Theorem brcgr3 36090

Description: Binary relation form of the three-place congruence predicate. (Contributed by Scott Fenton, 4-Oct-2013.)

Assertion

Ref	Expression
brcgr3	⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩ ↔ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)))

Proof of Theorem brcgr3

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑛 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 4822	. . . 4 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
2	1	breq1d 5099	. . 3 ⊢ (𝑎 = 𝐴 → (⟨𝑎, 𝑏⟩Cgr⟨𝑑, 𝑒⟩ ↔ ⟨𝐴, 𝑏⟩Cgr⟨𝑑, 𝑒⟩))
3		opeq1 4822	. . . 4 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑐⟩ = ⟨𝐴, 𝑐⟩)
4	3	breq1d 5099	. . 3 ⊢ (𝑎 = 𝐴 → (⟨𝑎, 𝑐⟩Cgr⟨𝑑, 𝑓⟩ ↔ ⟨𝐴, 𝑐⟩Cgr⟨𝑑, 𝑓⟩))
5	2, 4	3anbi12d 1439	. 2 ⊢ (𝑎 = 𝐴 → ((⟨𝑎, 𝑏⟩Cgr⟨𝑑, 𝑒⟩ ∧ ⟨𝑎, 𝑐⟩Cgr⟨𝑑, 𝑓⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑒, 𝑓⟩) ↔ (⟨𝐴, 𝑏⟩Cgr⟨𝑑, 𝑒⟩ ∧ ⟨𝐴, 𝑐⟩Cgr⟨𝑑, 𝑓⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑒, 𝑓⟩)))
6		opeq2 4823	. . . 4 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
7	6	breq1d 5099	. . 3 ⊢ (𝑏 = 𝐵 → (⟨𝐴, 𝑏⟩Cgr⟨𝑑, 𝑒⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝑑, 𝑒⟩))
8		opeq1 4822	. . . 4 ⊢ (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
9	8	breq1d 5099	. . 3 ⊢ (𝑏 = 𝐵 → (⟨𝑏, 𝑐⟩Cgr⟨𝑒, 𝑓⟩ ↔ ⟨𝐵, 𝑐⟩Cgr⟨𝑒, 𝑓⟩))
10	7, 9	3anbi13d 1440	. 2 ⊢ (𝑏 = 𝐵 → ((⟨𝐴, 𝑏⟩Cgr⟨𝑑, 𝑒⟩ ∧ ⟨𝐴, 𝑐⟩Cgr⟨𝑑, 𝑓⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑒, 𝑓⟩) ↔ (⟨𝐴, 𝐵⟩Cgr⟨𝑑, 𝑒⟩ ∧ ⟨𝐴, 𝑐⟩Cgr⟨𝑑, 𝑓⟩ ∧ ⟨𝐵, 𝑐⟩Cgr⟨𝑒, 𝑓⟩)))
11		opeq2 4823	. . . 4 ⊢ (𝑐 = 𝐶 → ⟨𝐴, 𝑐⟩ = ⟨𝐴, 𝐶⟩)
12	11	breq1d 5099	. . 3 ⊢ (𝑐 = 𝐶 → (⟨𝐴, 𝑐⟩Cgr⟨𝑑, 𝑓⟩ ↔ ⟨𝐴, 𝐶⟩Cgr⟨𝑑, 𝑓⟩))
13		opeq2 4823	. . . 4 ⊢ (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
14	13	breq1d 5099	. . 3 ⊢ (𝑐 = 𝐶 → (⟨𝐵, 𝑐⟩Cgr⟨𝑒, 𝑓⟩ ↔ ⟨𝐵, 𝐶⟩Cgr⟨𝑒, 𝑓⟩))
15	12, 14	3anbi23d 1441	. 2 ⊢ (𝑐 = 𝐶 → ((⟨𝐴, 𝐵⟩Cgr⟨𝑑, 𝑒⟩ ∧ ⟨𝐴, 𝑐⟩Cgr⟨𝑑, 𝑓⟩ ∧ ⟨𝐵, 𝑐⟩Cgr⟨𝑒, 𝑓⟩) ↔ (⟨𝐴, 𝐵⟩Cgr⟨𝑑, 𝑒⟩ ∧ ⟨𝐴, 𝐶⟩Cgr⟨𝑑, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑒, 𝑓⟩)))
16		opeq1 4822	. . . 4 ⊢ (𝑑 = 𝐷 → ⟨𝑑, 𝑒⟩ = ⟨𝐷, 𝑒⟩)
17	16	breq2d 5101	. . 3 ⊢ (𝑑 = 𝐷 → (⟨𝐴, 𝐵⟩Cgr⟨𝑑, 𝑒⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝑒⟩))
18		opeq1 4822	. . . 4 ⊢ (𝑑 = 𝐷 → ⟨𝑑, 𝑓⟩ = ⟨𝐷, 𝑓⟩)
19	18	breq2d 5101	. . 3 ⊢ (𝑑 = 𝐷 → (⟨𝐴, 𝐶⟩Cgr⟨𝑑, 𝑓⟩ ↔ ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝑓⟩))
20	17, 19	3anbi12d 1439	. 2 ⊢ (𝑑 = 𝐷 → ((⟨𝐴, 𝐵⟩Cgr⟨𝑑, 𝑒⟩ ∧ ⟨𝐴, 𝐶⟩Cgr⟨𝑑, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑒, 𝑓⟩) ↔ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝑒⟩ ∧ ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑒, 𝑓⟩)))
21		opeq2 4823	. . . 4 ⊢ (𝑒 = 𝐸 → ⟨𝐷, 𝑒⟩ = ⟨𝐷, 𝐸⟩)
22	21	breq2d 5101	. . 3 ⊢ (𝑒 = 𝐸 → (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝑒⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩))
23		opeq1 4822	. . . 4 ⊢ (𝑒 = 𝐸 → ⟨𝑒, 𝑓⟩ = ⟨𝐸, 𝑓⟩)
24	23	breq2d 5101	. . 3 ⊢ (𝑒 = 𝐸 → (⟨𝐵, 𝐶⟩Cgr⟨𝑒, 𝑓⟩ ↔ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝑓⟩))
25	22, 24	3anbi13d 1440	. 2 ⊢ (𝑒 = 𝐸 → ((⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝑒⟩ ∧ ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑒, 𝑓⟩) ↔ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝑓⟩)))
26		opeq2 4823	. . . 4 ⊢ (𝑓 = 𝐹 → ⟨𝐷, 𝑓⟩ = ⟨𝐷, 𝐹⟩)
27	26	breq2d 5101	. . 3 ⊢ (𝑓 = 𝐹 → (⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝑓⟩ ↔ ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩))
28		opeq2 4823	. . . 4 ⊢ (𝑓 = 𝐹 → ⟨𝐸, 𝑓⟩ = ⟨𝐸, 𝐹⟩)
29	28	breq2d 5101	. . 3 ⊢ (𝑓 = 𝐹 → (⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝑓⟩ ↔ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩))
30	27, 29	3anbi23d 1441	. 2 ⊢ (𝑓 = 𝐹 → ((⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝑓⟩) ↔ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)))
31		fveq2 6822	. 2 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
32		df-cgr3 36085	. 2 ⊢ Cgr3 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑒 ∈ (𝔼‘𝑛)∃𝑓 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, ⟨𝑏, 𝑐⟩⟩ ∧ 𝑞 = ⟨𝑑, ⟨𝑒, 𝑓⟩⟩ ∧ (⟨𝑎, 𝑏⟩Cgr⟨𝑑, 𝑒⟩ ∧ ⟨𝑎, 𝑐⟩Cgr⟨𝑑, 𝑓⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑒, 𝑓⟩))}
33	5, 10, 15, 20, 25, 30, 31, 32	br6 35801	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩ ↔ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ⟨cop 4579   class class class wbr 5089  ‘cfv 6481  ℕcn 12125  𝔼cee 28866  Cgrccgr 28868  Cgr3ccgr3 36080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-iota 6437  df-fv 6489  df-cgr3 36085

This theorem is referenced by:  cgr3permute3  36091  cgr3permute1  36092  cgr3tr4  36096  cgr3com  36097  cgr3rflx  36098  cgrxfr  36099  btwnxfr  36100  lineext  36120  brofs2  36121  brifs2  36122  endofsegid  36129  btwnconn1lem4  36134  btwnconn1lem8  36138  btwnconn1lem11  36141  brsegle2  36153  seglecgr12im  36154  segletr  36158