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Theorem ecopover 6611
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 4737 . . . 4 Rel
32a1i 9 . . 3 (⊤ → Rel )
4 ecopopr.com . . . . 5 (𝑥 + 𝑦) = (𝑦 + 𝑥)
51, 4ecopovsym 6609 . . . 4 (𝑓 𝑔𝑔 𝑓)
65adantl 275 . . 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 6610 . . . 4 ((𝑓 𝑔𝑔 ) → 𝑓 )
1110adantl 275 . . 3 ((⊤ ∧ (𝑓 𝑔𝑔 )) → 𝑓 )
12 vex 2733 . . . . . . . . . . 11 𝑔 ∈ V
13 vex 2733 . . . . . . . . . . 11 ∈ V
1412, 13, 4caovcom 6010 . . . . . . . . . 10 (𝑔 + ) = ( + 𝑔)
151ecopoveq 6608 . . . . . . . . . 10 (((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) → (⟨𝑔, 𝑔, ⟩ ↔ (𝑔 + ) = ( + 𝑔)))
1614, 15mpbiri 167 . . . . . . . . 9 (((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) → ⟨𝑔, 𝑔, ⟩)
1716anidms 395 . . . . . . . 8 ((𝑔𝑆𝑆) → ⟨𝑔, 𝑔, ⟩)
1817rgen2a 2524 . . . . . . 7 𝑔𝑆𝑆𝑔, 𝑔,
19 breq12 3994 . . . . . . . . 9 ((𝑓 = ⟨𝑔, ⟩ ∧ 𝑓 = ⟨𝑔, ⟩) → (𝑓 𝑓 ↔ ⟨𝑔, 𝑔, ⟩))
2019anidms 395 . . . . . . . 8 (𝑓 = ⟨𝑔, ⟩ → (𝑓 𝑓 ↔ ⟨𝑔, 𝑔, ⟩))
2120ralxp 4754 . . . . . . 7 (∀𝑓 ∈ (𝑆 × 𝑆)𝑓 𝑓 ↔ ∀𝑔𝑆𝑆𝑔, 𝑔, ⟩)
2218, 21mpbir 145 . . . . . 6 𝑓 ∈ (𝑆 × 𝑆)𝑓 𝑓
2322rspec 2522 . . . . 5 (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 𝑓)
2423a1i 9 . . . 4 (⊤ → (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 𝑓))
25 opabssxp 4685 . . . . . . 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 3179 . . . . . 6 ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
2726ssbri 4033 . . . . 5 (𝑓 𝑓𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓)
28 brxp 4642 . . . . . 6 (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓 ↔ (𝑓 ∈ (𝑆 × 𝑆) ∧ 𝑓 ∈ (𝑆 × 𝑆)))
2928simplbi 272 . . . . 5 (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓𝑓 ∈ (𝑆 × 𝑆))
3027, 29syl 14 . . . 4 (𝑓 𝑓𝑓 ∈ (𝑆 × 𝑆))
3124, 30impbid1 141 . . 3 (⊤ → (𝑓 ∈ (𝑆 × 𝑆) ↔ 𝑓 𝑓))
323, 6, 11, 31iserd 6539 . 2 (⊤ → Er (𝑆 × 𝑆))
3332mptru 1357 1 Er (𝑆 × 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wtru 1349  wex 1485  wcel 2141  wral 2448  cop 3586   class class class wbr 3989  {copab 4049   × cxp 4609  Rel wrel 4616  (class class class)co 5853   Er wer 6510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fv 5206  df-ov 5856  df-er 6513
This theorem is referenced by: (None)
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