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Theorem cmtvalN 33975
Description: Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 28289 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 33974 . . . . 5 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
7 df-3an 1038 . . . . . 6 ((𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))) ↔ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))))
87opabbii 4679 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}
96, 8syl6eq 2671 . . . 4 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
109breqd 4624 . . 3 (𝐾𝐴 → (𝑋𝐶𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}𝑌))
11103ad2ant1 1080 . 2 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}𝑌))
12 df-br 4614 . . . 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 6611 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
15 oveq1 6611 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 ( 𝑦)) = (𝑋 ( 𝑦)))
1614, 15oveq12d 6622 . . . . . 6 (𝑥 = 𝑋 → ((𝑥 𝑦) (𝑥 ( 𝑦))) = ((𝑋 𝑦) (𝑋 ( 𝑦))))
1713, 16eqeq12d 2636 . . . . 5 (𝑥 = 𝑋 → (𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))) ↔ 𝑋 = ((𝑋 𝑦) (𝑋 ( 𝑦)))))
18 oveq2 6612 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
19 fveq2 6148 . . . . . . . 8 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
2019oveq2d 6620 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 ( 𝑦)) = (𝑋 ( 𝑌)))
2118, 20oveq12d 6622 . . . . . 6 (𝑦 = 𝑌 → ((𝑋 𝑦) (𝑋 ( 𝑦))) = ((𝑋 𝑌) (𝑋 ( 𝑌))))
2221eqeq2d 2631 . . . . 5 (𝑦 = 𝑌 → (𝑋 = ((𝑋 𝑦) (𝑋 ( 𝑦))) ↔ 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
2317, 22opelopab2 4956 . . . 4 ((𝑋𝐵𝑌𝐵) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} ↔ 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
2412, 23syl5bb 272 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
25243adant1 1077 . 2 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
2611, 25bitrd 268 1 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  cop 4154   class class class wbr 4613  {copab 4672  cfv 5847  (class class class)co 6604  Basecbs 15781  occoc 15870  joincjn 16865  meetcmee 16866  cmccmtN 33937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-cmtN 33941
This theorem is referenced by:  cmtcomlemN  34012  cmt2N  34014  cmtbr2N  34017  cmtbr3N  34018
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