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Theorem ecopover 6232
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 NM, 16-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-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)
ecopopr.cl ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
ecopopr.ass ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))
ecopopr.can ((𝑥𝑆𝑦𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))
Assertion
Ref Expression
ecopover 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 ecopover
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 4488 . . . 4 Rel
32a1i 9 . . 3 (⊤ → Rel )
4 ecopopr.com . . . . 5 (𝑥 + 𝑦) = (𝑦 + 𝑥)
51, 4ecopovsym 6230 . . . 4 (𝑓 𝑔𝑔 𝑓)
65adantl 266 . . 3 ((⊤ ∧ 𝑓 𝑔) → 𝑔 𝑓)
7 ecopopr.cl . . . . 5 ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
8 ecopopr.ass . . . . 5 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))
9 ecopopr.can . . . . 5 ((𝑥𝑆𝑦𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))
101, 4, 7, 8, 9ecopovtrn 6231 . . . 4 ((𝑓 𝑔𝑔 ) → 𝑓 )
1110adantl 266 . . 3 ((⊤ ∧ (𝑓 𝑔𝑔 )) → 𝑓 )
12 vex 2575 . . . . . . . . . . 11 𝑔 ∈ V
13 vex 2575 . . . . . . . . . . 11 ∈ V
1412, 13, 4caovcom 5683 . . . . . . . . . 10 (𝑔 + ) = ( + 𝑔)
151ecopoveq 6229 . . . . . . . . . 10 (((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) → (⟨𝑔, 𝑔, ⟩ ↔ (𝑔 + ) = ( + 𝑔)))
1614, 15mpbiri 161 . . . . . . . . 9 (((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) → ⟨𝑔, 𝑔, ⟩)
1716anidms 383 . . . . . . . 8 ((𝑔𝑆𝑆) → ⟨𝑔, 𝑔, ⟩)
1817rgen2a 2390 . . . . . . 7 𝑔𝑆𝑆𝑔, 𝑔,
19 breq12 3794 . . . . . . . . 9 ((𝑓 = ⟨𝑔, ⟩ ∧ 𝑓 = ⟨𝑔, ⟩) → (𝑓 𝑓 ↔ ⟨𝑔, 𝑔, ⟩))
2019anidms 383 . . . . . . . 8 (𝑓 = ⟨𝑔, ⟩ → (𝑓 𝑓 ↔ ⟨𝑔, 𝑔, ⟩))
2120ralxp 4504 . . . . . . 7 (∀𝑓 ∈ (𝑆 × 𝑆)𝑓 𝑓 ↔ ∀𝑔𝑆𝑆𝑔, 𝑔, ⟩)
2218, 21mpbir 138 . . . . . 6 𝑓 ∈ (𝑆 × 𝑆)𝑓 𝑓
2322rspec 2388 . . . . 5 (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 𝑓)
2423a1i 9 . . . 4 (⊤ → (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 𝑓))
25 opabssxp 4439 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
261, 25eqsstri 3000 . . . . . 6 ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
2726ssbri 3831 . . . . 5 (𝑓 𝑓𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓)
28 brxp 4400 . . . . . 6 (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓 ↔ (𝑓 ∈ (𝑆 × 𝑆) ∧ 𝑓 ∈ (𝑆 × 𝑆)))
2928simplbi 263 . . . . 5 (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓𝑓 ∈ (𝑆 × 𝑆))
3027, 29syl 14 . . . 4 (𝑓 𝑓𝑓 ∈ (𝑆 × 𝑆))
3124, 30impbid1 134 . . 3 (⊤ → (𝑓 ∈ (𝑆 × 𝑆) ↔ 𝑓 𝑓))
323, 6, 11, 31iserd 6160 . 2 (⊤ → Er (𝑆 × 𝑆))
3332trud 1266 1 Er (𝑆 × 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1257  wtru 1258  wex 1395  wcel 1407  wral 2321  cop 3403   class class class wbr 3789  {copab 3842   × cxp 4368  Rel wrel 4375  (class class class)co 5537   Er wer 6131
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-sbc 2785  df-csb 2878  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-iun 3684  df-br 3790  df-opab 3844  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-iota 4892  df-fv 4935  df-ov 5540  df-er 6134
This theorem is referenced by: (None)
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