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Theorem cmtvalN 38813
Description: Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 31466 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
cmtfval.b 𝐵 = (Base‘𝐾)
cmtfval.j = (join‘𝐾)
cmtfval.m = (meet‘𝐾)
cmtfval.o = (oc‘𝐾)
cmtfval.c 𝐶 = (cm‘𝐾)
Assertion
Ref Expression
cmtvalN ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))

Proof of Theorem cmtvalN
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cmtfval.b . . . . . 6 𝐵 = (Base‘𝐾)
2 cmtfval.j . . . . . 6 = (join‘𝐾)
3 cmtfval.m . . . . . 6 = (meet‘𝐾)
4 cmtfval.o . . . . . 6 = (oc‘𝐾)
5 cmtfval.c . . . . . 6 𝐶 = (cm‘𝐾)
61, 2, 3, 4, 5cmtfvalN 38812 . . . . 5 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
7 df-3an 1086 . . . . . 6 ((𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))) ↔ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))))
87opabbii 5216 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}
96, 8eqtrdi 2781 . . . 4 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
109breqd 5160 . . 3 (𝐾𝐴 → (𝑋𝐶𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}𝑌))
11103ad2ant1 1130 . 2 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}𝑌))
12 df-br 5150 . . . 4 (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
13 id 22 . . . . . 6 (𝑥 = 𝑋𝑥 = 𝑋)
14 oveq1 7426 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
15 oveq1 7426 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 ( 𝑦)) = (𝑋 ( 𝑦)))
1614, 15oveq12d 7437 . . . . . 6 (𝑥 = 𝑋 → ((𝑥 𝑦) (𝑥 ( 𝑦))) = ((𝑋 𝑦) (𝑋 ( 𝑦))))
1713, 16eqeq12d 2741 . . . . 5 (𝑥 = 𝑋 → (𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))) ↔ 𝑋 = ((𝑋 𝑦) (𝑋 ( 𝑦)))))
18 oveq2 7427 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
19 fveq2 6896 . . . . . . . 8 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
2019oveq2d 7435 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 ( 𝑦)) = (𝑋 ( 𝑌)))
2118, 20oveq12d 7437 . . . . . 6 (𝑦 = 𝑌 → ((𝑋 𝑦) (𝑋 ( 𝑦))) = ((𝑋 𝑌) (𝑋 ( 𝑌))))
2221eqeq2d 2736 . . . . 5 (𝑦 = 𝑌 → (𝑋 = ((𝑋 𝑦) (𝑋 ( 𝑦))) ↔ 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
2317, 22opelopab2 5543 . . . 4 ((𝑋𝐵𝑌𝐵) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} ↔ 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
2412, 23bitrid 282 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
25243adant1 1127 . 2 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
2611, 25bitrd 278 1 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  cop 4636   class class class wbr 5149  {copab 5211  cfv 6549  (class class class)co 7419  Basecbs 17183  occoc 17244  joincjn 18306  meetcmee 18307  cmccmtN 38775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-cmtN 38779
This theorem is referenced by:  cmtcomlemN  38850  cmt2N  38852  cmtbr2N  38855  cmtbr3N  38856
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