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Theorem ecopover 7796
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.) (Proof shortened by AV, 1-May-2021.)
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 . . 3 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}
21relopabi 5205 . 2 Rel
3 ecopopr.com . . 3 (𝑥 + 𝑦) = (𝑦 + 𝑥)
41, 3ecopovsym 7794 . 2 (𝑓 𝑔𝑔 𝑓)
5 ecopopr.cl . . 3 ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
6 ecopopr.ass . . 3 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))
7 ecopopr.can . . 3 ((𝑥𝑆𝑦𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))
81, 3, 5, 6, 7ecopovtrn 7795 . 2 ((𝑓 𝑔𝑔 ) → 𝑓 )
9 vex 3189 . . . . . . . . 9 𝑔 ∈ V
10 vex 3189 . . . . . . . . 9 ∈ V
119, 10, 3caovcom 6784 . . . . . . . 8 (𝑔 + ) = ( + 𝑔)
121ecopoveq 7793 . . . . . . . 8 (((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) → (⟨𝑔, 𝑔, ⟩ ↔ (𝑔 + ) = ( + 𝑔)))
1311, 12mpbiri 248 . . . . . . 7 (((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) → ⟨𝑔, 𝑔, ⟩)
1413anidms 676 . . . . . 6 ((𝑔𝑆𝑆) → ⟨𝑔, 𝑔, ⟩)
1514rgen2a 2971 . . . . 5 𝑔𝑆𝑆𝑔, 𝑔,
16 breq12 4618 . . . . . . 7 ((𝑓 = ⟨𝑔, ⟩ ∧ 𝑓 = ⟨𝑔, ⟩) → (𝑓 𝑓 ↔ ⟨𝑔, 𝑔, ⟩))
1716anidms 676 . . . . . 6 (𝑓 = ⟨𝑔, ⟩ → (𝑓 𝑓 ↔ ⟨𝑔, 𝑔, ⟩))
1817ralxp 5223 . . . . 5 (∀𝑓 ∈ (𝑆 × 𝑆)𝑓 𝑓 ↔ ∀𝑔𝑆𝑆𝑔, 𝑔, ⟩)
1915, 18mpbir 221 . . . 4 𝑓 ∈ (𝑆 × 𝑆)𝑓 𝑓
2019rspec 2926 . . 3 (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 𝑓)
21 opabssxp 5154 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
221, 21eqsstri 3614 . . . . 5 ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
2322ssbri 4657 . . . 4 (𝑓 𝑓𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓)
24 brxp 5107 . . . . 5 (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓 ↔ (𝑓 ∈ (𝑆 × 𝑆) ∧ 𝑓 ∈ (𝑆 × 𝑆)))
2524simplbi 476 . . . 4 (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓𝑓 ∈ (𝑆 × 𝑆))
2623, 25syl 17 . . 3 (𝑓 𝑓𝑓 ∈ (𝑆 × 𝑆))
2720, 26impbii 199 . 2 (𝑓 ∈ (𝑆 × 𝑆) ↔ 𝑓 𝑓)
282, 4, 8, 27iseri 7714 1 Er (𝑆 × 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wral 2907  cop 4154   class class class wbr 4613  {copab 4672   × cxp 5072  (class class class)co 6604   Er wer 7684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fv 5855  df-ov 6607  df-er 7687
This theorem is referenced by:  enqer  9687  enrer  9830
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