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Theorem ecopovsymg 6233
Description: Assuming the operation 𝐹 is commutative, show that the relation , specified by the first hypothesis, is symmetric. (Contributed by Jim Kingdon, 1-Sep-2019.)
Hypotheses
Ref Expression
ecopopr.1 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}
ecopoprg.com ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
Assertion
Ref Expression
ecopovsymg (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 ecopovsymg
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 4439 . . . . 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 3000 . . . 4 ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
43brel 4417 . . 3 (𝐴 𝐵 → (𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆)))
5 eqid 2054 . . . 4 (𝑆 × 𝑆) = (𝑆 × 𝑆)
6 breq1 3792 . . . . 5 (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨𝑓, 𝑔, 𝑡⟩ ↔ 𝐴 , 𝑡⟩))
7 breq2 3793 . . . . 5 (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨, 𝑡𝑓, 𝑔⟩ ↔ ⟨, 𝑡 𝐴))
86, 7bibi12d 228 . . . 4 (⟨𝑓, 𝑔⟩ = 𝐴 → ((⟨𝑓, 𝑔, 𝑡⟩ ↔ ⟨, 𝑡𝑓, 𝑔⟩) ↔ (𝐴 , 𝑡⟩ ↔ ⟨, 𝑡 𝐴)))
9 breq2 3793 . . . . 5 (⟨, 𝑡⟩ = 𝐵 → (𝐴 , 𝑡⟩ ↔ 𝐴 𝐵))
10 breq1 3792 . . . . 5 (⟨, 𝑡⟩ = 𝐵 → (⟨, 𝑡 𝐴𝐵 𝐴))
119, 10bibi12d 228 . . . 4 (⟨, 𝑡⟩ = 𝐵 → ((𝐴 , 𝑡⟩ ↔ ⟨, 𝑡 𝐴) ↔ (𝐴 𝐵𝐵 𝐴)))
12 ecopoprg.com . . . . . . . . 9 ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
1312adantl 266 . . . . . . . 8 ((((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
14 simpll 489 . . . . . . . 8 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → 𝑓𝑆)
15 simprr 492 . . . . . . . 8 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → 𝑡𝑆)
1613, 14, 15caovcomd 5682 . . . . . . 7 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (𝑓 + 𝑡) = (𝑡 + 𝑓))
17 simplr 490 . . . . . . . 8 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → 𝑔𝑆)
18 simprl 491 . . . . . . . 8 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → 𝑆)
1913, 17, 18caovcomd 5682 . . . . . . 7 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (𝑔 + ) = ( + 𝑔))
2016, 19eqeq12d 2068 . . . . . 6 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → ((𝑓 + 𝑡) = (𝑔 + ) ↔ (𝑡 + 𝑓) = ( + 𝑔)))
21 eqcom 2056 . . . . . 6 ((𝑡 + 𝑓) = ( + 𝑔) ↔ ( + 𝑔) = (𝑡 + 𝑓))
2220, 21syl6bb 189 . . . . 5 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → ((𝑓 + 𝑡) = (𝑔 + ) ↔ ( + 𝑔) = (𝑡 + 𝑓)))
231ecopoveq 6229 . . . . 5 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (⟨𝑓, 𝑔, 𝑡⟩ ↔ (𝑓 + 𝑡) = (𝑔 + )))
241ecopoveq 6229 . . . . . 6 (((𝑆𝑡𝑆) ∧ (𝑓𝑆𝑔𝑆)) → (⟨, 𝑡𝑓, 𝑔⟩ ↔ ( + 𝑔) = (𝑡 + 𝑓)))
2524ancoms 259 . . . . 5 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (⟨, 𝑡𝑓, 𝑔⟩ ↔ ( + 𝑔) = (𝑡 + 𝑓)))
2622, 23, 253bitr4d 213 . . . 4 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (⟨𝑓, 𝑔, 𝑡⟩ ↔ ⟨, 𝑡𝑓, 𝑔⟩))
275, 8, 11, 262optocl 4442 . . 3 ((𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆)) → (𝐴 𝐵𝐵 𝐴))
284, 27syl 14 . 2 (𝐴 𝐵 → (𝐴 𝐵𝐵 𝐴))
2928ibi 169 1 (𝐴 𝐵𝐵 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1257  wex 1395  wcel 1407  cop 3403   class class class wbr 3789  {copab 3842   × cxp 4368  (class class class)co 5537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-br 3790  df-opab 3844  df-xp 4376  df-iota 4892  df-fv 4935  df-ov 5540
This theorem is referenced by:  ecopoverg  6235
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