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Theorem cmtfvalN 33977
Description: Value of commutes relation. (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
cmtfvalN (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐶(𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)

Proof of Theorem cmtfvalN
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 elex 3198 . 2 (𝐾𝐴𝐾 ∈ V)
2 cmtfval.c . . 3 𝐶 = (cm‘𝐾)
3 fveq2 6148 . . . . . . . 8 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4 cmtfval.b . . . . . . . 8 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2673 . . . . . . 7 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
65eleq2d 2684 . . . . . 6 (𝑝 = 𝐾 → (𝑥 ∈ (Base‘𝑝) ↔ 𝑥𝐵))
75eleq2d 2684 . . . . . 6 (𝑝 = 𝐾 → (𝑦 ∈ (Base‘𝑝) ↔ 𝑦𝐵))
8 fveq2 6148 . . . . . . . . 9 (𝑝 = 𝐾 → (join‘𝑝) = (join‘𝐾))
9 cmtfval.j . . . . . . . . 9 = (join‘𝐾)
108, 9syl6eqr 2673 . . . . . . . 8 (𝑝 = 𝐾 → (join‘𝑝) = )
11 fveq2 6148 . . . . . . . . . 10 (𝑝 = 𝐾 → (meet‘𝑝) = (meet‘𝐾))
12 cmtfval.m . . . . . . . . . 10 = (meet‘𝐾)
1311, 12syl6eqr 2673 . . . . . . . . 9 (𝑝 = 𝐾 → (meet‘𝑝) = )
1413oveqd 6621 . . . . . . . 8 (𝑝 = 𝐾 → (𝑥(meet‘𝑝)𝑦) = (𝑥 𝑦))
15 eqidd 2622 . . . . . . . . 9 (𝑝 = 𝐾𝑥 = 𝑥)
16 fveq2 6148 . . . . . . . . . . 11 (𝑝 = 𝐾 → (oc‘𝑝) = (oc‘𝐾))
17 cmtfval.o . . . . . . . . . . 11 = (oc‘𝐾)
1816, 17syl6eqr 2673 . . . . . . . . . 10 (𝑝 = 𝐾 → (oc‘𝑝) = )
1918fveq1d 6150 . . . . . . . . 9 (𝑝 = 𝐾 → ((oc‘𝑝)‘𝑦) = ( 𝑦))
2013, 15, 19oveq123d 6625 . . . . . . . 8 (𝑝 = 𝐾 → (𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦)) = (𝑥 ( 𝑦)))
2110, 14, 20oveq123d 6625 . . . . . . 7 (𝑝 = 𝐾 → ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))) = ((𝑥 𝑦) (𝑥 ( 𝑦))))
2221eqeq2d 2631 . . . . . 6 (𝑝 = 𝐾 → (𝑥 = ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))) ↔ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))))
236, 7, 223anbi123d 1396 . . . . 5 (𝑝 = 𝐾 → ((𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝) ∧ 𝑥 = ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦)))) ↔ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))))
2423opabbidv 4678 . . . 4 (𝑝 = 𝐾 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝) ∧ 𝑥 = ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
25 df-cmtN 33944 . . . 4 cm = (𝑝 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝) ∧ 𝑥 = ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))))})
26 df-3an 1038 . . . . . 6 ((𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))) ↔ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦)))))
2726opabbii 4679 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))}
28 fvex 6158 . . . . . . . 8 (Base‘𝐾) ∈ V
294, 28eqeltri 2694 . . . . . . 7 𝐵 ∈ V
3029, 29xpex 6915 . . . . . 6 (𝐵 × 𝐵) ∈ V
31 opabssxp 5154 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} ⊆ (𝐵 × 𝐵)
3230, 31ssexi 4763 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} ∈ V
3327, 32eqeltri 2694 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))} ∈ V
3424, 25, 33fvmpt 6239 . . 3 (𝐾 ∈ V → (cm‘𝐾) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
352, 34syl5eq 2667 . 2 (𝐾 ∈ V → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
361, 35syl 17 1 (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3186  {copab 4672   × cxp 5072  cfv 5847  (class class class)co 6604  Basecbs 15781  occoc 15870  joincjn 16865  meetcmee 16866  cmccmtN 33940
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 33944
This theorem is referenced by:  cmtvalN  33978
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