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Theorem pexmidN 38835
Description: Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 38819. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 38833. (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 765 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ 𝐾 ∈ HL)
2 simplr 767 . . . . 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 38774 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘‹) βŠ† 𝐴)
65adantr 481 . . . . 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 38794 . . . . 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 1371 . . . 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 38799 . . . . 5 ((𝐾 ∈ HL ∧ ( βŠ₯ β€˜π‘‹) βŠ† 𝐴) β†’ (( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹))) = βˆ…)
111, 6, 10syl2anc 584 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ (( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹))) = βˆ…)
129, 11eqtrd 2772 . . 3 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ ( βŠ₯ β€˜(𝑋 + ( βŠ₯ β€˜π‘‹))) = βˆ…)
1312fveq2d 6895 . 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 485 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)
15 eqid 2732 . . . . . . 7 (PSubClβ€˜πΎ) = (PSubClβ€˜πΎ)
163, 4, 15ispsubclN 38803 . . . . . 6 (𝐾 ∈ HL β†’ (𝑋 ∈ (PSubClβ€˜πΎ) ↔ (𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)))
1716ad2antrr 724 . . . . 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 711 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ 𝑋 ∈ (PSubClβ€˜πΎ))
193, 4, 15polsubclN 38818 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘‹) ∈ (PSubClβ€˜πΎ))
2019adantr 481 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ ( βŠ₯ β€˜π‘‹) ∈ (PSubClβ€˜πΎ))
213, 42polssN 38781 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ 𝑋 βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)))
2221adantr 481 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ 𝑋 βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)))
237, 4, 15osumclN 38833 . . . 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 1373 . . 3 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ (𝑋 + ( βŠ₯ β€˜π‘‹)) ∈ (PSubClβ€˜πΎ))
254, 15psubcli2N 38805 . . 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 584 . 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 38775 . . 3 (𝐾 ∈ HL β†’ ( βŠ₯ β€˜βˆ…) = 𝐴)
2827ad2antrr 724 . 2 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋) β†’ ( βŠ₯ β€˜βˆ…) = 𝐴)
2913, 26, 283eqtr3d 2780 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 396   = wceq 1541   ∈ wcel 2106   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  β€˜cfv 6543  (class class class)co 7408  Atomscatm 38128  HLchlt 38215  +𝑃cpadd 38661  βŠ₯𝑃cpolN 38768  PSubClcpscN 38800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-proset 18247  df-poset 18265  df-plt 18282  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-p0 18377  df-p1 18378  df-lat 18384  df-clat 18451  df-oposet 38041  df-ol 38043  df-oml 38044  df-covers 38131  df-ats 38132  df-atl 38163  df-cvlat 38187  df-hlat 38216  df-psubsp 38369  df-pmap 38370  df-padd 38662  df-polarityN 38769  df-psubclN 38801
This theorem is referenced by: (None)
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