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Theorem hlmod1i 36874
Description: A version of the modular law pmod1i 36866 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 36381 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
433ad2ant1 1125 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝐾 ∈ Lat)
5 simp21 1198 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝑋𝐵)
6 simp22 1199 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝑌𝐵)
7 hlmod.j . . . . . 6 = (join‘𝐾)
81, 7latjcl 17651 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
94, 5, 6, 8syl3anc 1363 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑋 𝑌) ∈ 𝐵)
10 simp23 1200 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝑍𝐵)
11 hlmod.m . . . . 5 = (meet‘𝐾)
121, 11latmcl 17652 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
134, 9, 10, 12syl3anc 1363 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
141, 11latmcl 17652 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
154, 6, 10, 14syl3anc 1363 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑌 𝑍) ∈ 𝐵)
161, 7latjcl 17651 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
174, 5, 15, 16syl3anc 1363 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
18 simp1 1128 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝐾 ∈ HL)
19 eqid 2821 . . . . . . . . 9 (Atoms‘𝐾) = (Atoms‘𝐾)
20 hlmod.f . . . . . . . . 9 𝐹 = (pmap‘𝐾)
211, 19, 20pmapssat 36777 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝐹𝑋) ⊆ (Atoms‘𝐾))
2218, 5, 21syl2anc 584 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹𝑋) ⊆ (Atoms‘𝐾))
231, 19, 20pmapssat 36777 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑌𝐵) → (𝐹𝑌) ⊆ (Atoms‘𝐾))
2418, 6, 23syl2anc 584 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹𝑌) ⊆ (Atoms‘𝐾))
25 eqid 2821 . . . . . . . . 9 (PSubSp‘𝐾) = (PSubSp‘𝐾)
261, 25, 20pmapsub 36786 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑍𝐵) → (𝐹𝑍) ∈ (PSubSp‘𝐾))
274, 10, 26syl2anc 584 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹𝑍) ∈ (PSubSp‘𝐾))
28 simp3l 1193 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝑋 𝑍)
291, 2, 20pmaple 36779 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍 ↔ (𝐹𝑋) ⊆ (𝐹𝑍)))
3018, 5, 10, 29syl3anc 1363 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑋 𝑍 ↔ (𝐹𝑋) ⊆ (𝐹𝑍)))
3128, 30mpbid 233 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹𝑋) ⊆ (𝐹𝑍))
32 hlmod.p . . . . . . . . 9 + = (+𝑃𝐾)
3319, 25, 32pmod1i 36866 . . . . . . . 8 ((𝐾 ∈ HL ∧ ((𝐹𝑋) ⊆ (Atoms‘𝐾) ∧ (𝐹𝑌) ⊆ (Atoms‘𝐾) ∧ (𝐹𝑍) ∈ (PSubSp‘𝐾))) → ((𝐹𝑋) ⊆ (𝐹𝑍) → (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) = ((𝐹𝑋) + ((𝐹𝑌) ∩ (𝐹𝑍)))))
34333impia 1109 . . . . . . 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 1375 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) = ((𝐹𝑋) + ((𝐹𝑌) ∩ (𝐹𝑍))))
361, 11, 19, 20pmapmeet 36791 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → (𝐹‘((𝑋 𝑌) 𝑍)) = ((𝐹‘(𝑋 𝑌)) ∩ (𝐹𝑍)))
3718, 9, 10, 36syl3anc 1363 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘((𝑋 𝑌) 𝑍)) = ((𝐹‘(𝑋 𝑌)) ∩ (𝐹𝑍)))
38 simp3r 1194 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))
3938ineq1d 4187 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝐹‘(𝑋 𝑌)) ∩ (𝐹𝑍)) = (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)))
4037, 39eqtrd 2856 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘((𝑋 𝑌) 𝑍)) = (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)))
411, 11, 19, 20pmapmeet 36791 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑌𝐵𝑍𝐵) → (𝐹‘(𝑌 𝑍)) = ((𝐹𝑌) ∩ (𝐹𝑍)))
4218, 6, 10, 41syl3anc 1363 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘(𝑌 𝑍)) = ((𝐹𝑌) ∩ (𝐹𝑍)))
4342oveq2d 7161 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝐹𝑋) + (𝐹‘(𝑌 𝑍))) = ((𝐹𝑋) + ((𝐹𝑌) ∩ (𝐹𝑍))))
4435, 40, 433eqtr4d 2866 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘((𝑋 𝑌) 𝑍)) = ((𝐹𝑋) + (𝐹‘(𝑌 𝑍))))
451, 7, 20, 32pmapjoin 36870 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → ((𝐹𝑋) + (𝐹‘(𝑌 𝑍))) ⊆ (𝐹‘(𝑋 (𝑌 𝑍))))
464, 5, 15, 45syl3anc 1363 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝐹𝑋) + (𝐹‘(𝑌 𝑍))) ⊆ (𝐹‘(𝑋 (𝑌 𝑍))))
4744, 46eqsstrd 4004 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘((𝑋 𝑌) 𝑍)) ⊆ (𝐹‘(𝑋 (𝑌 𝑍))))
481, 2, 20pmaple 36779 . . . . 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 1363 . . . 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 258 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝑋 𝑌) 𝑍) (𝑋 (𝑌 𝑍)))
511, 2, 7, 11mod1ile 17705 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍 → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍)))
52513impia 1109 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑍) → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍))
534, 5, 6, 10, 28, 52syl131anc 1375 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍))
541, 2, 4, 13, 17, 50, 53latasymd 17657 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
55543expia 1113 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  cin 3934  wss 3935   class class class wbr 5058  cfv 6349  (class class class)co 7145  Basecbs 16473  lecple 16562  joincjn 17544  meetcmee 17545  Latclat 17645  Atomscatm 36281  HLchlt 36368  PSubSpcpsubsp 36514  pmapcpmap 36515  +𝑃cpadd 36813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-iun 4914  df-iin 4915  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7680  df-2nd 7681  df-proset 17528  df-poset 17546  df-plt 17558  df-lub 17574  df-glb 17575  df-join 17576  df-meet 17577  df-p0 17639  df-lat 17646  df-clat 17708  df-oposet 36194  df-ol 36196  df-oml 36197  df-covers 36284  df-ats 36285  df-atl 36316  df-cvlat 36340  df-hlat 36369  df-psubsp 36521  df-pmap 36522  df-padd 36814
This theorem is referenced by:  atmod1i1  36875  atmod1i2  36877  llnmod1i2  36878
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