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Theorem ecopovsym 6525
 Description: Assuming the operation 𝐹 is commutative, show that the relation ∼, specified by the first hypothesis, is symmetric. (Contributed by NM, 27-Aug-1995.) (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)
Assertion
Ref Expression
ecopovsym (𝐴 𝐵𝐵 𝐴)
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)   (𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem ecopovsym
Dummy variables 𝑓 𝑔 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecopopr.1 . . . . 5 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}
2 opabssxp 4613 . . . . 5 {⟨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 3129 . . . 4 ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
43brel 4591 . . 3 (𝐴 𝐵 → (𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆)))
5 eqid 2139 . . . 4 (𝑆 × 𝑆) = (𝑆 × 𝑆)
6 breq1 3932 . . . . 5 (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨𝑓, 𝑔, 𝑡⟩ ↔ 𝐴 , 𝑡⟩))
7 breq2 3933 . . . . 5 (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨, 𝑡𝑓, 𝑔⟩ ↔ ⟨, 𝑡 𝐴))
86, 7bibi12d 234 . . . 4 (⟨𝑓, 𝑔⟩ = 𝐴 → ((⟨𝑓, 𝑔, 𝑡⟩ ↔ ⟨, 𝑡𝑓, 𝑔⟩) ↔ (𝐴 , 𝑡⟩ ↔ ⟨, 𝑡 𝐴)))
9 breq2 3933 . . . . 5 (⟨, 𝑡⟩ = 𝐵 → (𝐴 , 𝑡⟩ ↔ 𝐴 𝐵))
10 breq1 3932 . . . . 5 (⟨, 𝑡⟩ = 𝐵 → (⟨, 𝑡 𝐴𝐵 𝐴))
119, 10bibi12d 234 . . . 4 (⟨, 𝑡⟩ = 𝐵 → ((𝐴 , 𝑡⟩ ↔ ⟨, 𝑡 𝐴) ↔ (𝐴 𝐵𝐵 𝐴)))
121ecopoveq 6524 . . . . . 6 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (⟨𝑓, 𝑔, 𝑡⟩ ↔ (𝑓 + 𝑡) = (𝑔 + )))
13 vex 2689 . . . . . . . . 9 𝑓 ∈ V
14 vex 2689 . . . . . . . . 9 𝑡 ∈ V
15 ecopopr.com . . . . . . . . 9 (𝑥 + 𝑦) = (𝑦 + 𝑥)
1613, 14, 15caovcom 5928 . . . . . . . 8 (𝑓 + 𝑡) = (𝑡 + 𝑓)
17 vex 2689 . . . . . . . . 9 𝑔 ∈ V
18 vex 2689 . . . . . . . . 9 ∈ V
1917, 18, 15caovcom 5928 . . . . . . . 8 (𝑔 + ) = ( + 𝑔)
2016, 19eqeq12i 2153 . . . . . . 7 ((𝑓 + 𝑡) = (𝑔 + ) ↔ (𝑡 + 𝑓) = ( + 𝑔))
21 eqcom 2141 . . . . . . 7 ((𝑡 + 𝑓) = ( + 𝑔) ↔ ( + 𝑔) = (𝑡 + 𝑓))
2220, 21bitri 183 . . . . . 6 ((𝑓 + 𝑡) = (𝑔 + ) ↔ ( + 𝑔) = (𝑡 + 𝑓))
2312, 22syl6bb 195 . . . . 5 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (⟨𝑓, 𝑔, 𝑡⟩ ↔ ( + 𝑔) = (𝑡 + 𝑓)))
241ecopoveq 6524 . . . . . 6 (((𝑆𝑡𝑆) ∧ (𝑓𝑆𝑔𝑆)) → (⟨, 𝑡𝑓, 𝑔⟩ ↔ ( + 𝑔) = (𝑡 + 𝑓)))
2524ancoms 266 . . . . 5 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (⟨, 𝑡𝑓, 𝑔⟩ ↔ ( + 𝑔) = (𝑡 + 𝑓)))
2623, 25bitr4d 190 . . . 4 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (⟨𝑓, 𝑔, 𝑡⟩ ↔ ⟨, 𝑡𝑓, 𝑔⟩))
275, 8, 11, 262optocl 4616 . . 3 ((𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆)) → (𝐴 𝐵𝐵 𝐴))
284, 27syl 14 . 2 (𝐴 𝐵 → (𝐴 𝐵𝐵 𝐴))
2928ibi 175 1 (𝐴 𝐵𝐵 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ⟨cop 3530   class class class wbr 3929  {copab 3988   × cxp 4537  (class class class)co 5774 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-iota 5088  df-fv 5131  df-ov 5777 This theorem is referenced by:  ecopover  6527
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