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Theorem ecopovsymg 6205
 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 4414 . . . . 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 2975 . . . 4 ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
43brel 4392 . . 3 (𝐴 𝐵 → (𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆)))
5 eqid 2040 . . . 4 (𝑆 × 𝑆) = (𝑆 × 𝑆)
6 breq1 3767 . . . . 5 (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨𝑓, 𝑔, 𝑡⟩ ↔ 𝐴 , 𝑡⟩))
7 breq2 3768 . . . . 5 (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨, 𝑡𝑓, 𝑔⟩ ↔ ⟨, 𝑡 𝐴))
86, 7bibi12d 224 . . . 4 (⟨𝑓, 𝑔⟩ = 𝐴 → ((⟨𝑓, 𝑔, 𝑡⟩ ↔ ⟨, 𝑡𝑓, 𝑔⟩) ↔ (𝐴 , 𝑡⟩ ↔ ⟨, 𝑡 𝐴)))
9 breq2 3768 . . . . 5 (⟨, 𝑡⟩ = 𝐵 → (𝐴 , 𝑡⟩ ↔ 𝐴 𝐵))
10 breq1 3767 . . . . 5 (⟨, 𝑡⟩ = 𝐵 → (⟨, 𝑡 𝐴𝐵 𝐴))
119, 10bibi12d 224 . . . 4 (⟨, 𝑡⟩ = 𝐵 → ((𝐴 , 𝑡⟩ ↔ ⟨, 𝑡 𝐴) ↔ (𝐴 𝐵𝐵 𝐴)))
12 ecopoprg.com . . . . . . . . 9 ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
1312adantl 262 . . . . . . . 8 ((((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
14 simpll 481 . . . . . . . 8 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → 𝑓𝑆)
15 simprr 484 . . . . . . . 8 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → 𝑡𝑆)
1613, 14, 15caovcomd 5657 . . . . . . 7 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (𝑓 + 𝑡) = (𝑡 + 𝑓))
17 simplr 482 . . . . . . . 8 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → 𝑔𝑆)
18 simprl 483 . . . . . . . 8 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → 𝑆)
1913, 17, 18caovcomd 5657 . . . . . . 7 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (𝑔 + ) = ( + 𝑔))
2016, 19eqeq12d 2054 . . . . . 6 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → ((𝑓 + 𝑡) = (𝑔 + ) ↔ (𝑡 + 𝑓) = ( + 𝑔)))
21 eqcom 2042 . . . . . 6 ((𝑡 + 𝑓) = ( + 𝑔) ↔ ( + 𝑔) = (𝑡 + 𝑓))
2220, 21syl6bb 185 . . . . 5 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → ((𝑓 + 𝑡) = (𝑔 + ) ↔ ( + 𝑔) = (𝑡 + 𝑓)))
231ecopoveq 6201 . . . . 5 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (⟨𝑓, 𝑔, 𝑡⟩ ↔ (𝑓 + 𝑡) = (𝑔 + )))
241ecopoveq 6201 . . . . . 6 (((𝑆𝑡𝑆) ∧ (𝑓𝑆𝑔𝑆)) → (⟨, 𝑡𝑓, 𝑔⟩ ↔ ( + 𝑔) = (𝑡 + 𝑓)))
2524ancoms 255 . . . . 5 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (⟨, 𝑡𝑓, 𝑔⟩ ↔ ( + 𝑔) = (𝑡 + 𝑓)))
2622, 23, 253bitr4d 209 . . . 4 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (⟨𝑓, 𝑔, 𝑡⟩ ↔ ⟨, 𝑡𝑓, 𝑔⟩))
275, 8, 11, 262optocl 4417 . . 3 ((𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆)) → (𝐴 𝐵𝐵 𝐴))
284, 27syl 14 . 2 (𝐴 𝐵 → (𝐴 𝐵𝐵 𝐴))
2928ibi 165 1 (𝐴 𝐵𝐵 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = 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:  ecopoverg  6207
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