ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ecopovsymg GIF version

Theorem ecopovsymg 6292
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 4460 . . . . 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 3038 . . . 4 ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
43brel 4438 . . 3 (𝐴 𝐵 → (𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆)))
5 eqid 2083 . . . 4 (𝑆 × 𝑆) = (𝑆 × 𝑆)
6 breq1 3808 . . . . 5 (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨𝑓, 𝑔, 𝑡⟩ ↔ 𝐴 , 𝑡⟩))
7 breq2 3809 . . . . 5 (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨, 𝑡𝑓, 𝑔⟩ ↔ ⟨, 𝑡 𝐴))
86, 7bibi12d 233 . . . 4 (⟨𝑓, 𝑔⟩ = 𝐴 → ((⟨𝑓, 𝑔, 𝑡⟩ ↔ ⟨, 𝑡𝑓, 𝑔⟩) ↔ (𝐴 , 𝑡⟩ ↔ ⟨, 𝑡 𝐴)))
9 breq2 3809 . . . . 5 (⟨, 𝑡⟩ = 𝐵 → (𝐴 , 𝑡⟩ ↔ 𝐴 𝐵))
10 breq1 3808 . . . . 5 (⟨, 𝑡⟩ = 𝐵 → (⟨, 𝑡 𝐴𝐵 𝐴))
119, 10bibi12d 233 . . . 4 (⟨, 𝑡⟩ = 𝐵 → ((𝐴 , 𝑡⟩ ↔ ⟨, 𝑡 𝐴) ↔ (𝐴 𝐵𝐵 𝐴)))
12 ecopoprg.com . . . . . . . . 9 ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
1312adantl 271 . . . . . . . 8 ((((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
14 simpll 496 . . . . . . . 8 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → 𝑓𝑆)
15 simprr 499 . . . . . . . 8 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → 𝑡𝑆)
1613, 14, 15caovcomd 5708 . . . . . . 7 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (𝑓 + 𝑡) = (𝑡 + 𝑓))
17 simplr 497 . . . . . . . 8 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → 𝑔𝑆)
18 simprl 498 . . . . . . . 8 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → 𝑆)
1913, 17, 18caovcomd 5708 . . . . . . 7 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (𝑔 + ) = ( + 𝑔))
2016, 19eqeq12d 2097 . . . . . 6 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → ((𝑓 + 𝑡) = (𝑔 + ) ↔ (𝑡 + 𝑓) = ( + 𝑔)))
21 eqcom 2085 . . . . . 6 ((𝑡 + 𝑓) = ( + 𝑔) ↔ ( + 𝑔) = (𝑡 + 𝑓))
2220, 21syl6bb 194 . . . . 5 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → ((𝑓 + 𝑡) = (𝑔 + ) ↔ ( + 𝑔) = (𝑡 + 𝑓)))
231ecopoveq 6288 . . . . 5 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (⟨𝑓, 𝑔, 𝑡⟩ ↔ (𝑓 + 𝑡) = (𝑔 + )))
241ecopoveq 6288 . . . . . 6 (((𝑆𝑡𝑆) ∧ (𝑓𝑆𝑔𝑆)) → (⟨, 𝑡𝑓, 𝑔⟩ ↔ ( + 𝑔) = (𝑡 + 𝑓)))
2524ancoms 264 . . . . 5 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (⟨, 𝑡𝑓, 𝑔⟩ ↔ ( + 𝑔) = (𝑡 + 𝑓)))
2622, 23, 253bitr4d 218 . . . 4 (((𝑓𝑆𝑔𝑆) ∧ (𝑆𝑡𝑆)) → (⟨𝑓, 𝑔, 𝑡⟩ ↔ ⟨, 𝑡𝑓, 𝑔⟩))
275, 8, 11, 262optocl 4463 . . 3 ((𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆)) → (𝐴 𝐵𝐵 𝐴))
284, 27syl 14 . 2 (𝐴 𝐵 → (𝐴 𝐵𝐵 𝐴))
2928ibi 174 1 (𝐴 𝐵𝐵 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wex 1422  wcel 1434  cop 3419   class class class wbr 3805  {copab 3858   × cxp 4389  (class class class)co 5563
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-xp 4397  df-iota 4917  df-fv 4960  df-ov 5566
This theorem is referenced by:  ecopoverg  6294
  Copyright terms: Public domain W3C validator