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Theorem dmdsl3 30250
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 30237 . . . . . 6 (((𝐵C𝐴C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶)) → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))
21exp32 424 . . . . 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 422 . . 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 29441 . . . . . 6 ((𝐴C𝐵C ) → (𝐴 𝐵) = (𝐵 𝐴))
76ineq2d 4104 . . . . 5 ((𝐴C𝐵C ) → (𝐶 ∩ (𝐴 𝐵)) = (𝐶 ∩ (𝐵 𝐴)))
873adant3 1133 . . . 4 ((𝐴C𝐵C𝐶C ) → (𝐶 ∩ (𝐴 𝐵)) = (𝐶 ∩ (𝐵 𝐴)))
9 df-ss 3861 . . . . 5 (𝐶 ⊆ (𝐴 𝐵) ↔ (𝐶 ∩ (𝐴 𝐵)) = 𝐶)
109biimpi 219 . . . 4 (𝐶 ⊆ (𝐴 𝐵) → (𝐶 ∩ (𝐴 𝐵)) = 𝐶)
118, 10sylan9req 2794 . . 3 (((𝐴C𝐵C𝐶C ) ∧ 𝐶 ⊆ (𝐴 𝐵)) → (𝐶 ∩ (𝐵 𝐴)) = 𝐶)
12113ad2antr3 1191 . 2 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶𝐶 ⊆ (𝐴 𝐵))) → (𝐶 ∩ (𝐵 𝐴)) = 𝐶)
135, 12eqtrd 2773 1 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶𝐶 ⊆ (𝐴 𝐵))) → ((𝐶𝐵) ∨ 𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2113  cin 3843  wss 3844   class class class wbr 5031  (class class class)co 7171   C cch 28864   chj 28868   𝑀* cdmd 28902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-hilex 28934
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3683  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fv 6348  df-ov 7174  df-oprab 7175  df-mpo 7176  df-sh 29142  df-ch 29156  df-chj 29245  df-dmd 30216
This theorem is referenced by:  mdslle1i  30252  mdslj1i  30254  mdslj2i  30255  mdslmd1lem1  30260
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