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Theorem ecopovtrn 6203
 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)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}
ecopopr.com (𝑥 + 𝑦) = (𝑦 + 𝑥)
ecopopr.cl ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
ecopopr.ass ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))
ecopopr.can ((𝑥𝑆𝑦𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))
Assertion
Ref Expression
ecopovtrn ((𝐴 𝐵𝐵 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 𝑡 𝑠 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecopopr.1 . . . . . . 7 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}
2 opabssxp 4414 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
31, 2eqsstri 2975 . . . . . 6 ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
43brel 4392 . . . . 5 (𝐴 𝐵 → (𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆)))
54simpld 105 . . . 4 (𝐴 𝐵𝐴 ∈ (𝑆 × 𝑆))
63brel 4392 . . . 4 (𝐵 𝐶 → (𝐵 ∈ (𝑆 × 𝑆) ∧ 𝐶 ∈ (𝑆 × 𝑆)))
75, 6anim12i 321 . . 3 ((𝐴 𝐵𝐵 𝐶) → (𝐴 ∈ (𝑆 × 𝑆) ∧ (𝐵 ∈ (𝑆 × 𝑆) ∧ 𝐶 ∈ (𝑆 × 𝑆))))
8 3anass 889 . . 3 ((𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆) ∧ 𝐶 ∈ (𝑆 × 𝑆)) ↔ (𝐴 ∈ (𝑆 × 𝑆) ∧ (𝐵 ∈ (𝑆 × 𝑆) ∧ 𝐶 ∈ (𝑆 × 𝑆))))
97, 8sylibr 137 . 2 ((𝐴 𝐵𝐵 𝐶) → (𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆) ∧ 𝐶 ∈ (𝑆 × 𝑆)))
10 eqid 2040 . . 3 (𝑆 × 𝑆) = (𝑆 × 𝑆)
11 breq1 3767 . . . . 5 (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨𝑓, 𝑔, 𝑡⟩ ↔ 𝐴 , 𝑡⟩))
1211anbi1d 438 . . . 4 (⟨𝑓, 𝑔⟩ = 𝐴 → ((⟨𝑓, 𝑔, 𝑡⟩ ∧ ⟨, 𝑡𝑠, 𝑟⟩) ↔ (𝐴 , 𝑡⟩ ∧ ⟨, 𝑡𝑠, 𝑟⟩)))
13 breq1 3767 . . . 4 (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨𝑓, 𝑔𝑠, 𝑟⟩ ↔ 𝐴 𝑠, 𝑟⟩))
1412, 13imbi12d 223 . . 3 (⟨𝑓, 𝑔⟩ = 𝐴 → (((⟨𝑓, 𝑔, 𝑡⟩ ∧ ⟨, 𝑡𝑠, 𝑟⟩) → ⟨𝑓, 𝑔𝑠, 𝑟⟩) ↔ ((𝐴 , 𝑡⟩ ∧ ⟨, 𝑡𝑠, 𝑟⟩) → 𝐴 𝑠, 𝑟⟩)))
15 breq2 3768 . . . . 5 (⟨, 𝑡⟩ = 𝐵 → (𝐴 , 𝑡⟩ ↔ 𝐴 𝐵))
16 breq1 3767 . . . . 5 (⟨, 𝑡⟩ = 𝐵 → (⟨, 𝑡𝑠, 𝑟⟩ ↔ 𝐵 𝑠, 𝑟⟩))
1715, 16anbi12d 442 . . . 4 (⟨, 𝑡⟩ = 𝐵 → ((𝐴 , 𝑡⟩ ∧ ⟨, 𝑡𝑠, 𝑟⟩) ↔ (𝐴 𝐵𝐵 𝑠, 𝑟⟩)))
1817imbi1d 220 . . 3 (⟨, 𝑡⟩ = 𝐵 → (((𝐴 , 𝑡⟩ ∧ ⟨, 𝑡𝑠, 𝑟⟩) → 𝐴 𝑠, 𝑟⟩) ↔ ((𝐴 𝐵𝐵 𝑠, 𝑟⟩) → 𝐴 𝑠, 𝑟⟩)))
19 breq2 3768 . . . . 5 (⟨𝑠, 𝑟⟩ = 𝐶 → (𝐵 𝑠, 𝑟⟩ ↔ 𝐵 𝐶))
2019anbi2d 437 . . . 4 (⟨𝑠, 𝑟⟩ = 𝐶 → ((𝐴 𝐵𝐵 𝑠, 𝑟⟩) ↔ (𝐴 𝐵𝐵 𝐶)))
21 breq2 3768 . . . 4 (⟨𝑠, 𝑟⟩ = 𝐶 → (𝐴 𝑠, 𝑟⟩ ↔ 𝐴 𝐶))
2220, 21imbi12d 223 . . 3 (⟨𝑠, 𝑟⟩ = 𝐶 → (((𝐴 𝐵𝐵 𝑠, 𝑟⟩) → 𝐴 𝑠, 𝑟⟩) ↔ ((𝐴 𝐵𝐵 𝐶) → 𝐴 𝐶)))
231ecopoveq 6201 . . . . . . . 8 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (⟨𝑓, 𝑔, 𝑡⟩ ↔ (𝑓 + 𝑡) = (𝑔 + )))
24233adant3 924 . . . . . . 7 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → (⟨𝑓, 𝑔, 𝑡⟩ ↔ (𝑓 + 𝑡) = (𝑔 + )))
251ecopoveq 6201 . . . . . . . 8 (((𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → (⟨, 𝑡𝑠, 𝑟⟩ ↔ ( + 𝑟) = (𝑡 + 𝑠)))
26253adant1 922 . . . . . . 7 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → (⟨, 𝑡𝑠, 𝑟⟩ ↔ ( + 𝑟) = (𝑡 + 𝑠)))
2724, 26anbi12d 442 . . . . . 6 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → ((⟨𝑓, 𝑔, 𝑡⟩ ∧ ⟨, 𝑡𝑠, 𝑟⟩) ↔ ((𝑓 + 𝑡) = (𝑔 + ) ∧ ( + 𝑟) = (𝑡 + 𝑠))))
28 oveq12 5521 . . . . . . 7 (((𝑓 + 𝑡) = (𝑔 + ) ∧ ( + 𝑟) = (𝑡 + 𝑠)) → ((𝑓 + 𝑡) + ( + 𝑟)) = ((𝑔 + ) + (𝑡 + 𝑠)))
29 simp2l 930 . . . . . . . . 9 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → 𝑆)
30 simp2r 931 . . . . . . . . 9 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → 𝑡𝑆)
31 simp1l 928 . . . . . . . . 9 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → 𝑓𝑆)
32 ecopopr.com . . . . . . . . . 10 (𝑥 + 𝑦) = (𝑦 + 𝑥)
3332a1i 9 . . . . . . . . 9 ((((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
34 ecopopr.ass . . . . . . . . . 10 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))
3534a1i 9 . . . . . . . . 9 ((((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
36 simp3r 933 . . . . . . . . 9 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → 𝑟𝑆)
37 ecopopr.cl . . . . . . . . . 10 ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
3837adantl 262 . . . . . . . . 9 ((((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
3929, 30, 31, 33, 35, 36, 38caov411d 5686 . . . . . . . 8 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → (( + 𝑡) + (𝑓 + 𝑟)) = ((𝑓 + 𝑡) + ( + 𝑟)))
40 simp1r 929 . . . . . . . . . 10 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → 𝑔𝑆)
41 simp3l 932 . . . . . . . . . 10 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → 𝑠𝑆)
4240, 30, 29, 33, 35, 41, 38caov411d 5686 . . . . . . . . 9 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → ((𝑔 + 𝑡) + ( + 𝑠)) = (( + 𝑡) + (𝑔 + 𝑠)))
4340, 30, 29, 33, 35, 41, 38caov4d 5685 . . . . . . . . 9 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → ((𝑔 + 𝑡) + ( + 𝑠)) = ((𝑔 + ) + (𝑡 + 𝑠)))
4442, 43eqtr3d 2074 . . . . . . . 8 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → (( + 𝑡) + (𝑔 + 𝑠)) = ((𝑔 + ) + (𝑡 + 𝑠)))
4539, 44eqeq12d 2054 . . . . . . 7 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → ((( + 𝑡) + (𝑓 + 𝑟)) = (( + 𝑡) + (𝑔 + 𝑠)) ↔ ((𝑓 + 𝑡) + ( + 𝑟)) = ((𝑔 + ) + (𝑡 + 𝑠))))
4628, 45syl5ibr 145 . . . . . 6 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → (((𝑓 + 𝑡) = (𝑔 + ) ∧ ( + 𝑟) = (𝑡 + 𝑠)) → (( + 𝑡) + (𝑓 + 𝑟)) = (( + 𝑡) + (𝑔 + 𝑠))))
4727, 46sylbid 139 . . . . 5 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → ((⟨𝑓, 𝑔, 𝑡⟩ ∧ ⟨, 𝑡𝑠, 𝑟⟩) → (( + 𝑡) + (𝑓 + 𝑟)) = (( + 𝑡) + (𝑔 + 𝑠))))
48 ecopopr.can . . . . . . . . 9 ((𝑥𝑆𝑦𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))
49483adant3 924 . . . . . . . 8 ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))
50 oveq2 5520 . . . . . . . 8 (𝑦 = 𝑧 → (𝑥 + 𝑦) = (𝑥 + 𝑧))
5149, 50impbid1 130 . . . . . . 7 ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) ↔ 𝑦 = 𝑧))
5251adantl 262 . . . . . 6 ((((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) ↔ 𝑦 = 𝑧))
5337caovcl 5655 . . . . . . 7 ((𝑆𝑡𝑆) → ( + 𝑡) ∈ 𝑆)
5429, 30, 53syl2anc 391 . . . . . 6 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → ( + 𝑡) ∈ 𝑆)
5537caovcl 5655 . . . . . . 7 ((𝑓𝑆𝑟𝑆) → (𝑓 + 𝑟) ∈ 𝑆)
5631, 36, 55syl2anc 391 . . . . . 6 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → (𝑓 + 𝑟) ∈ 𝑆)
5738, 40, 41caovcld 5654 . . . . . 6 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → (𝑔 + 𝑠) ∈ 𝑆)
5852, 54, 56, 57caovcand 5663 . . . . 5 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → ((( + 𝑡) + (𝑓 + 𝑟)) = (( + 𝑡) + (𝑔 + 𝑠)) ↔ (𝑓 + 𝑟) = (𝑔 + 𝑠)))
5947, 58sylibd 138 . . . 4 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → ((⟨𝑓, 𝑔, 𝑡⟩ ∧ ⟨, 𝑡𝑠, 𝑟⟩) → (𝑓 + 𝑟) = (𝑔 + 𝑠)))
601ecopoveq 6201 . . . . 5 (((𝑓𝑆𝑔𝑆) ∧ (𝑠𝑆𝑟𝑆)) → (⟨𝑓, 𝑔𝑠, 𝑟⟩ ↔ (𝑓 + 𝑟) = (𝑔 + 𝑠)))
61603adant2 923 . . . 4 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → (⟨𝑓, 𝑔𝑠, 𝑟⟩ ↔ (𝑓 + 𝑟) = (𝑔 + 𝑠)))
6259, 61sylibrd 158 . . 3 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆) ∧ (𝑠𝑆𝑟𝑆)) → ((⟨𝑓, 𝑔, 𝑡⟩ ∧ ⟨, 𝑡𝑠, 𝑟⟩) → ⟨𝑓, 𝑔𝑠, 𝑟⟩))
6310, 14, 18, 22, 623optocl 4418 . 2 ((𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆) ∧ 𝐶 ∈ (𝑆 × 𝑆)) → ((𝐴 𝐵𝐵 𝐶) → 𝐴 𝐶))
649, 63mpcom 32 1 ((𝐴 𝐵𝐵 𝐶) → 𝐴 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ⟨cop 3378   class class class wbr 3764  {copab 3817   × cxp 4343  (class class class)co 5512 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-iota 4867  df-fv 4910  df-ov 5515 This theorem is referenced by:  ecopover  6204
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