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Theorem pl42lem2N 37154
 Description: Lemma for pl42N 37157. (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 1188 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝐾 ∈ HL)
21hllatd 36538 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝐾 ∈ Lat)
3 simpl2 1189 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑋𝐵)
4 simpl3 1190 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑌𝐵)
5 pl42lem.b . . . . . . 7 𝐵 = (Base‘𝐾)
6 pl42lem.j . . . . . . 7 = (join‘𝐾)
75, 6latjcl 17639 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
82, 3, 4, 7syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝑋 𝑌) ∈ 𝐵)
9 eqid 2821 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
10 pl42lem.f . . . . . 6 𝐹 = (pmap‘𝐾)
115, 9, 10pmapssat 36933 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 𝑌) ∈ 𝐵) → (𝐹‘(𝑋 𝑌)) ⊆ (Atoms‘𝐾))
121, 8, 11syl2anc 587 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹‘(𝑋 𝑌)) ⊆ (Atoms‘𝐾))
13 simpr2 1192 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑊𝐵)
145, 6latjcl 17639 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
152, 3, 13, 14syl3anc 1368 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝑋 𝑊) ∈ 𝐵)
16 simpr3 1193 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑉𝐵)
175, 6latjcl 17639 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑉𝐵) → (𝑌 𝑉) ∈ 𝐵)
182, 4, 16, 17syl3anc 1368 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝑌 𝑉) ∈ 𝐵)
19 pl42lem.m . . . . . . 7 = (meet‘𝐾)
205, 19latmcl 17640 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑋 𝑊) ∈ 𝐵 ∧ (𝑌 𝑉) ∈ 𝐵) → ((𝑋 𝑊) (𝑌 𝑉)) ∈ 𝐵)
212, 15, 18, 20syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝑋 𝑊) (𝑌 𝑉)) ∈ 𝐵)
225, 9, 10pmapssat 36933 . . . . 5 ((𝐾 ∈ HL ∧ ((𝑋 𝑊) (𝑌 𝑉)) ∈ 𝐵) → (𝐹‘((𝑋 𝑊) (𝑌 𝑉))) ⊆ (Atoms‘𝐾))
231, 21, 22syl2anc 587 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹‘((𝑋 𝑊) (𝑌 𝑉))) ⊆ (Atoms‘𝐾))
241, 12, 233jca 1125 . . 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 37026 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝐹𝑋) + (𝐹𝑌)) ⊆ (𝐹‘(𝑋 𝑌)))
272, 3, 4, 26syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝐹𝑋) + (𝐹𝑌)) ⊆ (𝐹‘(𝑋 𝑌)))
285, 6, 10, 25pmapjoin 37026 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → ((𝐹𝑋) + (𝐹𝑊)) ⊆ (𝐹‘(𝑋 𝑊)))
292, 3, 13, 28syl3anc 1368 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝐹𝑋) + (𝐹𝑊)) ⊆ (𝐹‘(𝑋 𝑊)))
305, 6, 10, 25pmapjoin 37026 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑉𝐵) → ((𝐹𝑌) + (𝐹𝑉)) ⊆ (𝐹‘(𝑌 𝑉)))
312, 4, 16, 30syl3anc 1368 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝐹𝑌) + (𝐹𝑉)) ⊆ (𝐹‘(𝑌 𝑉)))
32 ss2in 4188 . . . . . 6 ((((𝐹𝑋) + (𝐹𝑊)) ⊆ (𝐹‘(𝑋 𝑊)) ∧ ((𝐹𝑌) + (𝐹𝑉)) ⊆ (𝐹‘(𝑌 𝑉))) → (((𝐹𝑋) + (𝐹𝑊)) ∩ ((𝐹𝑌) + (𝐹𝑉))) ⊆ ((𝐹‘(𝑋 𝑊)) ∩ (𝐹‘(𝑌 𝑉))))
3329, 31, 32syl2anc 587 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (((𝐹𝑋) + (𝐹𝑊)) ∩ ((𝐹𝑌) + (𝐹𝑉))) ⊆ ((𝐹‘(𝑋 𝑊)) ∩ (𝐹‘(𝑌 𝑉))))
345, 19, 9, 10pmapmeet 36947 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 𝑊) ∈ 𝐵 ∧ (𝑌 𝑉) ∈ 𝐵) → (𝐹‘((𝑋 𝑊) (𝑌 𝑉))) = ((𝐹‘(𝑋 𝑊)) ∩ (𝐹‘(𝑌 𝑉))))
351, 15, 18, 34syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹‘((𝑋 𝑊) (𝑌 𝑉))) = ((𝐹‘(𝑋 𝑊)) ∩ (𝐹‘(𝑌 𝑉))))
3633, 35sseqtrrd 3984 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (((𝐹𝑋) + (𝐹𝑊)) ∩ ((𝐹𝑌) + (𝐹𝑉))) ⊆ (𝐹‘((𝑋 𝑊) (𝑌 𝑉))))
3727, 36jca 515 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (((𝐹𝑋) + (𝐹𝑌)) ⊆ (𝐹‘(𝑋 𝑌)) ∧ (((𝐹𝑋) + (𝐹𝑊)) ∩ ((𝐹𝑌) + (𝐹𝑉))) ⊆ (𝐹‘((𝑋 𝑊) (𝑌 𝑉)))))
389, 25paddss12 36993 . . 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 37026 . . 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 1368 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝐹‘(𝑋 𝑌)) + (𝐹‘((𝑋 𝑊) (𝑌 𝑉)))) ⊆ (𝐹‘((𝑋 𝑌) ((𝑋 𝑊) (𝑌 𝑉)))))
4239, 41sstrd 3953 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   ∩ cin 3909   ⊆ wss 3910  ‘cfv 6328  (class class class)co 7130  Basecbs 16461  lecple 16550  occoc 16551  joincjn 17532  meetcmee 17533  Latclat 17633  Atomscatm 36437  HLchlt 36524  pmapcpmap 36671  +𝑃cpadd 36969 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-poset 17534  df-lub 17562  df-glb 17563  df-join 17564  df-meet 17565  df-lat 17634  df-clat 17696  df-ats 36441  df-atl 36472  df-cvlat 36496  df-hlat 36525  df-pmap 36678  df-padd 36970 This theorem is referenced by:  pl42lem4N  37156
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