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Theorem cmtvalN 39193
Description: Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 31613 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 39192 . . . . 5 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
7 df-3an 1088 . . . . . 6 ((𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))) ↔ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))))
87opabbii 5215 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}
96, 8eqtrdi 2791 . . . 4 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
109breqd 5159 . . 3 (𝐾𝐴 → (𝑋𝐶𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}𝑌))
11103ad2ant1 1132 . 2 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}𝑌))
12 df-br 5149 . . . 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 7438 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
15 oveq1 7438 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 ( 𝑦)) = (𝑋 ( 𝑦)))
1614, 15oveq12d 7449 . . . . . 6 (𝑥 = 𝑋 → ((𝑥 𝑦) (𝑥 ( 𝑦))) = ((𝑋 𝑦) (𝑋 ( 𝑦))))
1713, 16eqeq12d 2751 . . . . 5 (𝑥 = 𝑋 → (𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))) ↔ 𝑋 = ((𝑋 𝑦) (𝑋 ( 𝑦)))))
18 oveq2 7439 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
19 fveq2 6907 . . . . . . . 8 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
2019oveq2d 7447 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 ( 𝑦)) = (𝑋 ( 𝑌)))
2118, 20oveq12d 7449 . . . . . 6 (𝑦 = 𝑌 → ((𝑋 𝑦) (𝑋 ( 𝑦))) = ((𝑋 𝑌) (𝑋 ( 𝑌))))
2221eqeq2d 2746 . . . . 5 (𝑦 = 𝑌 → (𝑋 = ((𝑋 𝑦) (𝑋 ( 𝑦))) ↔ 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
2317, 22opelopab2 5551 . . . 4 ((𝑋𝐵𝑌𝐵) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} ↔ 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
2412, 23bitrid 283 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
25243adant1 1129 . 2 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
2611, 25bitrd 279 1 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  cop 4637   class class class wbr 5148  {copab 5210  cfv 6563  (class class class)co 7431  Basecbs 17245  occoc 17306  joincjn 18369  meetcmee 18370  cmccmtN 39155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-cmtN 39159
This theorem is referenced by:  cmtcomlemN  39230  cmt2N  39232  cmtbr2N  39235  cmtbr3N  39236
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