Theorem ecopovtrng 6694

Description: Assuming that operation 𝐹 is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation ∼, specified by the first hypothesis, is transitive. (Contributed by Jim Kingdon, 1-Sep-2019.)

Hypotheses

Ref	Expression
ecopopr.1	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}
ecopoprg.com	⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
ecopoprg.cl	⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
ecopoprg.ass	⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
ecopoprg.can	⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))

Assertion

Ref	Expression
ecopovtrng	⊢ ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ 𝐶) → 𝐴 ∼ 𝐶)

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢, +   𝑥,𝑆,𝑦,𝑧,𝑤,𝑣,𝑢

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐵(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐶(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   ∼ (𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem ecopovtrng

Dummy variables 𝑓 𝑔 ℎ 𝑡 𝑠 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ecopopr.1	. . . . . . 7 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}
2		opabssxp 4737	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
3	1, 2	eqsstri 3215	. . . . . 6 ⊢ ∼ ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
4	3	brel 4715	. . . . 5 ⊢ (𝐴 ∼ 𝐵 → (𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆)))
5	4	simpld 112	. . . 4 ⊢ (𝐴 ∼ 𝐵 → 𝐴 ∈ (𝑆 × 𝑆))
6	3	brel 4715	. . . 4 ⊢ (𝐵 ∼ 𝐶 → (𝐵 ∈ (𝑆 × 𝑆) ∧ 𝐶 ∈ (𝑆 × 𝑆)))
7	5, 6	anim12i 338	. . 3 ⊢ ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ 𝐶) → (𝐴 ∈ (𝑆 × 𝑆) ∧ (𝐵 ∈ (𝑆 × 𝑆) ∧ 𝐶 ∈ (𝑆 × 𝑆))))
8		3anass 984	. . 3 ⊢ ((𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆) ∧ 𝐶 ∈ (𝑆 × 𝑆)) ↔ (𝐴 ∈ (𝑆 × 𝑆) ∧ (𝐵 ∈ (𝑆 × 𝑆) ∧ 𝐶 ∈ (𝑆 × 𝑆))))
9	7, 8	sylibr 134	. 2 ⊢ ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ 𝐶) → (𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆) ∧ 𝐶 ∈ (𝑆 × 𝑆)))
10		eqid 2196	. . 3 ⊢ (𝑆 × 𝑆) = (𝑆 × 𝑆)
11		breq1 4036	. . . . 5 ⊢ (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ↔ 𝐴 ∼ ⟨ℎ, 𝑡⟩))
12	11	anbi1d 465	. . . 4 ⊢ (⟨𝑓, 𝑔⟩ = 𝐴 → ((⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ∧ ⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩) ↔ (𝐴 ∼ ⟨ℎ, 𝑡⟩ ∧ ⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩)))
13		breq1 4036	. . . 4 ⊢ (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨𝑓, 𝑔⟩ ∼ ⟨𝑠, 𝑟⟩ ↔ 𝐴 ∼ ⟨𝑠, 𝑟⟩))
14	12, 13	imbi12d 234	. . 3 ⊢ (⟨𝑓, 𝑔⟩ = 𝐴 → (((⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ∧ ⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩) → ⟨𝑓, 𝑔⟩ ∼ ⟨𝑠, 𝑟⟩) ↔ ((𝐴 ∼ ⟨ℎ, 𝑡⟩ ∧ ⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩) → 𝐴 ∼ ⟨𝑠, 𝑟⟩)))
15		breq2 4037	. . . . 5 ⊢ (⟨ℎ, 𝑡⟩ = 𝐵 → (𝐴 ∼ ⟨ℎ, 𝑡⟩ ↔ 𝐴 ∼ 𝐵))
16		breq1 4036	. . . . 5 ⊢ (⟨ℎ, 𝑡⟩ = 𝐵 → (⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩ ↔ 𝐵 ∼ ⟨𝑠, 𝑟⟩))
17	15, 16	anbi12d 473	. . . 4 ⊢ (⟨ℎ, 𝑡⟩ = 𝐵 → ((𝐴 ∼ ⟨ℎ, 𝑡⟩ ∧ ⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩) ↔ (𝐴 ∼ 𝐵 ∧ 𝐵 ∼ ⟨𝑠, 𝑟⟩)))
18	17	imbi1d 231	. . 3 ⊢ (⟨ℎ, 𝑡⟩ = 𝐵 → (((𝐴 ∼ ⟨ℎ, 𝑡⟩ ∧ ⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩) → 𝐴 ∼ ⟨𝑠, 𝑟⟩) ↔ ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ ⟨𝑠, 𝑟⟩) → 𝐴 ∼ ⟨𝑠, 𝑟⟩)))
19		breq2 4037	. . . . 5 ⊢ (⟨𝑠, 𝑟⟩ = 𝐶 → (𝐵 ∼ ⟨𝑠, 𝑟⟩ ↔ 𝐵 ∼ 𝐶))
20	19	anbi2d 464	. . . 4 ⊢ (⟨𝑠, 𝑟⟩ = 𝐶 → ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ ⟨𝑠, 𝑟⟩) ↔ (𝐴 ∼ 𝐵 ∧ 𝐵 ∼ 𝐶)))
21		breq2 4037	. . . 4 ⊢ (⟨𝑠, 𝑟⟩ = 𝐶 → (𝐴 ∼ ⟨𝑠, 𝑟⟩ ↔ 𝐴 ∼ 𝐶))
22	20, 21	imbi12d 234	. . 3 ⊢ (⟨𝑠, 𝑟⟩ = 𝐶 → (((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ ⟨𝑠, 𝑟⟩) → 𝐴 ∼ ⟨𝑠, 𝑟⟩) ↔ ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ 𝐶) → 𝐴 ∼ 𝐶)))
23	1	ecopoveq 6689	. . . . . . . 8 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → (⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ↔ (𝑓 + 𝑡) = (𝑔 + ℎ)))
24	23	3adant3 1019	. . . . . . 7 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → (⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ↔ (𝑓 + 𝑡) = (𝑔 + ℎ)))
25	1	ecopoveq 6689	. . . . . . . 8 ⊢ (((ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → (⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩ ↔ (ℎ + 𝑟) = (𝑡 + 𝑠)))
26	25	3adant1 1017	. . . . . . 7 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → (⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩ ↔ (ℎ + 𝑟) = (𝑡 + 𝑠)))
27	24, 26	anbi12d 473	. . . . . 6 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → ((⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ∧ ⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩) ↔ ((𝑓 + 𝑡) = (𝑔 + ℎ) ∧ (ℎ + 𝑟) = (𝑡 + 𝑠))))
28		oveq12 5931	. . . . . . 7 ⊢ (((𝑓 + 𝑡) = (𝑔 + ℎ) ∧ (ℎ + 𝑟) = (𝑡 + 𝑠)) → ((𝑓 + 𝑡) + (ℎ + 𝑟)) = ((𝑔 + ℎ) + (𝑡 + 𝑠)))
29		simp2l 1025	. . . . . . . . 9 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → ℎ ∈ 𝑆)
30		simp2r 1026	. . . . . . . . 9 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → 𝑡 ∈ 𝑆)
31		simp1l 1023	. . . . . . . . 9 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → 𝑓 ∈ 𝑆)
32		ecopoprg.com	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
33	32	adantl 277	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
34		ecopoprg.ass	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
35	34	adantl 277	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
36		simp3r 1028	. . . . . . . . 9 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → 𝑟 ∈ 𝑆)
37		ecopoprg.cl	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
38	37	adantl 277	. . . . . . . . 9 ⊢ ((((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
39	29, 30, 31, 33, 35, 36, 38	caov411d 6109	. . . . . . . 8 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → ((ℎ + 𝑡) + (𝑓 + 𝑟)) = ((𝑓 + 𝑡) + (ℎ + 𝑟)))
40		simp1r 1024	. . . . . . . . . 10 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → 𝑔 ∈ 𝑆)
41		simp3l 1027	. . . . . . . . . 10 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → 𝑠 ∈ 𝑆)
42	40, 30, 29, 33, 35, 41, 38	caov411d 6109	. . . . . . . . 9 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → ((𝑔 + 𝑡) + (ℎ + 𝑠)) = ((ℎ + 𝑡) + (𝑔 + 𝑠)))
43	40, 30, 29, 33, 35, 41, 38	caov4d 6108	. . . . . . . . 9 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → ((𝑔 + 𝑡) + (ℎ + 𝑠)) = ((𝑔 + ℎ) + (𝑡 + 𝑠)))
44	42, 43	eqtr3d 2231	. . . . . . . 8 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → ((ℎ + 𝑡) + (𝑔 + 𝑠)) = ((𝑔 + ℎ) + (𝑡 + 𝑠)))
45	39, 44	eqeq12d 2211	. . . . . . 7 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → (((ℎ + 𝑡) + (𝑓 + 𝑟)) = ((ℎ + 𝑡) + (𝑔 + 𝑠)) ↔ ((𝑓 + 𝑡) + (ℎ + 𝑟)) = ((𝑔 + ℎ) + (𝑡 + 𝑠))))
46	28, 45	imbitrrid 156	. . . . . 6 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → (((𝑓 + 𝑡) = (𝑔 + ℎ) ∧ (ℎ + 𝑟) = (𝑡 + 𝑠)) → ((ℎ + 𝑡) + (𝑓 + 𝑟)) = ((ℎ + 𝑡) + (𝑔 + 𝑠))))
47	27, 46	sylbid 150	. . . . 5 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → ((⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ∧ ⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩) → ((ℎ + 𝑡) + (𝑓 + 𝑟)) = ((ℎ + 𝑡) + (𝑔 + 𝑠))))
48		ecopoprg.can	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))
49		oveq2 5930	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑥 + 𝑦) = (𝑥 + 𝑧))
50	48, 49	impbid1 142	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) ↔ 𝑦 = 𝑧))
51	50	adantl 277	. . . . . 6 ⊢ ((((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) ↔ 𝑦 = 𝑧))
52	37	caovcl 6078	. . . . . . 7 ⊢ ((ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) → (ℎ + 𝑡) ∈ 𝑆)
53	29, 30, 52	syl2anc 411	. . . . . 6 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → (ℎ + 𝑡) ∈ 𝑆)
54	37	caovcl 6078	. . . . . . 7 ⊢ ((𝑓 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆) → (𝑓 + 𝑟) ∈ 𝑆)
55	31, 36, 54	syl2anc 411	. . . . . 6 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → (𝑓 + 𝑟) ∈ 𝑆)
56	38, 40, 41	caovcld 6077	. . . . . 6 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → (𝑔 + 𝑠) ∈ 𝑆)
57	51, 53, 55, 56	caovcand 6086	. . . . 5 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → (((ℎ + 𝑡) + (𝑓 + 𝑟)) = ((ℎ + 𝑡) + (𝑔 + 𝑠)) ↔ (𝑓 + 𝑟) = (𝑔 + 𝑠)))
58	47, 57	sylibd 149	. . . 4 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → ((⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ∧ ⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩) → (𝑓 + 𝑟) = (𝑔 + 𝑠)))
59	1	ecopoveq 6689	. . . . 5 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → (⟨𝑓, 𝑔⟩ ∼ ⟨𝑠, 𝑟⟩ ↔ (𝑓 + 𝑟) = (𝑔 + 𝑠)))
60	59	3adant2 1018	. . . 4 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → (⟨𝑓, 𝑔⟩ ∼ ⟨𝑠, 𝑟⟩ ↔ (𝑓 + 𝑟) = (𝑔 + 𝑠)))
61	58, 60	sylibrd 169	. . 3 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → ((⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ∧ ⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩) → ⟨𝑓, 𝑔⟩ ∼ ⟨𝑠, 𝑟⟩))
62	10, 14, 18, 22, 61	3optocl 4741	. 2 ⊢ ((𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆) ∧ 𝐶 ∈ (𝑆 × 𝑆)) → ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ 𝐶) → 𝐴 ∼ 𝐶))
63	9, 62	mpcom 36	1 ⊢ ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ 𝐶) → 𝐴 ∼ 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ⟨cop 3625   class class class wbr 4033  {copab 4093   × cxp 4661  (class class class)co 5922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-xp 4669  df-iota 5219  df-fv 5266  df-ov 5925

This theorem is referenced by:  ecopoverg  6695