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Theorem ecopoverOLD 7798
 Description: Obsolete proof of ecopover 7797 as of 1-May-2021. 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 modification is discouraged.) (New usage is discouraged.)
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
ecopoverOLD 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 ecopoverOLD
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 5210 . . . 4 Rel
32a1i 11 . . 3 (⊤ → Rel )
4 ecopopr.com . . . . 5 (𝑥 + 𝑦) = (𝑦 + 𝑥)
51, 4ecopovsym 7795 . . . 4 (𝑓 𝑔𝑔 𝑓)
65adantl 482 . . 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 7796 . . . 4 ((𝑓 𝑔𝑔 ) → 𝑓 )
1110adantl 482 . . 3 ((⊤ ∧ (𝑓 𝑔𝑔 )) → 𝑓 )
12 vex 3194 . . . . . . . . . . 11 𝑔 ∈ V
13 vex 3194 . . . . . . . . . . 11 ∈ V
1412, 13, 4caovcom 6785 . . . . . . . . . 10 (𝑔 + ) = ( + 𝑔)
151ecopoveq 7794 . . . . . . . . . 10 (((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) → (⟨𝑔, 𝑔, ⟩ ↔ (𝑔 + ) = ( + 𝑔)))
1614, 15mpbiri 248 . . . . . . . . 9 (((𝑔𝑆𝑆) ∧ (𝑔𝑆𝑆)) → ⟨𝑔, 𝑔, ⟩)
1716anidms 676 . . . . . . . 8 ((𝑔𝑆𝑆) → ⟨𝑔, 𝑔, ⟩)
1817rgen2a 2976 . . . . . . 7 𝑔𝑆𝑆𝑔, 𝑔,
19 breq12 4623 . . . . . . . . 9 ((𝑓 = ⟨𝑔, ⟩ ∧ 𝑓 = ⟨𝑔, ⟩) → (𝑓 𝑓 ↔ ⟨𝑔, 𝑔, ⟩))
2019anidms 676 . . . . . . . 8 (𝑓 = ⟨𝑔, ⟩ → (𝑓 𝑓 ↔ ⟨𝑔, 𝑔, ⟩))
2120ralxp 5228 . . . . . . 7 (∀𝑓 ∈ (𝑆 × 𝑆)𝑓 𝑓 ↔ ∀𝑔𝑆𝑆𝑔, 𝑔, ⟩)
2218, 21mpbir 221 . . . . . 6 𝑓 ∈ (𝑆 × 𝑆)𝑓 𝑓
2322rspec 2931 . . . . 5 (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 𝑓)
2423a1i 11 . . . 4 (⊤ → (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 𝑓))
25 opabssxp 5159 . . . . . . 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 3619 . . . . . 6 ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
2726ssbri 4662 . . . . 5 (𝑓 𝑓𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓)
28 brxp 5112 . . . . . 6 (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓 ↔ (𝑓 ∈ (𝑆 × 𝑆) ∧ 𝑓 ∈ (𝑆 × 𝑆)))
2928simplbi 476 . . . . 5 (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓𝑓 ∈ (𝑆 × 𝑆))
3027, 29syl 17 . . . 4 (𝑓 𝑓𝑓 ∈ (𝑆 × 𝑆))
3124, 30impbid1 215 . . 3 (⊤ → (𝑓 ∈ (𝑆 × 𝑆) ↔ 𝑓 𝑓))
323, 6, 11, 31iserd 7714 . 2 (⊤ → Er (𝑆 × 𝑆))
3332trud 1490 1 Er (𝑆 × 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480  ⊤wtru 1481  ∃wex 1701   ∈ wcel 1992  ∀wral 2912  ⟨cop 4159   class class class wbr 4618  {copab 4677   × cxp 5077  Rel wrel 5084  (class class class)co 6605   Er wer 7685 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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872 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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fv 5858  df-ov 6608  df-er 7688 This theorem is referenced by: (None)
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