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Theorem dmdsl3 32344
Description: Sublattice mapping for a dual-modular pair. Part of Theorem 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 26-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
dmdsl3 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶𝐶 ⊆ (𝐴 𝐵))) → ((𝐶𝐵) ∨ 𝐴) = 𝐶)

Proof of Theorem dmdsl3
StepHypRef Expression
1 dmdi 32331 . . . . . 6 (((𝐵C𝐴C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶)) → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))
21exp32 420 . . . . 5 ((𝐵C𝐴C𝐶C ) → (𝐵 𝑀* 𝐴 → (𝐴𝐶 → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))))
323com12 1122 . . . 4 ((𝐴C𝐵C𝐶C ) → (𝐵 𝑀* 𝐴 → (𝐴𝐶 → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))))
43imp32 418 . . 3 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶)) → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))
543adantr3 1170 . 2 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶𝐶 ⊆ (𝐴 𝐵))) → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))
6 chjcom 31535 . . . . . 6 ((𝐴C𝐵C ) → (𝐴 𝐵) = (𝐵 𝐴))
76ineq2d 4228 . . . . 5 ((𝐴C𝐵C ) → (𝐶 ∩ (𝐴 𝐵)) = (𝐶 ∩ (𝐵 𝐴)))
873adant3 1131 . . . 4 ((𝐴C𝐵C𝐶C ) → (𝐶 ∩ (𝐴 𝐵)) = (𝐶 ∩ (𝐵 𝐴)))
9 dfss2 3981 . . . . 5 (𝐶 ⊆ (𝐴 𝐵) ↔ (𝐶 ∩ (𝐴 𝐵)) = 𝐶)
109biimpi 216 . . . 4 (𝐶 ⊆ (𝐴 𝐵) → (𝐶 ∩ (𝐴 𝐵)) = 𝐶)
118, 10sylan9req 2796 . . 3 (((𝐴C𝐵C𝐶C ) ∧ 𝐶 ⊆ (𝐴 𝐵)) → (𝐶 ∩ (𝐵 𝐴)) = 𝐶)
12113ad2antr3 1189 . 2 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶𝐶 ⊆ (𝐴 𝐵))) → (𝐶 ∩ (𝐵 𝐴)) = 𝐶)
135, 12eqtrd 2775 1 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶𝐶 ⊆ (𝐴 𝐵))) → ((𝐶𝐵) ∨ 𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  cin 3962  wss 3963   class class class wbr 5148  (class class class)co 7431   C cch 30958   chj 30962   𝑀* cdmd 30996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-hilex 31028
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-sh 31236  df-ch 31250  df-chj 31339  df-dmd 32310
This theorem is referenced by:  mdslle1i  32346  mdslj1i  32348  mdslj2i  32349  mdslmd1lem1  32354
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