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Theorem hlmod1i 33963
Description: A version of the modular law pmod1i 33955 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 33471 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
433ad2ant1 1074 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝐾 ∈ Lat)
5 simp21 1086 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝑋𝐵)
6 simp22 1087 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝑌𝐵)
7 hlmod.j . . . . . 6 = (join‘𝐾)
81, 7latjcl 16820 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
94, 5, 6, 8syl3anc 1317 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑋 𝑌) ∈ 𝐵)
10 simp23 1088 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝑍𝐵)
11 hlmod.m . . . . 5 = (meet‘𝐾)
121, 11latmcl 16821 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
134, 9, 10, 12syl3anc 1317 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
141, 11latmcl 16821 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
154, 6, 10, 14syl3anc 1317 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑌 𝑍) ∈ 𝐵)
161, 7latjcl 16820 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
174, 5, 15, 16syl3anc 1317 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
18 simp1 1053 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝐾 ∈ HL)
19 eqid 2609 . . . . . . . . 9 (Atoms‘𝐾) = (Atoms‘𝐾)
20 hlmod.f . . . . . . . . 9 𝐹 = (pmap‘𝐾)
211, 19, 20pmapssat 33866 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝐹𝑋) ⊆ (Atoms‘𝐾))
2218, 5, 21syl2anc 690 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹𝑋) ⊆ (Atoms‘𝐾))
231, 19, 20pmapssat 33866 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑌𝐵) → (𝐹𝑌) ⊆ (Atoms‘𝐾))
2418, 6, 23syl2anc 690 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹𝑌) ⊆ (Atoms‘𝐾))
25 eqid 2609 . . . . . . . . 9 (PSubSp‘𝐾) = (PSubSp‘𝐾)
261, 25, 20pmapsub 33875 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑍𝐵) → (𝐹𝑍) ∈ (PSubSp‘𝐾))
274, 10, 26syl2anc 690 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹𝑍) ∈ (PSubSp‘𝐾))
28 simp3l 1081 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝑋 𝑍)
291, 2, 20pmaple 33868 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍 ↔ (𝐹𝑋) ⊆ (𝐹𝑍)))
3018, 5, 10, 29syl3anc 1317 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑋 𝑍 ↔ (𝐹𝑋) ⊆ (𝐹𝑍)))
3128, 30mpbid 220 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹𝑋) ⊆ (𝐹𝑍))
32 hlmod.p . . . . . . . . 9 + = (+𝑃𝐾)
3319, 25, 32pmod1i 33955 . . . . . . . 8 ((𝐾 ∈ HL ∧ ((𝐹𝑋) ⊆ (Atoms‘𝐾) ∧ (𝐹𝑌) ⊆ (Atoms‘𝐾) ∧ (𝐹𝑍) ∈ (PSubSp‘𝐾))) → ((𝐹𝑋) ⊆ (𝐹𝑍) → (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) = ((𝐹𝑋) + ((𝐹𝑌) ∩ (𝐹𝑍)))))
34333impia 1252 . . . . . . 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 1330 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) = ((𝐹𝑋) + ((𝐹𝑌) ∩ (𝐹𝑍))))
361, 11, 19, 20pmapmeet 33880 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → (𝐹‘((𝑋 𝑌) 𝑍)) = ((𝐹‘(𝑋 𝑌)) ∩ (𝐹𝑍)))
3718, 9, 10, 36syl3anc 1317 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘((𝑋 𝑌) 𝑍)) = ((𝐹‘(𝑋 𝑌)) ∩ (𝐹𝑍)))
38 simp3r 1082 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))
3938ineq1d 3774 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝐹‘(𝑋 𝑌)) ∩ (𝐹𝑍)) = (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)))
4037, 39eqtrd 2643 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘((𝑋 𝑌) 𝑍)) = (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)))
411, 11, 19, 20pmapmeet 33880 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑌𝐵𝑍𝐵) → (𝐹‘(𝑌 𝑍)) = ((𝐹𝑌) ∩ (𝐹𝑍)))
4218, 6, 10, 41syl3anc 1317 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘(𝑌 𝑍)) = ((𝐹𝑌) ∩ (𝐹𝑍)))
4342oveq2d 6543 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝐹𝑋) + (𝐹‘(𝑌 𝑍))) = ((𝐹𝑋) + ((𝐹𝑌) ∩ (𝐹𝑍))))
4435, 40, 433eqtr4d 2653 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘((𝑋 𝑌) 𝑍)) = ((𝐹𝑋) + (𝐹‘(𝑌 𝑍))))
451, 7, 20, 32pmapjoin 33959 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → ((𝐹𝑋) + (𝐹‘(𝑌 𝑍))) ⊆ (𝐹‘(𝑋 (𝑌 𝑍))))
464, 5, 15, 45syl3anc 1317 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝐹𝑋) + (𝐹‘(𝑌 𝑍))) ⊆ (𝐹‘(𝑋 (𝑌 𝑍))))
4744, 46eqsstrd 3601 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘((𝑋 𝑌) 𝑍)) ⊆ (𝐹‘(𝑋 (𝑌 𝑍))))
481, 2, 20pmaple 33868 . . . . 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 1317 . . . 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 245 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝑋 𝑌) 𝑍) (𝑋 (𝑌 𝑍)))
511, 2, 7, 11mod1ile 16874 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍 → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍)))
52513impia 1252 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑍) → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍))
534, 5, 6, 10, 28, 52syl131anc 1330 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍))
541, 2, 4, 13, 17, 50, 53latasymd 16826 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
55543expia 1258 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  cin 3538  wss 3539   class class class wbr 4577  cfv 5790  (class class class)co 6527  Basecbs 15641  lecple 15721  joincjn 16713  meetcmee 16714  Latclat 16814  Atomscatm 33371  HLchlt 33458  PSubSpcpsubsp 33603  pmapcpmap 33604  +𝑃cpadd 33902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-preset 16697  df-poset 16715  df-plt 16727  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-p0 16808  df-lat 16815  df-clat 16877  df-oposet 33284  df-ol 33286  df-oml 33287  df-covers 33374  df-ats 33375  df-atl 33406  df-cvlat 33430  df-hlat 33459  df-psubsp 33610  df-pmap 33611  df-padd 33903
This theorem is referenced by:  atmod1i1  33964  atmod1i2  33966  llnmod1i2  33967
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