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Theorem pl42lem2N 36001
Description: Lemma for pl42N 36004. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pl42lem.b 𝐵 = (Base‘𝐾)
pl42lem.l = (le‘𝐾)
pl42lem.j = (join‘𝐾)
pl42lem.m = (meet‘𝐾)
pl42lem.o = (oc‘𝐾)
pl42lem.f 𝐹 = (pmap‘𝐾)
pl42lem.p + = (+𝑃𝐾)
Assertion
Ref Expression
pl42lem2N (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (((𝐹𝑋) + (𝐹𝑌)) + (((𝐹𝑋) + (𝐹𝑊)) ∩ ((𝐹𝑌) + (𝐹𝑉)))) ⊆ (𝐹‘((𝑋 𝑌) ((𝑋 𝑊) (𝑌 𝑉)))))

Proof of Theorem pl42lem2N
StepHypRef Expression
1 simpl1 1243 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝐾 ∈ HL)
21hllatd 35385 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝐾 ∈ Lat)
3 simpl2 1245 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑋𝐵)
4 simpl3 1247 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑌𝐵)
5 pl42lem.b . . . . . . 7 𝐵 = (Base‘𝐾)
6 pl42lem.j . . . . . . 7 = (join‘𝐾)
75, 6latjcl 17366 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
82, 3, 4, 7syl3anc 1491 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝑋 𝑌) ∈ 𝐵)
9 eqid 2799 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
10 pl42lem.f . . . . . 6 𝐹 = (pmap‘𝐾)
115, 9, 10pmapssat 35780 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 𝑌) ∈ 𝐵) → (𝐹‘(𝑋 𝑌)) ⊆ (Atoms‘𝐾))
121, 8, 11syl2anc 580 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹‘(𝑋 𝑌)) ⊆ (Atoms‘𝐾))
13 simpr2 1251 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑊𝐵)
145, 6latjcl 17366 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
152, 3, 13, 14syl3anc 1491 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝑋 𝑊) ∈ 𝐵)
16 simpr3 1253 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑉𝐵)
175, 6latjcl 17366 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑉𝐵) → (𝑌 𝑉) ∈ 𝐵)
182, 4, 16, 17syl3anc 1491 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝑌 𝑉) ∈ 𝐵)
19 pl42lem.m . . . . . . 7 = (meet‘𝐾)
205, 19latmcl 17367 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑋 𝑊) ∈ 𝐵 ∧ (𝑌 𝑉) ∈ 𝐵) → ((𝑋 𝑊) (𝑌 𝑉)) ∈ 𝐵)
212, 15, 18, 20syl3anc 1491 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝑋 𝑊) (𝑌 𝑉)) ∈ 𝐵)
225, 9, 10pmapssat 35780 . . . . 5 ((𝐾 ∈ HL ∧ ((𝑋 𝑊) (𝑌 𝑉)) ∈ 𝐵) → (𝐹‘((𝑋 𝑊) (𝑌 𝑉))) ⊆ (Atoms‘𝐾))
231, 21, 22syl2anc 580 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹‘((𝑋 𝑊) (𝑌 𝑉))) ⊆ (Atoms‘𝐾))
241, 12, 233jca 1159 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐾 ∈ HL ∧ (𝐹‘(𝑋 𝑌)) ⊆ (Atoms‘𝐾) ∧ (𝐹‘((𝑋 𝑊) (𝑌 𝑉))) ⊆ (Atoms‘𝐾)))
25 pl42lem.p . . . . . 6 + = (+𝑃𝐾)
265, 6, 10, 25pmapjoin 35873 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝐹𝑋) + (𝐹𝑌)) ⊆ (𝐹‘(𝑋 𝑌)))
272, 3, 4, 26syl3anc 1491 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝐹𝑋) + (𝐹𝑌)) ⊆ (𝐹‘(𝑋 𝑌)))
285, 6, 10, 25pmapjoin 35873 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → ((𝐹𝑋) + (𝐹𝑊)) ⊆ (𝐹‘(𝑋 𝑊)))
292, 3, 13, 28syl3anc 1491 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝐹𝑋) + (𝐹𝑊)) ⊆ (𝐹‘(𝑋 𝑊)))
305, 6, 10, 25pmapjoin 35873 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑉𝐵) → ((𝐹𝑌) + (𝐹𝑉)) ⊆ (𝐹‘(𝑌 𝑉)))
312, 4, 16, 30syl3anc 1491 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝐹𝑌) + (𝐹𝑉)) ⊆ (𝐹‘(𝑌 𝑉)))
32 ss2in 4036 . . . . . 6 ((((𝐹𝑋) + (𝐹𝑊)) ⊆ (𝐹‘(𝑋 𝑊)) ∧ ((𝐹𝑌) + (𝐹𝑉)) ⊆ (𝐹‘(𝑌 𝑉))) → (((𝐹𝑋) + (𝐹𝑊)) ∩ ((𝐹𝑌) + (𝐹𝑉))) ⊆ ((𝐹‘(𝑋 𝑊)) ∩ (𝐹‘(𝑌 𝑉))))
3329, 31, 32syl2anc 580 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (((𝐹𝑋) + (𝐹𝑊)) ∩ ((𝐹𝑌) + (𝐹𝑉))) ⊆ ((𝐹‘(𝑋 𝑊)) ∩ (𝐹‘(𝑌 𝑉))))
345, 19, 9, 10pmapmeet 35794 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 𝑊) ∈ 𝐵 ∧ (𝑌 𝑉) ∈ 𝐵) → (𝐹‘((𝑋 𝑊) (𝑌 𝑉))) = ((𝐹‘(𝑋 𝑊)) ∩ (𝐹‘(𝑌 𝑉))))
351, 15, 18, 34syl3anc 1491 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹‘((𝑋 𝑊) (𝑌 𝑉))) = ((𝐹‘(𝑋 𝑊)) ∩ (𝐹‘(𝑌 𝑉))))
3633, 35sseqtr4d 3838 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (((𝐹𝑋) + (𝐹𝑊)) ∩ ((𝐹𝑌) + (𝐹𝑉))) ⊆ (𝐹‘((𝑋 𝑊) (𝑌 𝑉))))
3727, 36jca 508 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (((𝐹𝑋) + (𝐹𝑌)) ⊆ (𝐹‘(𝑋 𝑌)) ∧ (((𝐹𝑋) + (𝐹𝑊)) ∩ ((𝐹𝑌) + (𝐹𝑉))) ⊆ (𝐹‘((𝑋 𝑊) (𝑌 𝑉)))))
389, 25paddss12 35840 . . 3 ((𝐾 ∈ HL ∧ (𝐹‘(𝑋 𝑌)) ⊆ (Atoms‘𝐾) ∧ (𝐹‘((𝑋 𝑊) (𝑌 𝑉))) ⊆ (Atoms‘𝐾)) → ((((𝐹𝑋) + (𝐹𝑌)) ⊆ (𝐹‘(𝑋 𝑌)) ∧ (((𝐹𝑋) + (𝐹𝑊)) ∩ ((𝐹𝑌) + (𝐹𝑉))) ⊆ (𝐹‘((𝑋 𝑊) (𝑌 𝑉)))) → (((𝐹𝑋) + (𝐹𝑌)) + (((𝐹𝑋) + (𝐹𝑊)) ∩ ((𝐹𝑌) + (𝐹𝑉)))) ⊆ ((𝐹‘(𝑋 𝑌)) + (𝐹‘((𝑋 𝑊) (𝑌 𝑉))))))
3924, 37, 38sylc 65 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (((𝐹𝑋) + (𝐹𝑌)) + (((𝐹𝑋) + (𝐹𝑊)) ∩ ((𝐹𝑌) + (𝐹𝑉)))) ⊆ ((𝐹‘(𝑋 𝑌)) + (𝐹‘((𝑋 𝑊) (𝑌 𝑉)))))
405, 6, 10, 25pmapjoin 35873 . . 3 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵 ∧ ((𝑋 𝑊) (𝑌 𝑉)) ∈ 𝐵) → ((𝐹‘(𝑋 𝑌)) + (𝐹‘((𝑋 𝑊) (𝑌 𝑉)))) ⊆ (𝐹‘((𝑋 𝑌) ((𝑋 𝑊) (𝑌 𝑉)))))
412, 8, 21, 40syl3anc 1491 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝐹‘(𝑋 𝑌)) + (𝐹‘((𝑋 𝑊) (𝑌 𝑉)))) ⊆ (𝐹‘((𝑋 𝑌) ((𝑋 𝑊) (𝑌 𝑉)))))
4239, 41sstrd 3808 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (((𝐹𝑋) + (𝐹𝑌)) + (((𝐹𝑋) + (𝐹𝑊)) ∩ ((𝐹𝑌) + (𝐹𝑉)))) ⊆ (𝐹‘((𝑋 𝑌) ((𝑋 𝑊) (𝑌 𝑉)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  cin 3768  wss 3769  cfv 6101  (class class class)co 6878  Basecbs 16184  lecple 16274  occoc 16275  joincjn 17259  meetcmee 17260  Latclat 17360  Atomscatm 35284  HLchlt 35371  pmapcpmap 35518  +𝑃cpadd 35816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-iin 4713  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-poset 17261  df-lub 17289  df-glb 17290  df-join 17291  df-meet 17292  df-lat 17361  df-clat 17423  df-ats 35288  df-atl 35319  df-cvlat 35343  df-hlat 35372  df-pmap 35525  df-padd 35817
This theorem is referenced by:  pl42lem4N  36003
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