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Theorem dmdsl3 32334
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 32321 . . . . . 6 (((𝐵C𝐴C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶)) → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))
21exp32 420 . . . . 5 ((𝐵C𝐴C𝐶C ) → (𝐵 𝑀* 𝐴 → (𝐴𝐶 → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))))
323com12 1124 . . . 4 ((𝐴C𝐵C𝐶C ) → (𝐵 𝑀* 𝐴 → (𝐴𝐶 → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))))
43imp32 418 . . 3 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶)) → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))
543adantr3 1172 . 2 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶𝐶 ⊆ (𝐴 𝐵))) → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))
6 chjcom 31525 . . . . . 6 ((𝐴C𝐵C ) → (𝐴 𝐵) = (𝐵 𝐴))
76ineq2d 4220 . . . . 5 ((𝐴C𝐵C ) → (𝐶 ∩ (𝐴 𝐵)) = (𝐶 ∩ (𝐵 𝐴)))
873adant3 1133 . . . 4 ((𝐴C𝐵C𝐶C ) → (𝐶 ∩ (𝐴 𝐵)) = (𝐶 ∩ (𝐵 𝐴)))
9 dfss2 3969 . . . . 5 (𝐶 ⊆ (𝐴 𝐵) ↔ (𝐶 ∩ (𝐴 𝐵)) = 𝐶)
109biimpi 216 . . . 4 (𝐶 ⊆ (𝐴 𝐵) → (𝐶 ∩ (𝐴 𝐵)) = 𝐶)
118, 10sylan9req 2798 . . 3 (((𝐴C𝐵C𝐶C ) ∧ 𝐶 ⊆ (𝐴 𝐵)) → (𝐶 ∩ (𝐵 𝐴)) = 𝐶)
12113ad2antr3 1191 . 2 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶𝐶 ⊆ (𝐴 𝐵))) → (𝐶 ∩ (𝐵 𝐴)) = 𝐶)
135, 12eqtrd 2777 1 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶𝐶 ⊆ (𝐴 𝐵))) → ((𝐶𝐵) ∨ 𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  cin 3950  wss 3951   class class class wbr 5143  (class class class)co 7431   C cch 30948   chj 30952   𝑀* cdmd 30986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-hilex 31018
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-sh 31226  df-ch 31240  df-chj 31329  df-dmd 32300
This theorem is referenced by:  mdslle1i  32336  mdslj1i  32338  mdslj2i  32339  mdslmd1lem1  32344
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