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Theorem ecopovsym 6109
 Description: Assuming the operation 𝐹 is commutative, show that the relation ∼, specified by the first hypothesis, is symmetric. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
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)
Assertion
Ref Expression
ecopovsym (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 ecopovsym
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 4337 . . . . 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 2948 . . . 4 ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
43brel 4315 . . 3 (A B → (A (𝑆 × 𝑆) B (𝑆 × 𝑆)))
5 eqid 2018 . . . 4 (𝑆 × 𝑆) = (𝑆 × 𝑆)
6 breq1 3737 . . . . 5 (⟨f, g⟩ = A → (⟨f, g, 𝑡⟩ ↔ A , 𝑡⟩))
7 breq2 3738 . . . . 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 3738 . . . . 5 (⟨, 𝑡⟩ = B → (A , 𝑡⟩ ↔ A B))
10 breq1 3737 . . . . 5 (⟨, 𝑡⟩ = B → (⟨, 𝑡 AB A))
119, 10bibi12d 224 . . . 4 (⟨, 𝑡⟩ = B → ((A , 𝑡⟩ ↔ ⟨, 𝑡 A) ↔ (A BB A)))
121ecopoveq 6108 . . . . . 6 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆)) → (⟨f, g, 𝑡⟩ ↔ (f + 𝑡) = (g + )))
13 vex 2534 . . . . . . . . 9 f V
14 vex 2534 . . . . . . . . 9 𝑡 V
15 ecopopr.com . . . . . . . . 9 (x + y) = (y + x)
1613, 14, 15caovcom 5577 . . . . . . . 8 (f + 𝑡) = (𝑡 + f)
17 vex 2534 . . . . . . . . 9 g V
18 vex 2534 . . . . . . . . 9 V
1917, 18, 15caovcom 5577 . . . . . . . 8 (g + ) = ( + g)
2016, 19eqeq12i 2031 . . . . . . 7 ((f + 𝑡) = (g + ) ↔ (𝑡 + f) = ( + g))
21 eqcom 2020 . . . . . . 7 ((𝑡 + f) = ( + g) ↔ ( + g) = (𝑡 + f))
2220, 21bitri 173 . . . . . 6 ((f + 𝑡) = (g + ) ↔ ( + g) = (𝑡 + f))
2312, 22syl6bb 185 . . . . 5 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆)) → (⟨f, g, 𝑡⟩ ↔ ( + g) = (𝑡 + f)))
241ecopoveq 6108 . . . . . 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)))
2623, 25bitr4d 180 . . . 4 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆)) → (⟨f, g, 𝑡⟩ ↔ ⟨, 𝑡f, g⟩))
275, 8, 11, 262optocl 4340 . . 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 1226  ∃wex 1358   ∈ wcel 1370  ⟨cop 3349   class class class wbr 3734  {copab 3787   × cxp 4266  (class class class)co 5432 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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-xp 4274  df-iota 4790  df-fv 4833  df-ov 5435 This theorem is referenced by:  ecopover  6111
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