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Theorem ecopoverg 6523
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 4660 . . . 4 Rel
32a1i 9 . . 3 (⊤ → Rel )
4 ecopoprg.com . . . . 5 ((𝑥𝑆𝑦𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
51, 4ecopovsymg 6521 . . . 4 (𝑓 𝑔𝑔 𝑓)
65adantl 275 . . 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 6522 . . . 4 ((𝑓 𝑔𝑔 ) → 𝑓 )
1110adantl 275 . . 3 ((⊤ ∧ (𝑓 𝑔𝑔 )) → 𝑓 )
124adantl 275 . . . . . . . . . . 11 ((((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
13 simpll 518 . . . . . . . . . . 11 (((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) → 𝑔𝑆)
14 simplr 519 . . . . . . . . . . 11 (((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) → 𝑆)
1512, 13, 14caovcomd 5920 . . . . . . . . . 10 (((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) → (𝑔 + ) = ( + 𝑔))
161ecopoveq 6517 . . . . . . . . . 10 (((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) → (⟨𝑔, 𝑔, ⟩ ↔ (𝑔 + ) = ( + 𝑔)))
1715, 16mpbird 166 . . . . . . . . 9 (((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) → ⟨𝑔, 𝑔, ⟩)
1817anidms 394 . . . . . . . 8 ((𝑔𝑆𝑆) → ⟨𝑔, 𝑔, ⟩)
1918rgen2a 2484 . . . . . . 7 𝑔𝑆𝑆𝑔, 𝑔,
20 breq12 3929 . . . . . . . . 9 ((𝑓 = ⟨𝑔, ⟩ ∧ 𝑓 = ⟨𝑔, ⟩) → (𝑓 𝑓 ↔ ⟨𝑔, 𝑔, ⟩))
2120anidms 394 . . . . . . . 8 (𝑓 = ⟨𝑔, ⟩ → (𝑓 𝑓 ↔ ⟨𝑔, 𝑔, ⟩))
2221ralxp 4677 . . . . . . 7 (∀𝑓 ∈ (𝑆 × 𝑆)𝑓 𝑓 ↔ ∀𝑔𝑆𝑆𝑔, 𝑔, ⟩)
2319, 22mpbir 145 . . . . . 6 𝑓 ∈ (𝑆 × 𝑆)𝑓 𝑓
2423rspec 2482 . . . . 5 (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 𝑓)
2524a1i 9 . . . 4 (⊤ → (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 𝑓))
26 opabssxp 4608 . . . . . . 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 3124 . . . . . 6 ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
2827ssbri 3967 . . . . 5 (𝑓 𝑓𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓)
29 brxp 4565 . . . . . 6 (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓 ↔ (𝑓 ∈ (𝑆 × 𝑆) ∧ 𝑓 ∈ (𝑆 × 𝑆)))
3029simplbi 272 . . . . 5 (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓𝑓 ∈ (𝑆 × 𝑆))
3128, 30syl 14 . . . 4 (𝑓 𝑓𝑓 ∈ (𝑆 × 𝑆))
3225, 31impbid1 141 . . 3 (⊤ → (𝑓 ∈ (𝑆 × 𝑆) ↔ 𝑓 𝑓))
333, 6, 11, 32iserd 6448 . 2 (⊤ → Er (𝑆 × 𝑆))
3433mptru 1340 1 Er (𝑆 × 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 962   = wceq 1331  wtru 1332  wex 1468  wcel 1480  wral 2414  cop 3525   class class class wbr 3924  {copab 3983   × cxp 4532  Rel wrel 4539  (class class class)co 5767   Er wer 6419
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fv 5126  df-ov 5770  df-er 6422
This theorem is referenced by:  enqer  7159  enrer  7536
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