Theorem brcgr3 36064

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 4849	. . . 4 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
2	1	breq1d 5129	. . 3 ⊢ (𝑎 = 𝐴 → (⟨𝑎, 𝑏⟩Cgr⟨𝑑, 𝑒⟩ ↔ ⟨𝐴, 𝑏⟩Cgr⟨𝑑, 𝑒⟩))
3		opeq1 4849	. . . 4 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑐⟩ = ⟨𝐴, 𝑐⟩)
4	3	breq1d 5129	. . 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 4850	. . . 4 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
7	6	breq1d 5129	. . 3 ⊢ (𝑏 = 𝐵 → (⟨𝐴, 𝑏⟩Cgr⟨𝑑, 𝑒⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝑑, 𝑒⟩))
8		opeq1 4849	. . . 4 ⊢ (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
9	8	breq1d 5129	. . 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 4850	. . . 4 ⊢ (𝑐 = 𝐶 → ⟨𝐴, 𝑐⟩ = ⟨𝐴, 𝐶⟩)
12	11	breq1d 5129	. . 3 ⊢ (𝑐 = 𝐶 → (⟨𝐴, 𝑐⟩Cgr⟨𝑑, 𝑓⟩ ↔ ⟨𝐴, 𝐶⟩Cgr⟨𝑑, 𝑓⟩))
13		opeq2 4850	. . . 4 ⊢ (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
14	13	breq1d 5129	. . 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 4849	. . . 4 ⊢ (𝑑 = 𝐷 → ⟨𝑑, 𝑒⟩ = ⟨𝐷, 𝑒⟩)
17	16	breq2d 5131	. . 3 ⊢ (𝑑 = 𝐷 → (⟨𝐴, 𝐵⟩Cgr⟨𝑑, 𝑒⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝑒⟩))
18		opeq1 4849	. . . 4 ⊢ (𝑑 = 𝐷 → ⟨𝑑, 𝑓⟩ = ⟨𝐷, 𝑓⟩)
19	18	breq2d 5131	. . 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 4850	. . . 4 ⊢ (𝑒 = 𝐸 → ⟨𝐷, 𝑒⟩ = ⟨𝐷, 𝐸⟩)
22	21	breq2d 5131	. . 3 ⊢ (𝑒 = 𝐸 → (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝑒⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩))
23		opeq1 4849	. . . 4 ⊢ (𝑒 = 𝐸 → ⟨𝑒, 𝑓⟩ = ⟨𝐸, 𝑓⟩)
24	23	breq2d 5131	. . 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 4850	. . . 4 ⊢ (𝑓 = 𝐹 → ⟨𝐷, 𝑓⟩ = ⟨𝐷, 𝐹⟩)
27	26	breq2d 5131	. . 3 ⊢ (𝑓 = 𝐹 → (⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝑓⟩ ↔ ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩))
28		opeq2 4850	. . . 4 ⊢ (𝑓 = 𝐹 → ⟨𝐸, 𝑓⟩ = ⟨𝐸, 𝐹⟩)
29	28	breq2d 5131	. . 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 6876	. 2 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
32		df-cgr3 36059	. 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 35774	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 1540   ∈ wcel 2108  ⟨cop 4607   class class class wbr 5119  ‘cfv 6531  ℕcn 12240  𝔼cee 28867  Cgrccgr 28869  Cgr3ccgr3 36054

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-iota 6484  df-fv 6539  df-cgr3 36059

This theorem is referenced by:  cgr3permute3  36065  cgr3permute1  36066  cgr3tr4  36070  cgr3com  36071  cgr3rflx  36072  cgrxfr  36073  btwnxfr  36074  lineext  36094  brofs2  36095  brifs2  36096  endofsegid  36103  btwnconn1lem4  36108  btwnconn1lem8  36112  btwnconn1lem11  36115  brsegle2  36127  seglecgr12im  36128  segletr  36132