Theorem ecopovtrn 8824

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 NM, 11-Feb-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)

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)))}
ecopopr.com	⊢ (𝑥 + 𝑦) = (𝑦 + 𝑥)
ecopopr.cl	⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
ecopopr.ass	⊢ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))
ecopopr.can	⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))

Assertion

Ref	Expression
ecopovtrn	⊢ ((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 ecopovtrn

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 5755	. . . . . . 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 3984	. . . . . 6 ⊢ ∼ ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
4	3	brel 5728	. . . . 5 ⊢ (𝐴 ∼ 𝐵 → (𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆)))
5	4	simpld 500	. . . 4 ⊢ (𝐴 ∼ 𝐵 → 𝐴 ∈ (𝑆 × 𝑆))
6	3	brel 5728	. . . 4 ⊢ (𝐵 ∼ 𝐶 → (𝐵 ∈ (𝑆 × 𝑆) ∧ 𝐶 ∈ (𝑆 × 𝑆)))
7	5, 6	anim12i 625	. . 3 ⊢ ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ 𝐶) → (𝐴 ∈ (𝑆 × 𝑆) ∧ (𝐵 ∈ (𝑆 × 𝑆) ∧ 𝐶 ∈ (𝑆 × 𝑆))))
8		3anass 1111	. . 3 ⊢ ((𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆) ∧ 𝐶 ∈ (𝑆 × 𝑆)) ↔ (𝐴 ∈ (𝑆 × 𝑆) ∧ (𝐵 ∈ (𝑆 × 𝑆) ∧ 𝐶 ∈ (𝑆 × 𝑆))))
9	7, 8	sylibr 237	. 2 ⊢ ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ 𝐶) → (𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆) ∧ 𝐶 ∈ (𝑆 × 𝑆)))
10		eqid 2765	. . 3 ⊢ (𝑆 × 𝑆) = (𝑆 × 𝑆)
11		breq1 5114	. . . . 5 ⊢ (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ↔ 𝐴 ∼ ⟨ℎ, 𝑡⟩))
12	11	anbi1d 643	. . . 4 ⊢ (⟨𝑓, 𝑔⟩ = 𝐴 → ((⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ∧ ⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩) ↔ (𝐴 ∼ ⟨ℎ, 𝑡⟩ ∧ ⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩)))
13		breq1 5114	. . . 4 ⊢ (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨𝑓, 𝑔⟩ ∼ ⟨𝑠, 𝑟⟩ ↔ 𝐴 ∼ ⟨𝑠, 𝑟⟩))
14	12, 13	imbi12d 347	. . 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 5115	. . . . 5 ⊢ (⟨ℎ, 𝑡⟩ = 𝐵 → (𝐴 ∼ ⟨ℎ, 𝑡⟩ ↔ 𝐴 ∼ 𝐵))
16		breq1 5114	. . . . 5 ⊢ (⟨ℎ, 𝑡⟩ = 𝐵 → (⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩ ↔ 𝐵 ∼ ⟨𝑠, 𝑟⟩))
17	15, 16	anbi12d 644	. . . 4 ⊢ (⟨ℎ, 𝑡⟩ = 𝐵 → ((𝐴 ∼ ⟨ℎ, 𝑡⟩ ∧ ⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩) ↔ (𝐴 ∼ 𝐵 ∧ 𝐵 ∼ ⟨𝑠, 𝑟⟩)))
18	17	imbi1d 344	. . 3 ⊢ (⟨ℎ, 𝑡⟩ = 𝐵 → (((𝐴 ∼ ⟨ℎ, 𝑡⟩ ∧ ⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩) → 𝐴 ∼ ⟨𝑠, 𝑟⟩) ↔ ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ ⟨𝑠, 𝑟⟩) → 𝐴 ∼ ⟨𝑠, 𝑟⟩)))
19		breq2 5115	. . . . 5 ⊢ (⟨𝑠, 𝑟⟩ = 𝐶 → (𝐵 ∼ ⟨𝑠, 𝑟⟩ ↔ 𝐵 ∼ 𝐶))
20	19	anbi2d 642	. . . 4 ⊢ (⟨𝑠, 𝑟⟩ = 𝐶 → ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ ⟨𝑠, 𝑟⟩) ↔ (𝐴 ∼ 𝐵 ∧ 𝐵 ∼ 𝐶)))
21		breq2 5115	. . . 4 ⊢ (⟨𝑠, 𝑟⟩ = 𝐶 → (𝐴 ∼ ⟨𝑠, 𝑟⟩ ↔ 𝐴 ∼ 𝐶))
22	20, 21	imbi12d 347	. . 3 ⊢ (⟨𝑠, 𝑟⟩ = 𝐶 → (((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ ⟨𝑠, 𝑟⟩) → 𝐴 ∼ ⟨𝑠, 𝑟⟩) ↔ ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ 𝐶) → 𝐴 ∼ 𝐶)))
23	1	ecopoveq 8822	. . . . . . . 8 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → (⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ↔ (𝑓 + 𝑡) = (𝑔 + ℎ)))
24	23	3adant3 1150	. . . . . . 7 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → (⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ↔ (𝑓 + 𝑡) = (𝑔 + ℎ)))
25	1	ecopoveq 8822	. . . . . . . 8 ⊢ (((ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → (⟨ℎ, 𝑡⟩ ∼ ⟨𝑠, 𝑟⟩ ↔ (ℎ + 𝑟) = (𝑡 + 𝑠)))
26	25	3adant1 1148	. . . . . . 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 644	. . . . . 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 7428	. . . . . . 7 ⊢ (((𝑓 + 𝑡) = (𝑔 + ℎ) ∧ (ℎ + 𝑟) = (𝑡 + 𝑠)) → ((𝑓 + 𝑡) + (ℎ + 𝑟)) = ((𝑔 + ℎ) + (𝑡 + 𝑠)))
29		vex 3461	. . . . . . . 8 ⊢ ℎ ∈ V
30		vex 3461	. . . . . . . 8 ⊢ 𝑡 ∈ V
31		vex 3461	. . . . . . . 8 ⊢ 𝑓 ∈ V
32		ecopopr.com	. . . . . . . 8 ⊢ (𝑥 + 𝑦) = (𝑦 + 𝑥)
33		ecopopr.ass	. . . . . . . 8 ⊢ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))
34		vex 3461	. . . . . . . 8 ⊢ 𝑟 ∈ V
35	29, 30, 31, 32, 33, 34	caov411 7652	. . . . . . 7 ⊢ ((ℎ + 𝑡) + (𝑓 + 𝑟)) = ((𝑓 + 𝑡) + (ℎ + 𝑟))
36		vex 3461	. . . . . . . . 9 ⊢ 𝑔 ∈ V
37		vex 3461	. . . . . . . . 9 ⊢ 𝑠 ∈ V
38	36, 30, 29, 32, 33, 37	caov411 7652	. . . . . . . 8 ⊢ ((𝑔 + 𝑡) + (ℎ + 𝑠)) = ((ℎ + 𝑡) + (𝑔 + 𝑠))
39	36, 30, 29, 32, 33, 37	caov4 7651	. . . . . . . 8 ⊢ ((𝑔 + 𝑡) + (ℎ + 𝑠)) = ((𝑔 + ℎ) + (𝑡 + 𝑠))
40	38, 39	eqtr3i 2790	. . . . . . 7 ⊢ ((ℎ + 𝑡) + (𝑔 + 𝑠)) = ((𝑔 + ℎ) + (𝑡 + 𝑠))
41	28, 35, 40	3eqtr4g 2825	. . . . . 6 ⊢ (((𝑓 + 𝑡) = (𝑔 + ℎ) ∧ (ℎ + 𝑟) = (𝑡 + 𝑠)) → ((ℎ + 𝑡) + (𝑓 + 𝑟)) = ((ℎ + 𝑡) + (𝑔 + 𝑠)))
42	27, 41	biimtrdi 256	. . . . 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))))
43		ecopopr.cl	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
44	43	caovcl 7614	. . . . . . . . . 10 ⊢ ((ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) → (ℎ + 𝑡) ∈ 𝑆)
45	43	caovcl 7614	. . . . . . . . . 10 ⊢ ((𝑓 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆) → (𝑓 + 𝑟) ∈ 𝑆)
46		ovex 7452	. . . . . . . . . . 11 ⊢ (𝑔 + 𝑠) ∈ V
47		ecopopr.can	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))
48	46, 47	caovcan 7624	. . . . . . . . . 10 ⊢ (((ℎ + 𝑡) ∈ 𝑆 ∧ (𝑓 + 𝑟) ∈ 𝑆) → (((ℎ + 𝑡) + (𝑓 + 𝑟)) = ((ℎ + 𝑡) + (𝑔 + 𝑠)) → (𝑓 + 𝑟) = (𝑔 + 𝑠)))
49	44, 45, 48	syl2an 608	. . . . . . . . 9 ⊢ (((ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑓 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → (((ℎ + 𝑡) + (𝑓 + 𝑟)) = ((ℎ + 𝑡) + (𝑔 + 𝑠)) → (𝑓 + 𝑟) = (𝑔 + 𝑠)))
50	49	3impb 1132	. . . . . . . 8 ⊢ (((ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ 𝑓 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆) → (((ℎ + 𝑡) + (𝑓 + 𝑟)) = ((ℎ + 𝑡) + (𝑔 + 𝑠)) → (𝑓 + 𝑟) = (𝑔 + 𝑠)))
51	50	3com12 1141	. . . . . . 7 ⊢ ((𝑓 ∈ 𝑆 ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ 𝑟 ∈ 𝑆) → (((ℎ + 𝑡) + (𝑓 + 𝑟)) = ((ℎ + 𝑡) + (𝑔 + 𝑠)) → (𝑓 + 𝑟) = (𝑔 + 𝑠)))
52	51	3adant3l 1199	. . . . . 6 ⊢ ((𝑓 ∈ 𝑆 ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → (((ℎ + 𝑡) + (𝑓 + 𝑟)) = ((ℎ + 𝑡) + (𝑔 + 𝑠)) → (𝑓 + 𝑟) = (𝑔 + 𝑠)))
53	52	3adant1r 1196	. . . . 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)))
54	42, 53	syld 48	. . . 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)))
55	1	ecopoveq 8822	. . . . 5 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → (⟨𝑓, 𝑔⟩ ∼ ⟨𝑠, 𝑟⟩ ↔ (𝑓 + 𝑟) = (𝑔 + 𝑠)))
56	55	3adant2 1149	. . . 4 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑠 ∈ 𝑆 ∧ 𝑟 ∈ 𝑆)) → (⟨𝑓, 𝑔⟩ ∼ ⟨𝑠, 𝑟⟩ ↔ (𝑓 + 𝑟) = (𝑔 + 𝑠)))
57	54, 56	sylibrd 262	. . 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⟩))
58	10, 14, 18, 22, 57	3optocl 5760	. 2 ⊢ ((𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆) ∧ 𝐶 ∈ (𝑆 × 𝑆)) → ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ 𝐶) → 𝐴 ∼ 𝐶))
59	9, 58	mpcom 39	1 ⊢ ((𝐴 ∼ 𝐵 ∧ 𝐵 ∼ 𝐶) → 𝐴 ∼ 𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ⟨cop 4597   class class class wbr 5111  {copab 5175   × cxp 5661  (class class class)co 7419

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-xp 5669  df-iota 6496  df-fv 6548  df-ov 7422

This theorem is used by:  ecopover  8825