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Theorem hlmod1i 37643
Description: A version of the modular law pmod1i 37635 that holds in a Hilbert lattice. (Contributed by NM, 13-May-2012.)
Hypotheses
Ref Expression
hlmod.b 𝐵 = (Base‘𝐾)
hlmod.l = (le‘𝐾)
hlmod.j = (join‘𝐾)
hlmod.m = (meet‘𝐾)
hlmod.f 𝐹 = (pmap‘𝐾)
hlmod.p + = (+𝑃𝐾)
Assertion
Ref Expression
hlmod1i ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌))) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))

Proof of Theorem hlmod1i
StepHypRef Expression
1 hlmod.b . . 3 𝐵 = (Base‘𝐾)
2 hlmod.l . . 3 = (le‘𝐾)
3 hllat 37150 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
433ad2ant1 1135 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝐾 ∈ Lat)
5 simp21 1208 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝑋𝐵)
6 simp22 1209 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝑌𝐵)
7 hlmod.j . . . . . 6 = (join‘𝐾)
81, 7latjcl 17977 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
94, 5, 6, 8syl3anc 1373 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑋 𝑌) ∈ 𝐵)
10 simp23 1210 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝑍𝐵)
11 hlmod.m . . . . 5 = (meet‘𝐾)
121, 11latmcl 17978 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
134, 9, 10, 12syl3anc 1373 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
141, 11latmcl 17978 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
154, 6, 10, 14syl3anc 1373 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑌 𝑍) ∈ 𝐵)
161, 7latjcl 17977 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
174, 5, 15, 16syl3anc 1373 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
18 simp1 1138 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝐾 ∈ HL)
19 eqid 2739 . . . . . . . . 9 (Atoms‘𝐾) = (Atoms‘𝐾)
20 hlmod.f . . . . . . . . 9 𝐹 = (pmap‘𝐾)
211, 19, 20pmapssat 37546 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝐹𝑋) ⊆ (Atoms‘𝐾))
2218, 5, 21syl2anc 587 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹𝑋) ⊆ (Atoms‘𝐾))
231, 19, 20pmapssat 37546 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑌𝐵) → (𝐹𝑌) ⊆ (Atoms‘𝐾))
2418, 6, 23syl2anc 587 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹𝑌) ⊆ (Atoms‘𝐾))
25 eqid 2739 . . . . . . . . 9 (PSubSp‘𝐾) = (PSubSp‘𝐾)
261, 25, 20pmapsub 37555 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑍𝐵) → (𝐹𝑍) ∈ (PSubSp‘𝐾))
274, 10, 26syl2anc 587 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹𝑍) ∈ (PSubSp‘𝐾))
28 simp3l 1203 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝑋 𝑍)
291, 2, 20pmaple 37548 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍 ↔ (𝐹𝑋) ⊆ (𝐹𝑍)))
3018, 5, 10, 29syl3anc 1373 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑋 𝑍 ↔ (𝐹𝑋) ⊆ (𝐹𝑍)))
3128, 30mpbid 235 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹𝑋) ⊆ (𝐹𝑍))
32 hlmod.p . . . . . . . . 9 + = (+𝑃𝐾)
3319, 25, 32pmod1i 37635 . . . . . . . 8 ((𝐾 ∈ HL ∧ ((𝐹𝑋) ⊆ (Atoms‘𝐾) ∧ (𝐹𝑌) ⊆ (Atoms‘𝐾) ∧ (𝐹𝑍) ∈ (PSubSp‘𝐾))) → ((𝐹𝑋) ⊆ (𝐹𝑍) → (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) = ((𝐹𝑋) + ((𝐹𝑌) ∩ (𝐹𝑍)))))
34333impia 1119 . . . . . . 7 ((𝐾 ∈ HL ∧ ((𝐹𝑋) ⊆ (Atoms‘𝐾) ∧ (𝐹𝑌) ⊆ (Atoms‘𝐾) ∧ (𝐹𝑍) ∈ (PSubSp‘𝐾)) ∧ (𝐹𝑋) ⊆ (𝐹𝑍)) → (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) = ((𝐹𝑋) + ((𝐹𝑌) ∩ (𝐹𝑍))))
3518, 22, 24, 27, 31, 34syl131anc 1385 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) = ((𝐹𝑋) + ((𝐹𝑌) ∩ (𝐹𝑍))))
361, 11, 19, 20pmapmeet 37560 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → (𝐹‘((𝑋 𝑌) 𝑍)) = ((𝐹‘(𝑋 𝑌)) ∩ (𝐹𝑍)))
3718, 9, 10, 36syl3anc 1373 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘((𝑋 𝑌) 𝑍)) = ((𝐹‘(𝑋 𝑌)) ∩ (𝐹𝑍)))
38 simp3r 1204 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))
3938ineq1d 4142 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝐹‘(𝑋 𝑌)) ∩ (𝐹𝑍)) = (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)))
4037, 39eqtrd 2779 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘((𝑋 𝑌) 𝑍)) = (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)))
411, 11, 19, 20pmapmeet 37560 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑌𝐵𝑍𝐵) → (𝐹‘(𝑌 𝑍)) = ((𝐹𝑌) ∩ (𝐹𝑍)))
4218, 6, 10, 41syl3anc 1373 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘(𝑌 𝑍)) = ((𝐹𝑌) ∩ (𝐹𝑍)))
4342oveq2d 7250 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝐹𝑋) + (𝐹‘(𝑌 𝑍))) = ((𝐹𝑋) + ((𝐹𝑌) ∩ (𝐹𝑍))))
4435, 40, 433eqtr4d 2789 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘((𝑋 𝑌) 𝑍)) = ((𝐹𝑋) + (𝐹‘(𝑌 𝑍))))
451, 7, 20, 32pmapjoin 37639 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → ((𝐹𝑋) + (𝐹‘(𝑌 𝑍))) ⊆ (𝐹‘(𝑋 (𝑌 𝑍))))
464, 5, 15, 45syl3anc 1373 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝐹𝑋) + (𝐹‘(𝑌 𝑍))) ⊆ (𝐹‘(𝑋 (𝑌 𝑍))))
4744, 46eqsstrd 3955 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘((𝑋 𝑌) 𝑍)) ⊆ (𝐹‘(𝑋 (𝑌 𝑍))))
481, 2, 20pmaple 37548 . . . . 5 ((𝐾 ∈ HL ∧ ((𝑋 𝑌) 𝑍) ∈ 𝐵 ∧ (𝑋 (𝑌 𝑍)) ∈ 𝐵) → (((𝑋 𝑌) 𝑍) (𝑋 (𝑌 𝑍)) ↔ (𝐹‘((𝑋 𝑌) 𝑍)) ⊆ (𝐹‘(𝑋 (𝑌 𝑍)))))
4918, 13, 17, 48syl3anc 1373 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (((𝑋 𝑌) 𝑍) (𝑋 (𝑌 𝑍)) ↔ (𝐹‘((𝑋 𝑌) 𝑍)) ⊆ (𝐹‘(𝑋 (𝑌 𝑍)))))
5047, 49mpbird 260 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝑋 𝑌) 𝑍) (𝑋 (𝑌 𝑍)))
511, 2, 7, 11mod1ile 18031 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍 → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍)))
52513impia 1119 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑍) → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍))
534, 5, 6, 10, 28, 52syl131anc 1385 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍))
541, 2, 4, 13, 17, 50, 53latasymd 17983 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
55543expia 1123 1 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌))) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2112  cin 3882  wss 3883   class class class wbr 5069  cfv 6400  (class class class)co 7234  Basecbs 16792  lecple 16841  joincjn 17850  meetcmee 17851  Latclat 17969  Atomscatm 37050  HLchlt 37137  PSubSpcpsubsp 37283  pmapcpmap 37284  +𝑃cpadd 37582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-rep 5195  ax-sep 5208  ax-nul 5215  ax-pow 5274  ax-pr 5338  ax-un 7544
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4456  df-pw 4531  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-iun 4922  df-iin 4923  df-br 5070  df-opab 5132  df-mpt 5152  df-id 5471  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-f1 6405  df-fo 6406  df-f1o 6407  df-fv 6408  df-riota 7191  df-ov 7237  df-oprab 7238  df-mpo 7239  df-1st 7782  df-2nd 7783  df-proset 17834  df-poset 17852  df-plt 17868  df-lub 17884  df-glb 17885  df-join 17886  df-meet 17887  df-p0 17963  df-lat 17970  df-clat 18037  df-oposet 36963  df-ol 36965  df-oml 36966  df-covers 37053  df-ats 37054  df-atl 37085  df-cvlat 37109  df-hlat 37138  df-psubsp 37290  df-pmap 37291  df-padd 37583
This theorem is referenced by:  atmod1i1  37644  atmod1i2  37646  llnmod1i2  37647
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