Theorem cmtfvalN 39203

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	⊢ (𝐾 ∈ 𝐴 → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ((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.

Step	Hyp	Ref	Expression
1		elex 3468	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		cmtfval.c	. . 3 ⊢ 𝐶 = (cm‘𝐾)
3		fveq2 6858	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4		cmtfval.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2782	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
6	5	eleq2d 2814	. . . . . 6 ⊢ (𝑝 = 𝐾 → (𝑥 ∈ (Base‘𝑝) ↔ 𝑥 ∈ 𝐵))
7	5	eleq2d 2814	. . . . . 6 ⊢ (𝑝 = 𝐾 → (𝑦 ∈ (Base‘𝑝) ↔ 𝑦 ∈ 𝐵))
8		fveq2 6858	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → (join‘𝑝) = (join‘𝐾))
9		cmtfval.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
10	8, 9	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (join‘𝑝) = ∨ )
11		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (meet‘𝑝) = (meet‘𝐾))
12		cmtfval.m	. . . . . . . . . 10 ⊢ ∧ = (meet‘𝐾)
13	11, 12	eqtr4di 2782	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → (meet‘𝑝) = ∧ )
14	13	oveqd 7404	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (𝑥(meet‘𝑝)𝑦) = (𝑥 ∧ 𝑦))
15		eqidd 2730	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → 𝑥 = 𝑥)
16		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (oc‘𝑝) = (oc‘𝐾))
17		cmtfval.o	. . . . . . . . . . 11 ⊢ ⊥ = (oc‘𝐾)
18	16, 17	eqtr4di 2782	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (oc‘𝑝) = ⊥ )
19	18	fveq1d 6860	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → ((oc‘𝑝)‘𝑦) = ( ⊥ ‘𝑦))
20	13, 15, 19	oveq123d 7408	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦)) = (𝑥 ∧ ( ⊥ ‘𝑦)))
21	10, 14, 20	oveq123d 7408	. . . . . . 7 ⊢ (𝑝 = 𝐾 → ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))
22	21	eqeq2d 2740	. . . . . 6 ⊢ (𝑝 = 𝐾 → (𝑥 = ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))) ↔ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦)))))
23	6, 7, 22	3anbi123d 1438	. . . . 5 ⊢ (𝑝 = 𝐾 → ((𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝) ∧ 𝑥 = ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦)))) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))))
24	23	opabbidv 5173	. . . 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⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))})
25		df-cmtN 39170	. . . 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 1088	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦)))) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦)))))
27	26	opabbii 5174	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))}
28	4	fvexi 6872	. . . . . . 7 ⊢ 𝐵 ∈ V
29	28, 28	xpex 7729	. . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V
30		opabssxp 5731	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))} ⊆ (𝐵 × 𝐵)
31	29, 30	ssexi 5277	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))} ∈ V
32	27, 31	eqeltri 2824	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))} ∈ V
33	24, 25, 32	fvmpt 6968	. . 3 ⊢ (𝐾 ∈ V → (cm‘𝐾) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))})
34	2, 33	eqtrid 2776	. 2 ⊢ (𝐾 ∈ V → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))})
35	1, 34	syl 17	1 ⊢ (𝐾 ∈ 𝐴 → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3447  {copab 5169   × cxp 5636  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  occoc 17228  joincjn 18272  meetcmee 18273  cmccmtN 39166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-cmtN 39170

This theorem is referenced by:  cmtvalN  39204