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

Theorem ecopoverg 6238
 Description: Assuming that operation 𝐹 is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation ∼, specified by the first hypothesis, is an equivalence relation. (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))
ecopoprg.cl ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
ecopoprg.ass ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
ecopoprg.can ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))
Assertion
Ref Expression
ecopoverg Er (𝑆 × 𝑆)
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢, +   𝑥,𝑆,𝑦,𝑧,𝑤,𝑣,𝑢
Allowed substitution hints:   (𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem ecopoverg
Dummy variables 𝑓 𝑔 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)))}
21relopabi 4491 . . . 4 Rel
32a1i 9 . . 3 (⊤ → Rel )
4 ecopoprg.com . . . . 5 ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
51, 4ecopovsymg 6236 . . . 4 (𝑓 𝑔𝑔 𝑓)
65adantl 266 . . 3 ((⊤ ∧ 𝑓 𝑔) → 𝑔 𝑓)
7 ecopoprg.cl . . . . 5 ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
8 ecopoprg.ass . . . . 5 ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
9 ecopoprg.can . . . . 5 ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))
101, 4, 7, 8, 9ecopovtrng 6237 . . . 4 ((𝑓 𝑔𝑔 ) → 𝑓 )
1110adantl 266 . . 3 ((⊤ ∧ (𝑓 𝑔𝑔 )) → 𝑓 )
124adantl 266 . . . . . . . . . . 11 ((((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
13 simpll 489 . . . . . . . . . . 11 (((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) → 𝑔𝑆)
14 simplr 490 . . . . . . . . . . 11 (((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) → 𝑆)
1512, 13, 14caovcomd 5685 . . . . . . . . . 10 (((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) → (𝑔 + ) = ( + 𝑔))
161ecopoveq 6232 . . . . . . . . . 10 (((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) → (⟨𝑔, 𝑔, ⟩ ↔ (𝑔 + ) = ( + 𝑔)))
1715, 16mpbird 160 . . . . . . . . 9 (((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) → ⟨𝑔, 𝑔, ⟩)
1817anidms 383 . . . . . . . 8 ((𝑔𝑆𝑆) → ⟨𝑔, 𝑔, ⟩)
1918rgen2a 2392 . . . . . . 7 𝑔𝑆𝑆𝑔, 𝑔,
20 breq12 3797 . . . . . . . . 9 ((𝑓 = ⟨𝑔, ⟩ ∧ 𝑓 = ⟨𝑔, ⟩) → (𝑓 𝑓 ↔ ⟨𝑔, 𝑔, ⟩))
2120anidms 383 . . . . . . . 8 (𝑓 = ⟨𝑔, ⟩ → (𝑓 𝑓 ↔ ⟨𝑔, 𝑔, ⟩))
2221ralxp 4507 . . . . . . 7 (∀𝑓 ∈ (𝑆 × 𝑆)𝑓 𝑓 ↔ ∀𝑔𝑆𝑆𝑔, 𝑔, ⟩)
2319, 22mpbir 138 . . . . . 6 𝑓 ∈ (𝑆 × 𝑆)𝑓 𝑓
2423rspec 2390 . . . . 5 (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 𝑓)
2524a1i 9 . . . 4 (⊤ → (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 𝑓))
26 opabssxp 4442 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
271, 26eqsstri 3003 . . . . . 6 ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
2827ssbri 3834 . . . . 5 (𝑓 𝑓𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓)
29 brxp 4403 . . . . . 6 (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓 ↔ (𝑓 ∈ (𝑆 × 𝑆) ∧ 𝑓 ∈ (𝑆 × 𝑆)))
3029simplbi 263 . . . . 5 (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓𝑓 ∈ (𝑆 × 𝑆))
3128, 30syl 14 . . . 4 (𝑓 𝑓𝑓 ∈ (𝑆 × 𝑆))
3225, 31impbid1 134 . . 3 (⊤ → (𝑓 ∈ (𝑆 × 𝑆) ↔ 𝑓 𝑓))
333, 6, 11, 32iserd 6163 . 2 (⊤ → Er (𝑆 × 𝑆))
3433trud 1268 1 Er (𝑆 × 𝑆)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   ∧ w3a 896   = wceq 1259  ⊤wtru 1260  ∃wex 1397   ∈ wcel 1409  ∀wral 2323  ⟨cop 3406   class class class wbr 3792  {copab 3845   × cxp 4371  Rel wrel 4378  (class class class)co 5540   Er wer 6134 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 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fv 4938  df-ov 5543  df-er 6137 This theorem is referenced by:  enqer  6514  enrer  6878
 Copyright terms: Public domain W3C validator