Theorem brcgr3 36047

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 4873	. . . 4 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
2	1	breq1d 5153	. . 3 ⊢ (𝑎 = 𝐴 → (⟨𝑎, 𝑏⟩Cgr⟨𝑑, 𝑒⟩ ↔ ⟨𝐴, 𝑏⟩Cgr⟨𝑑, 𝑒⟩))
3		opeq1 4873	. . . 4 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑐⟩ = ⟨𝐴, 𝑐⟩)
4	3	breq1d 5153	. . 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 4874	. . . 4 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
7	6	breq1d 5153	. . 3 ⊢ (𝑏 = 𝐵 → (⟨𝐴, 𝑏⟩Cgr⟨𝑑, 𝑒⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝑑, 𝑒⟩))
8		opeq1 4873	. . . 4 ⊢ (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
9	8	breq1d 5153	. . 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 4874	. . . 4 ⊢ (𝑐 = 𝐶 → ⟨𝐴, 𝑐⟩ = ⟨𝐴, 𝐶⟩)
12	11	breq1d 5153	. . 3 ⊢ (𝑐 = 𝐶 → (⟨𝐴, 𝑐⟩Cgr⟨𝑑, 𝑓⟩ ↔ ⟨𝐴, 𝐶⟩Cgr⟨𝑑, 𝑓⟩))
13		opeq2 4874	. . . 4 ⊢ (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
14	13	breq1d 5153	. . 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 4873	. . . 4 ⊢ (𝑑 = 𝐷 → ⟨𝑑, 𝑒⟩ = ⟨𝐷, 𝑒⟩)
17	16	breq2d 5155	. . 3 ⊢ (𝑑 = 𝐷 → (⟨𝐴, 𝐵⟩Cgr⟨𝑑, 𝑒⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝑒⟩))
18		opeq1 4873	. . . 4 ⊢ (𝑑 = 𝐷 → ⟨𝑑, 𝑓⟩ = ⟨𝐷, 𝑓⟩)
19	18	breq2d 5155	. . 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 4874	. . . 4 ⊢ (𝑒 = 𝐸 → ⟨𝐷, 𝑒⟩ = ⟨𝐷, 𝐸⟩)
22	21	breq2d 5155	. . 3 ⊢ (𝑒 = 𝐸 → (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝑒⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩))
23		opeq1 4873	. . . 4 ⊢ (𝑒 = 𝐸 → ⟨𝑒, 𝑓⟩ = ⟨𝐸, 𝑓⟩)
24	23	breq2d 5155	. . 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 4874	. . . 4 ⊢ (𝑓 = 𝐹 → ⟨𝐷, 𝑓⟩ = ⟨𝐷, 𝐹⟩)
27	26	breq2d 5155	. . 3 ⊢ (𝑓 = 𝐹 → (⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝑓⟩ ↔ ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩))
28		opeq2 4874	. . . 4 ⊢ (𝑓 = 𝐹 → ⟨𝐸, 𝑓⟩ = ⟨𝐸, 𝐹⟩)
29	28	breq2d 5155	. . 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 6906	. 2 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
32		df-cgr3 36042	. 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 35757	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 1087   = wceq 1540   ∈ wcel 2108  ⟨cop 4632   class class class wbr 5143  ‘cfv 6561  ℕcn 12266  𝔼cee 28903  Cgrccgr 28905  Cgr3ccgr3 36037

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-iota 6514  df-fv 6569  df-cgr3 36042

This theorem is referenced by:  cgr3permute3  36048  cgr3permute1  36049  cgr3tr4  36053  cgr3com  36054  cgr3rflx  36055  cgrxfr  36056  btwnxfr  36057  lineext  36077  brofs2  36078  brifs2  36079  endofsegid  36086  btwnconn1lem4  36091  btwnconn1lem8  36095  btwnconn1lem11  36098  brsegle2  36110  seglecgr12im  36111  segletr  36115