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Theorem ecopovsymg 6116
Description: Assuming the operation 𝐹 is commutative, show that the relation , specified by the first hypothesis, is symmetric. (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))
Assertion
Ref Expression
ecopovsymg (A BB A)
Distinct variable groups:   x,y,z,w,v,u, +   x,𝑆,y,z,w,v,u
Allowed substitution hints:   A(x,y,z,w,v,u)   B(x,y,z,w,v,u)   (x,y,z,w,v,u)

Proof of Theorem ecopovsymg
Dummy variables f g 𝑡 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)))}
2 opabssxp 4341 . . . . 5 {⟨x, y⟩ ∣ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z + u) = (w + v)))} ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
31, 2eqsstri 2952 . . . 4 ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
43brel 4319 . . 3 (A B → (A (𝑆 × 𝑆) B (𝑆 × 𝑆)))
5 eqid 2022 . . . 4 (𝑆 × 𝑆) = (𝑆 × 𝑆)
6 breq1 3741 . . . . 5 (⟨f, g⟩ = A → (⟨f, g, 𝑡⟩ ↔ A , 𝑡⟩))
7 breq2 3742 . . . . 5 (⟨f, g⟩ = A → (⟨, 𝑡f, g⟩ ↔ ⟨, 𝑡 A))
86, 7bibi12d 224 . . . 4 (⟨f, g⟩ = A → ((⟨f, g, 𝑡⟩ ↔ ⟨, 𝑡f, g⟩) ↔ (A , 𝑡⟩ ↔ ⟨, 𝑡 A)))
9 breq2 3742 . . . . 5 (⟨, 𝑡⟩ = B → (A , 𝑡⟩ ↔ A B))
10 breq1 3741 . . . . 5 (⟨, 𝑡⟩ = B → (⟨, 𝑡 AB A))
119, 10bibi12d 224 . . . 4 (⟨, 𝑡⟩ = B → ((A , 𝑡⟩ ↔ ⟨, 𝑡 A) ↔ (A BB A)))
12 ecopoprg.com . . . . . . . . 9 ((x 𝑆 y 𝑆) → (x + y) = (y + x))
1312adantl 262 . . . . . . . 8 ((((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆)) (x 𝑆 y 𝑆)) → (x + y) = (y + x))
14 simpll 469 . . . . . . . 8 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆)) → f 𝑆)
15 simprr 472 . . . . . . . 8 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆)) → 𝑡 𝑆)
1613, 14, 15caovcomd 5580 . . . . . . 7 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆)) → (f + 𝑡) = (𝑡 + f))
17 simplr 470 . . . . . . . 8 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆)) → g 𝑆)
18 simprl 471 . . . . . . . 8 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆)) → 𝑆)
1913, 17, 18caovcomd 5580 . . . . . . 7 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆)) → (g + ) = ( + g))
2016, 19eqeq12d 2036 . . . . . 6 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆)) → ((f + 𝑡) = (g + ) ↔ (𝑡 + f) = ( + g)))
21 eqcom 2024 . . . . . 6 ((𝑡 + f) = ( + g) ↔ ( + g) = (𝑡 + f))
2220, 21syl6bb 185 . . . . 5 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆)) → ((f + 𝑡) = (g + ) ↔ ( + g) = (𝑡 + f)))
231ecopoveq 6112 . . . . 5 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆)) → (⟨f, g, 𝑡⟩ ↔ (f + 𝑡) = (g + )))
241ecopoveq 6112 . . . . . 6 ((( 𝑆 𝑡 𝑆) (f 𝑆 g 𝑆)) → (⟨, 𝑡f, g⟩ ↔ ( + g) = (𝑡 + f)))
2524ancoms 255 . . . . 5 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆)) → (⟨, 𝑡f, g⟩ ↔ ( + g) = (𝑡 + f)))
2622, 23, 253bitr4d 209 . . . 4 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆)) → (⟨f, g, 𝑡⟩ ↔ ⟨, 𝑡f, g⟩))
275, 8, 11, 262optocl 4344 . . 3 ((A (𝑆 × 𝑆) B (𝑆 × 𝑆)) → (A BB A))
284, 27syl 14 . 2 (A B → (A BB A))
2928ibi 165 1 (A BB A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wex 1362   wcel 1374  cop 3353   class class class wbr 3738  {copab 3791   × cxp 4270  (class class class)co 5436
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-xp 4278  df-iota 4794  df-fv 4837  df-ov 5439
This theorem is referenced by:  ecopoverg  6118
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