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Theorem cmtvalN 35231
Description: Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 28967 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 35230 . . . . 5 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
7 df-3an 1110 . . . . . 6 ((𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))) ↔ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))))
87opabbii 4911 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}
96, 8syl6eq 2850 . . . 4 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
109breqd 4855 . . 3 (𝐾𝐴 → (𝑋𝐶𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}𝑌))
11103ad2ant1 1164 . 2 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}𝑌))
12 df-br 4845 . . . 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 6886 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
15 oveq1 6886 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 ( 𝑦)) = (𝑋 ( 𝑦)))
1614, 15oveq12d 6897 . . . . . 6 (𝑥 = 𝑋 → ((𝑥 𝑦) (𝑥 ( 𝑦))) = ((𝑋 𝑦) (𝑋 ( 𝑦))))
1713, 16eqeq12d 2815 . . . . 5 (𝑥 = 𝑋 → (𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))) ↔ 𝑋 = ((𝑋 𝑦) (𝑋 ( 𝑦)))))
18 oveq2 6887 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
19 fveq2 6412 . . . . . . . 8 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
2019oveq2d 6895 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 ( 𝑦)) = (𝑋 ( 𝑌)))
2118, 20oveq12d 6897 . . . . . 6 (𝑦 = 𝑌 → ((𝑋 𝑦) (𝑋 ( 𝑦))) = ((𝑋 𝑌) (𝑋 ( 𝑌))))
2221eqeq2d 2810 . . . . 5 (𝑦 = 𝑌 → (𝑋 = ((𝑋 𝑦) (𝑋 ( 𝑦))) ↔ 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
2317, 22opelopab2 5193 . . . 4 ((𝑋𝐵𝑌𝐵) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} ↔ 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
2412, 23syl5bb 275 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
25243adant1 1161 . 2 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
2611, 25bitrd 271 1 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  cop 4375   class class class wbr 4844  {copab 4906  cfv 6102  (class class class)co 6879  Basecbs 16183  occoc 16274  joincjn 17258  meetcmee 17259  cmccmtN 35193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-iota 6065  df-fun 6104  df-fv 6110  df-ov 6882  df-cmtN 35197
This theorem is referenced by:  cmtcomlemN  35268  cmt2N  35270  cmtbr2N  35273  cmtbr3N  35274
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