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Theorem pexmidlem8N 39152
Description: Lemma for pexmidN 39144. The contradiction of pexmidlem6N 39150 and pexmidlem7N 39151 shows that there can be no atom 𝑝 that is not in 𝑋 + ( βŠ₯ β€˜π‘‹), which is therefore the whole atom space. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pexmidALT.a 𝐴 = (Atomsβ€˜πΎ)
pexmidALT.p + = (+π‘ƒβ€˜πΎ)
pexmidALT.o βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
Assertion
Ref Expression
pexmidlem8N (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ…)) β†’ (𝑋 + ( βŠ₯ β€˜π‘‹)) = 𝐴)

Proof of Theorem pexmidlem8N
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 nonconne 2951 . 2 Β¬ (𝑋 = 𝑋 ∧ 𝑋 β‰  𝑋)
2 simpll 764 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ…)) β†’ 𝐾 ∈ HL)
3 simplr 766 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ…)) β†’ 𝑋 βŠ† 𝐴)
4 pexmidALT.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
5 pexmidALT.o . . . . . . 7 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
64, 5polssatN 39083 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘‹) βŠ† 𝐴)
76adantr 480 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ…)) β†’ ( βŠ₯ β€˜π‘‹) βŠ† 𝐴)
8 pexmidALT.p . . . . . 6 + = (+π‘ƒβ€˜πΎ)
94, 8paddssat 38989 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ ( βŠ₯ β€˜π‘‹) βŠ† 𝐴) β†’ (𝑋 + ( βŠ₯ β€˜π‘‹)) βŠ† 𝐴)
102, 3, 7, 9syl3anc 1370 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ…)) β†’ (𝑋 + ( βŠ₯ β€˜π‘‹)) βŠ† 𝐴)
11 df-pss 3967 . . . . . . 7 ((𝑋 + ( βŠ₯ β€˜π‘‹)) ⊊ 𝐴 ↔ ((𝑋 + ( βŠ₯ β€˜π‘‹)) βŠ† 𝐴 ∧ (𝑋 + ( βŠ₯ β€˜π‘‹)) β‰  𝐴))
12 pssnel 4470 . . . . . . 7 ((𝑋 + ( βŠ₯ β€˜π‘‹)) ⊊ 𝐴 β†’ βˆƒπ‘(𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹))))
1311, 12sylbir 234 . . . . . 6 (((𝑋 + ( βŠ₯ β€˜π‘‹)) βŠ† 𝐴 ∧ (𝑋 + ( βŠ₯ β€˜π‘‹)) β‰  𝐴) β†’ βˆƒπ‘(𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹))))
14 df-rex 3070 . . . . . 6 (βˆƒπ‘ ∈ 𝐴 Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)) ↔ βˆƒπ‘(𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹))))
1513, 14sylibr 233 . . . . 5 (((𝑋 + ( βŠ₯ β€˜π‘‹)) βŠ† 𝐴 ∧ (𝑋 + ( βŠ₯ β€˜π‘‹)) β‰  𝐴) β†’ βˆƒπ‘ ∈ 𝐴 Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))
16 simplll 772 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ…)) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ 𝐾 ∈ HL)
17 simpllr 773 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ…)) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ 𝑋 βŠ† 𝐴)
18 simprl 768 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ…)) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ 𝑝 ∈ 𝐴)
19 simplrl 774 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ…)) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)
20 simplrr 775 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ…)) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ 𝑋 β‰  βˆ…)
21 simprr 770 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ…)) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))
22 eqid 2731 . . . . . . . . . 10 (leβ€˜πΎ) = (leβ€˜πΎ)
23 eqid 2731 . . . . . . . . . 10 (joinβ€˜πΎ) = (joinβ€˜πΎ)
24 eqid 2731 . . . . . . . . . 10 (𝑋 + {𝑝}) = (𝑋 + {𝑝})
2522, 23, 4, 8, 5, 24pexmidlem6N 39150 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ (𝑋 + {𝑝}) = 𝑋)
2622, 23, 4, 8, 5, 24pexmidlem7N 39151 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ (𝑋 + {𝑝}) β‰  𝑋)
2725, 26jca 511 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ… ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ ((𝑋 + {𝑝}) = 𝑋 ∧ (𝑋 + {𝑝}) β‰  𝑋))
2816, 17, 18, 19, 20, 21, 27syl33anc 1384 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ…)) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ ((𝑋 + {𝑝}) = 𝑋 ∧ (𝑋 + {𝑝}) β‰  𝑋))
29 nonconne 2951 . . . . . . . 8 Β¬ ((𝑋 + {𝑝}) = 𝑋 ∧ (𝑋 + {𝑝}) β‰  𝑋)
3029, 12false 375 . . . . . . 7 (((𝑋 + {𝑝}) = 𝑋 ∧ (𝑋 + {𝑝}) β‰  𝑋) ↔ (𝑋 = 𝑋 ∧ 𝑋 β‰  𝑋))
3128, 30sylib 217 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ…)) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)))) β†’ (𝑋 = 𝑋 ∧ 𝑋 β‰  𝑋))
3231rexlimdvaa 3155 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ…)) β†’ (βˆƒπ‘ ∈ 𝐴 Β¬ 𝑝 ∈ (𝑋 + ( βŠ₯ β€˜π‘‹)) β†’ (𝑋 = 𝑋 ∧ 𝑋 β‰  𝑋)))
3315, 32syl5 34 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ…)) β†’ (((𝑋 + ( βŠ₯ β€˜π‘‹)) βŠ† 𝐴 ∧ (𝑋 + ( βŠ₯ β€˜π‘‹)) β‰  𝐴) β†’ (𝑋 = 𝑋 ∧ 𝑋 β‰  𝑋)))
3410, 33mpand 692 . . 3 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ…)) β†’ ((𝑋 + ( βŠ₯ β€˜π‘‹)) β‰  𝐴 β†’ (𝑋 = 𝑋 ∧ 𝑋 β‰  𝑋)))
3534necon1bd 2957 . 2 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ…)) β†’ (Β¬ (𝑋 = 𝑋 ∧ 𝑋 β‰  𝑋) β†’ (𝑋 + ( βŠ₯ β€˜π‘‹)) = 𝐴))
361, 35mpi 20 1 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋 ∧ 𝑋 β‰  βˆ…)) β†’ (𝑋 + ( βŠ₯ β€˜π‘‹)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  βˆƒwex 1780   ∈ wcel 2105   β‰  wne 2939  βˆƒwrex 3069   βŠ† wss 3948   ⊊ wpss 3949  βˆ…c0 4322  {csn 4628  β€˜cfv 6543  (class class class)co 7412  lecple 17209  joincjn 18269  Atomscatm 38437  HLchlt 38524  +𝑃cpadd 38970  βŠ₯𝑃cpolN 39077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-proset 18253  df-poset 18271  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-p1 18384  df-lat 18390  df-clat 18457  df-oposet 38350  df-ol 38352  df-oml 38353  df-covers 38440  df-ats 38441  df-atl 38472  df-cvlat 38496  df-hlat 38525  df-psubsp 38678  df-pmap 38679  df-padd 38971  df-polarityN 39078  df-psubclN 39110
This theorem is referenced by:  pexmidALTN  39153
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