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Theorem dmdsl3 29504
 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 29491 . . . . . 6 (((𝐵C𝐴C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶)) → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))
21exp32 632 . . . . 5 ((𝐵C𝐴C𝐶C ) → (𝐵 𝑀* 𝐴 → (𝐴𝐶 → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))))
323com12 1118 . . . 4 ((𝐴C𝐵C𝐶C ) → (𝐵 𝑀* 𝐴 → (𝐴𝐶 → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))))
43imp32 448 . . 3 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶)) → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))
543adantr3 1177 . 2 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶𝐶 ⊆ (𝐴 𝐵))) → ((𝐶𝐵) ∨ 𝐴) = (𝐶 ∩ (𝐵 𝐴)))
6 chjcom 28695 . . . . . 6 ((𝐴C𝐵C ) → (𝐴 𝐵) = (𝐵 𝐴))
76ineq2d 3957 . . . . 5 ((𝐴C𝐵C ) → (𝐶 ∩ (𝐴 𝐵)) = (𝐶 ∩ (𝐵 𝐴)))
873adant3 1127 . . . 4 ((𝐴C𝐵C𝐶C ) → (𝐶 ∩ (𝐴 𝐵)) = (𝐶 ∩ (𝐵 𝐴)))
9 df-ss 3729 . . . . 5 (𝐶 ⊆ (𝐴 𝐵) ↔ (𝐶 ∩ (𝐴 𝐵)) = 𝐶)
109biimpi 206 . . . 4 (𝐶 ⊆ (𝐴 𝐵) → (𝐶 ∩ (𝐴 𝐵)) = 𝐶)
118, 10sylan9req 2815 . . 3 (((𝐴C𝐵C𝐶C ) ∧ 𝐶 ⊆ (𝐴 𝐵)) → (𝐶 ∩ (𝐵 𝐴)) = 𝐶)
12113ad2antr3 1206 . 2 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶𝐶 ⊆ (𝐴 𝐵))) → (𝐶 ∩ (𝐵 𝐴)) = 𝐶)
135, 12eqtrd 2794 1 (((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶𝐶 ⊆ (𝐴 𝐵))) → ((𝐶𝐵) ∨ 𝐴) = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139   ∩ cin 3714   ⊆ wss 3715   class class class wbr 4804  (class class class)co 6814   Cℋ cch 28116   ∨ℋ chj 28120   𝑀ℋ* cdmd 28154 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-hilex 28186 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-sh 28394  df-ch 28408  df-chj 28499  df-dmd 29470 This theorem is referenced by:  mdslle1i  29506  mdslj1i  29508  mdslj2i  29509  mdslmd1lem1  29514
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