsβ€˜πΎ)
pexmid.p + = (+π‘ƒβ€˜πΎ)
pexmid.o βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
Assertion
Ref Expression
pexmidN (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ (𝑋 + ( βŠ₯ β€˜π‘‹)) = 𝐴)

Proof of Theorem pexmidN
StepHypRef Expression
1 simpll 766 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ 𝐾 ∈ HL)
2 simplr 768 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ 𝑋 βŠ† 𝐴)
3 pexmid.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
4 pexmid.o . . . . . . 7 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
53, 4polssatN 38779 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘‹) βŠ† 𝐴)
65adantr 482 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ ( βŠ₯ β€˜π‘‹) βŠ† 𝐴)
7 pexmid.p . . . . . 6 + = (+π‘ƒβ€˜πΎ)
83, 7, 4poldmj1N 38799 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜π‘‹) βŠ† 𝐴) β†’ ( βŠ₯ β€˜(𝑋 + ( βŠ₯ β€˜π‘‹))) = (( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹))))
91, 2, 6, 8syl3anc 1372 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ ( βŠ₯ β€˜(𝑋 + ( βŠ₯ β€˜π‘‹))) = (( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹))))
103, 4pnonsingN 38804 . . . . 5 ((𝐾 ∈ HL ∧ ( βŠ₯ β€˜π‘‹) βŠ† 𝐴) β†’ (( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹))) = βˆ…)
111, 6, 10syl2anc 585 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ (( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹))) = βˆ…)
129, 11eqtrd 2773 . . 3 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ ( βŠ₯ β€˜(𝑋 + ( βŠ₯ β€˜π‘‹))) = βˆ…)
1312fveq2d 6896 . 2 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜(𝑋 + ( βŠ₯ β€˜π‘‹)))) = ( βŠ₯ β€˜βˆ…))
14 simpr 486 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)
15 eqid 2733 . . . . . . 7 (PSubClβ€˜πΎ) = (PSubClβ€˜πΎ)
163, 4, 15ispsubclN 38808 . . . . . 6 (𝐾 ∈ HL β†’ (𝑋 ∈ (PSubClβ€˜πΎ) ↔ (𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)))
1716ad2antrr 725 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ (𝑋 ∈ (PSubClβ€˜πΎ) ↔ (𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)))
182, 14, 17mpbir2and 712 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ 𝑋 ∈ (PSubClβ€˜πΎ))
193, 4, 15polsubclN 38823 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘‹) ∈ (PSubClβ€˜πΎ))
2019adantr 482 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ ( βŠ₯ β€˜π‘‹) ∈ (PSubClβ€˜πΎ))
213, 42polssN 38786 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ 𝑋 βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)))
2221adantr 482 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ 𝑋 βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)))
237, 4, 15osumclN 38838 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ (PSubClβ€˜πΎ) ∧ ( βŠ₯ β€˜π‘‹) ∈ (PSubClβ€˜πΎ)) ∧ 𝑋 βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹))) β†’ (𝑋 + ( βŠ₯ β€˜π‘‹)) ∈ (PSubClβ€˜πΎ))
241, 18, 20, 22, 23syl31anc 1374 . . 3 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ (𝑋 + ( βŠ₯ β€˜π‘‹)) ∈ (PSubClβ€˜πΎ))
254, 15psubcli2N 38810 . . 3 ((𝐾 ∈ HL ∧ (𝑋 + ( βŠ₯ β€˜π‘‹)) ∈ (PSubClβ€˜πΎ)) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜(𝑋 + ( βŠ₯ β€˜π‘‹)))) = (𝑋 + ( βŠ₯ β€˜π‘‹)))
261, 24, 25syl2anc 585 . 2 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜(𝑋 + ( βŠ₯ β€˜π‘‹)))) = (𝑋 + ( βŠ₯ β€˜π‘‹)))
273, 4pol0N 38780 . . 3 (𝐾 ∈ HL β†’ ( βŠ₯ β€˜βˆ…) = 𝐴)
2827ad2antrr 725 . 2 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ ( βŠ₯ β€˜βˆ…) = 𝐴)
2913, 26, 283eqtr3d 2781 1 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ (𝑋 + ( βŠ₯ β€˜π‘‹)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  β€˜cfv 6544  (class class class)co 7409  Atomscatm 38133  HLchlt 38220  +𝑃cpadd 38666  βŠ₯𝑃cpolN 38773  PSubClcpscN 38805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-p1 18379  df-lat 18385  df-clat 18452  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-psubsp 38374  df-pmap 38375  df-padd 38667  df-polarityN 38774  df-psubclN 38806
This theorem is referenced by: (None)
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