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Theorem ecopovsym 8053
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 5363 . . . . 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 3795 . . . 4 ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
43brel 5336 . . 3 (𝐴 𝐵 → (𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆)))
5 eqid 2765 . . . 4 (𝑆 × 𝑆) = (𝑆 × 𝑆)
6 breq1 4812 . . . . 5 (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨𝑓, 𝑔, 𝑡⟩ ↔ 𝐴 , 𝑡⟩))
7 breq2 4813 . . . . 5 (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨, 𝑡𝑓, 𝑔⟩ ↔ ⟨, 𝑡 𝐴))
86, 7bibi12d 336 . . . 4 (⟨𝑓, 𝑔⟩ = 𝐴 → ((⟨𝑓, 𝑔, 𝑡⟩ ↔ ⟨, 𝑡𝑓, 𝑔⟩) ↔ (𝐴 , 𝑡⟩ ↔ ⟨, 𝑡 𝐴)))
9 breq2 4813 . . . . 5 (⟨, 𝑡⟩ = 𝐵 → (𝐴 , 𝑡⟩ ↔ 𝐴 𝐵))
10 breq1 4812 . . . . 5 (⟨, 𝑡⟩ = 𝐵 → (⟨, 𝑡 𝐴𝐵 𝐴))
119, 10bibi12d 336 . . . 4 (⟨, 𝑡⟩ = 𝐵 → ((𝐴 , 𝑡⟩ ↔ ⟨, 𝑡 𝐴) ↔ (𝐴 𝐵𝐵 𝐴)))
121ecopoveq 8052 . . . . . 6 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (⟨𝑓, 𝑔, 𝑡⟩ ↔ (𝑓 + 𝑡) = (𝑔 + )))
13 vex 3353 . . . . . . . . 9 𝑓 ∈ V
14 vex 3353 . . . . . . . . 9 𝑡 ∈ V
15 ecopopr.com . . . . . . . . 9 (𝑥 + 𝑦) = (𝑦 + 𝑥)
1613, 14, 15caovcom 7029 . . . . . . . 8 (𝑓 + 𝑡) = (𝑡 + 𝑓)
17 vex 3353 . . . . . . . . 9 𝑔 ∈ V
18 vex 3353 . . . . . . . . 9 ∈ V
1917, 18, 15caovcom 7029 . . . . . . . 8 (𝑔 + ) = ( + 𝑔)
2016, 19eqeq12i 2779 . . . . . . 7 ((𝑓 + 𝑡) = (𝑔 + ) ↔ (𝑡 + 𝑓) = ( + 𝑔))
21 eqcom 2772 . . . . . . 7 ((𝑡 + 𝑓) = ( + 𝑔) ↔ ( + 𝑔) = (𝑡 + 𝑓))
2220, 21bitri 266 . . . . . 6 ((𝑓 + 𝑡) = (𝑔 + ) ↔ ( + 𝑔) = (𝑡 + 𝑓))
2312, 22syl6bb 278 . . . . 5 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (⟨𝑓, 𝑔, 𝑡⟩ ↔ ( + 𝑔) = (𝑡 + 𝑓)))
241ecopoveq 8052 . . . . . 6 (((𝑆𝑡𝑆) ∧ (𝑓𝑆𝑔𝑆)) → (⟨, 𝑡𝑓, 𝑔⟩ ↔ ( + 𝑔) = (𝑡 + 𝑓)))
2524ancoms 450 . . . . 5 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (⟨, 𝑡𝑓, 𝑔⟩ ↔ ( + 𝑔) = (𝑡 + 𝑓)))
2623, 25bitr4d 273 . . . 4 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (⟨𝑓, 𝑔, 𝑡⟩ ↔ ⟨, 𝑡𝑓, 𝑔⟩))
275, 8, 11, 262optocl 5366 . . 3 ((𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆)) → (𝐴 𝐵𝐵 𝐴))
284, 27syl 17 . 2 (𝐴 𝐵 → (𝐴 𝐵𝐵 𝐴))
2928ibi 258 1 (𝐴 𝐵𝐵 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  cop 4340   class class class wbr 4809  {copab 4871   × cxp 5275  (class class class)co 6842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-xp 5283  df-iota 6031  df-fv 6076  df-ov 6845
This theorem is referenced by:  ecopover  8055
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