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Theorem dmdsl3 28360
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 28347 . . . . . 6 (((𝐵C𝐴C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶)) → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))
21exp32 628 . . . . 5 ((𝐵C𝐴C𝐶C ) → (𝐵 𝑀* 𝐴 → (𝐴𝐶 → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))))
323com12 1260 . . . 4 ((𝐴C𝐵C𝐶C ) → (𝐵 𝑀* 𝐴 → (𝐴𝐶 → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))))
43imp32 447 . . 3 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶)) → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))
543adantr3 1214 . 2 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶𝐶 ⊆ (𝐴 𝐵))) → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))
6 chjcom 27551 . . . . . 6 ((𝐴C𝐵C ) → (𝐴 𝐵) = (𝐵 𝐴))
76ineq2d 3771 . . . . 5 ((𝐴C𝐵C ) → (𝐶 ∩ (𝐴 𝐵)) = (𝐶 ∩ (𝐵 𝐴)))
873adant3 1073 . . . 4 ((𝐴C𝐵C𝐶C ) → (𝐶 ∩ (𝐴 𝐵)) = (𝐶 ∩ (𝐵 𝐴)))
9 df-ss 3549 . . . . 5 (𝐶 ⊆ (𝐴 𝐵) ↔ (𝐶 ∩ (𝐴 𝐵)) = 𝐶)
109biimpi 204 . . . 4 (𝐶 ⊆ (𝐴 𝐵) → (𝐶 ∩ (𝐴 𝐵)) = 𝐶)
118, 10sylan9req 2660 . . 3 (((𝐴C𝐵C𝐶C ) ∧ 𝐶 ⊆ (𝐴 𝐵)) → (𝐶 ∩ (𝐵 𝐴)) = 𝐶)
12113ad2antr3 1220 . 2 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶𝐶 ⊆ (𝐴 𝐵))) → (𝐶 ∩ (𝐵 𝐴)) = 𝐶)
135, 12eqtrd 2639 1 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶𝐶 ⊆ (𝐴 𝐵))) → ((𝐶𝐵) ∨ 𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1975  cin 3534  wss 3535   class class class wbr 4573  (class class class)co 6523   C cch 26972   chj 26976   𝑀* cdmd 27010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824  ax-hilex 27042
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 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-sh 27250  df-ch 27264  df-chj 27355  df-dmd 28326
This theorem is referenced by:  mdslle1i  28362  mdslj1i  28364  mdslj2i  28365  mdslmd1lem1  28370
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