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Theorem cmtvalN 36407
Description: Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 29356 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 36406 . . . . 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 5114 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}
96, 8syl6eq 2875 . . . 4 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
109breqd 5058 . . 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 5048 . . . 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 7145 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
15 oveq1 7145 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 ( 𝑦)) = (𝑋 ( 𝑦)))
1614, 15oveq12d 7156 . . . . . 6 (𝑥 = 𝑋 → ((𝑥 𝑦) (𝑥 ( 𝑦))) = ((𝑋 𝑦) (𝑋 ( 𝑦))))
1713, 16eqeq12d 2840 . . . . 5 (𝑥 = 𝑋 → (𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))) ↔ 𝑋 = ((𝑋 𝑦) (𝑋 ( 𝑦)))))
18 oveq2 7146 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
19 fveq2 6651 . . . . . . . 8 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
2019oveq2d 7154 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 ( 𝑦)) = (𝑋 ( 𝑌)))
2118, 20oveq12d 7156 . . . . . 6 (𝑦 = 𝑌 → ((𝑋 𝑦) (𝑋 ( 𝑦))) = ((𝑋 𝑌) (𝑋 ( 𝑌))))
2221eqeq2d 2835 . . . . 5 (𝑦 = 𝑌 → (𝑋 = ((𝑋 𝑦) (𝑋 ( 𝑦))) ↔ 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
2317, 22opelopab2 5409 . . . 4 ((𝑋𝐵𝑌𝐵) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} ↔ 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
2412, 23syl5bb 286 . . 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 282 1 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  cop 4554   class class class wbr 5047  {copab 5109  cfv 6336  (class class class)co 7138  Basecbs 16472  occoc 16562  joincjn 17543  meetcmee 17544  cmccmtN 36369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-iota 6295  df-fun 6338  df-fv 6344  df-ov 7141  df-cmtN 36373
This theorem is referenced by:  cmtcomlemN  36444  cmt2N  36446  cmtbr2N  36449  cmtbr3N  36450
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