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Theorem hlmod1i 35926
Description: A version of the modular law pmod1i 35918 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 35433 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
433ad2ant1 1167 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝐾 ∈ Lat)
5 simp21 1267 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝑋𝐵)
6 simp22 1268 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝑌𝐵)
7 hlmod.j . . . . . 6 = (join‘𝐾)
81, 7latjcl 17411 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
94, 5, 6, 8syl3anc 1494 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑋 𝑌) ∈ 𝐵)
10 simp23 1269 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝑍𝐵)
11 hlmod.m . . . . 5 = (meet‘𝐾)
121, 11latmcl 17412 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
134, 9, 10, 12syl3anc 1494 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
141, 11latmcl 17412 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
154, 6, 10, 14syl3anc 1494 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑌 𝑍) ∈ 𝐵)
161, 7latjcl 17411 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
174, 5, 15, 16syl3anc 1494 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
18 simp1 1170 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝐾 ∈ HL)
19 eqid 2825 . . . . . . . . 9 (Atoms‘𝐾) = (Atoms‘𝐾)
20 hlmod.f . . . . . . . . 9 𝐹 = (pmap‘𝐾)
211, 19, 20pmapssat 35829 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝐹𝑋) ⊆ (Atoms‘𝐾))
2218, 5, 21syl2anc 579 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹𝑋) ⊆ (Atoms‘𝐾))
231, 19, 20pmapssat 35829 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑌𝐵) → (𝐹𝑌) ⊆ (Atoms‘𝐾))
2418, 6, 23syl2anc 579 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹𝑌) ⊆ (Atoms‘𝐾))
25 eqid 2825 . . . . . . . . 9 (PSubSp‘𝐾) = (PSubSp‘𝐾)
261, 25, 20pmapsub 35838 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑍𝐵) → (𝐹𝑍) ∈ (PSubSp‘𝐾))
274, 10, 26syl2anc 579 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹𝑍) ∈ (PSubSp‘𝐾))
28 simp3l 1262 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → 𝑋 𝑍)
291, 2, 20pmaple 35831 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍 ↔ (𝐹𝑋) ⊆ (𝐹𝑍)))
3018, 5, 10, 29syl3anc 1494 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑋 𝑍 ↔ (𝐹𝑋) ⊆ (𝐹𝑍)))
3128, 30mpbid 224 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹𝑋) ⊆ (𝐹𝑍))
32 hlmod.p . . . . . . . . 9 + = (+𝑃𝐾)
3319, 25, 32pmod1i 35918 . . . . . . . 8 ((𝐾 ∈ HL ∧ ((𝐹𝑋) ⊆ (Atoms‘𝐾) ∧ (𝐹𝑌) ⊆ (Atoms‘𝐾) ∧ (𝐹𝑍) ∈ (PSubSp‘𝐾))) → ((𝐹𝑋) ⊆ (𝐹𝑍) → (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) = ((𝐹𝑋) + ((𝐹𝑌) ∩ (𝐹𝑍)))))
34333impia 1149 . . . . . . 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 1506 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) = ((𝐹𝑋) + ((𝐹𝑌) ∩ (𝐹𝑍))))
361, 11, 19, 20pmapmeet 35843 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → (𝐹‘((𝑋 𝑌) 𝑍)) = ((𝐹‘(𝑋 𝑌)) ∩ (𝐹𝑍)))
3718, 9, 10, 36syl3anc 1494 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘((𝑋 𝑌) 𝑍)) = ((𝐹‘(𝑋 𝑌)) ∩ (𝐹𝑍)))
38 simp3r 1263 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))
3938ineq1d 4042 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝐹‘(𝑋 𝑌)) ∩ (𝐹𝑍)) = (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)))
4037, 39eqtrd 2861 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘((𝑋 𝑌) 𝑍)) = (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)))
411, 11, 19, 20pmapmeet 35843 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑌𝐵𝑍𝐵) → (𝐹‘(𝑌 𝑍)) = ((𝐹𝑌) ∩ (𝐹𝑍)))
4218, 6, 10, 41syl3anc 1494 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘(𝑌 𝑍)) = ((𝐹𝑌) ∩ (𝐹𝑍)))
4342oveq2d 6926 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝐹𝑋) + (𝐹‘(𝑌 𝑍))) = ((𝐹𝑋) + ((𝐹𝑌) ∩ (𝐹𝑍))))
4435, 40, 433eqtr4d 2871 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘((𝑋 𝑌) 𝑍)) = ((𝐹𝑋) + (𝐹‘(𝑌 𝑍))))
451, 7, 20, 32pmapjoin 35922 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → ((𝐹𝑋) + (𝐹‘(𝑌 𝑍))) ⊆ (𝐹‘(𝑋 (𝑌 𝑍))))
464, 5, 15, 45syl3anc 1494 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝐹𝑋) + (𝐹‘(𝑌 𝑍))) ⊆ (𝐹‘(𝑋 (𝑌 𝑍))))
4744, 46eqsstrd 3864 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝐹‘((𝑋 𝑌) 𝑍)) ⊆ (𝐹‘(𝑋 (𝑌 𝑍))))
481, 2, 20pmaple 35831 . . . . 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 1494 . . . 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 249 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝑋 𝑌) 𝑍) (𝑋 (𝑌 𝑍)))
511, 2, 7, 11mod1ile 17465 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍 → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍)))
52513impia 1149 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑍) → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍))
534, 5, 6, 10, 28, 52syl131anc 1506 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍))
541, 2, 4, 13, 17, 50, 53latasymd 17417 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌)))) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
55543expia 1154 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 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  cin 3797  wss 3798   class class class wbr 4875  cfv 6127  (class class class)co 6910  Basecbs 16229  lecple 16319  joincjn 17304  meetcmee 17305  Latclat 17405  Atomscatm 35333  HLchlt 35420  PSubSpcpsubsp 35566  pmapcpmap 35567  +𝑃cpadd 35865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-proset 17288  df-poset 17306  df-plt 17318  df-lub 17334  df-glb 17335  df-join 17336  df-meet 17337  df-p0 17399  df-lat 17406  df-clat 17468  df-oposet 35246  df-ol 35248  df-oml 35249  df-covers 35336  df-ats 35337  df-atl 35368  df-cvlat 35392  df-hlat 35421  df-psubsp 35573  df-pmap 35574  df-padd 35866
This theorem is referenced by:  atmod1i1  35927  atmod1i2  35929  llnmod1i2  35930
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