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Theorem ecopoverg 6143
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 ((x 𝑆 y 𝑆) → (x + y) = (y + x))
ecopoprg.cl ((x 𝑆 y 𝑆) → (x + y) 𝑆)
ecopoprg.ass ((x 𝑆 y 𝑆 z 𝑆) → ((x + y) + z) = (x + (y + z)))
ecopoprg.can ((x 𝑆 y 𝑆 z 𝑆) → ((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 4406 . . . 4 Rel
32a1i 9 . . 3 ( ⊤ → Rel )
4 ecopoprg.com . . . . 5 ((x 𝑆 y 𝑆) → (x + y) = (y + x))
51, 4ecopovsymg 6141 . . . 4 (f gg f)
65adantl 262 . . 3 (( ⊤ f g) → g f)
7 ecopoprg.cl . . . . 5 ((x 𝑆 y 𝑆) → (x + y) 𝑆)
8 ecopoprg.ass . . . . 5 ((x 𝑆 y 𝑆 z 𝑆) → ((x + y) + z) = (x + (y + z)))
9 ecopoprg.can . . . . 5 ((x 𝑆 y 𝑆 z 𝑆) → ((x + y) = (x + z) → y = z))
101, 4, 7, 8, 9ecopovtrng 6142 . . . 4 ((f g g ) → f )
1110adantl 262 . . 3 (( ⊤ (f g g )) → f )
124adantl 262 . . . . . . . . . . 11 ((((g 𝑆 𝑆) (g 𝑆 𝑆)) (x 𝑆 y 𝑆)) → (x + y) = (y + x))
13 simpll 481 . . . . . . . . . . 11 (((g 𝑆 𝑆) (g 𝑆 𝑆)) → g 𝑆)
14 simplr 482 . . . . . . . . . . 11 (((g 𝑆 𝑆) (g 𝑆 𝑆)) → 𝑆)
1512, 13, 14caovcomd 5599 . . . . . . . . . 10 (((g 𝑆 𝑆) (g 𝑆 𝑆)) → (g + ) = ( + g))
161ecopoveq 6137 . . . . . . . . . 10 (((g 𝑆 𝑆) (g 𝑆 𝑆)) → (⟨g, g, ⟩ ↔ (g + ) = ( + g)))
1715, 16mpbird 156 . . . . . . . . 9 (((g 𝑆 𝑆) (g 𝑆 𝑆)) → ⟨g, g, ⟩)
1817anidms 377 . . . . . . . 8 ((g 𝑆 𝑆) → ⟨g, g, ⟩)
1918rgen2a 2369 . . . . . . 7 g 𝑆 𝑆g, g,
20 breq12 3760 . . . . . . . . 9 ((f = ⟨g, f = ⟨g, ⟩) → (f f ↔ ⟨g, g, ⟩))
2120anidms 377 . . . . . . . 8 (f = ⟨g, ⟩ → (f f ↔ ⟨g, g, ⟩))
2221ralxp 4422 . . . . . . 7 (f (𝑆 × 𝑆)f fg 𝑆 𝑆g, g, ⟩)
2319, 22mpbir 134 . . . . . 6 f (𝑆 × 𝑆)f f
2423rspec 2367 . . . . 5 (f (𝑆 × 𝑆) → f f)
2524a1i 9 . . . 4 ( ⊤ → (f (𝑆 × 𝑆) → f f))
26 opabssxp 4357 . . . . . . 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 2969 . . . . . 6 ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
2827ssbri 3797 . . . . 5 (f ff((𝑆 × 𝑆) × (𝑆 × 𝑆))f)
29 brxp 4318 . . . . . 6 (f((𝑆 × 𝑆) × (𝑆 × 𝑆))f ↔ (f (𝑆 × 𝑆) f (𝑆 × 𝑆)))
3029simplbi 259 . . . . 5 (f((𝑆 × 𝑆) × (𝑆 × 𝑆))ff (𝑆 × 𝑆))
3128, 30syl 14 . . . 4 (f ff (𝑆 × 𝑆))
3225, 31impbid1 130 . . 3 ( ⊤ → (f (𝑆 × 𝑆) ↔ f f))
333, 6, 11, 32iserd 6068 . 2 ( ⊤ → Er (𝑆 × 𝑆))
3433trud 1251 1 Er (𝑆 × 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242  wtru 1243  wex 1378   wcel 1390  wral 2300  cop 3370   class class class wbr 3755  {copab 3808   × cxp 4286  Rel wrel 4293  (class class class)co 5455   Er wer 6039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fv 4853  df-ov 5458  df-er 6042
This theorem is referenced by:  enqer  6342  enrer  6623
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