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Theorem ecopovtrng 6142
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 transitive. (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
ecopovtrng ((A B B 𝐶) → A 𝐶)
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)   𝐶(x,y,z,w,v,u)   (x,y,z,w,v,u)

Proof of Theorem ecopovtrng
Dummy variables f g 𝑡 𝑠 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecopopr.1 . . . . . . 7 = {⟨x, y⟩ ∣ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z + u) = (w + v)))}
2 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))
31, 2eqsstri 2969 . . . . . 6 ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
43brel 4335 . . . . 5 (A B → (A (𝑆 × 𝑆) B (𝑆 × 𝑆)))
54simpld 105 . . . 4 (A BA (𝑆 × 𝑆))
63brel 4335 . . . 4 (B 𝐶 → (B (𝑆 × 𝑆) 𝐶 (𝑆 × 𝑆)))
75, 6anim12i 321 . . 3 ((A B B 𝐶) → (A (𝑆 × 𝑆) (B (𝑆 × 𝑆) 𝐶 (𝑆 × 𝑆))))
8 3anass 888 . . 3 ((A (𝑆 × 𝑆) B (𝑆 × 𝑆) 𝐶 (𝑆 × 𝑆)) ↔ (A (𝑆 × 𝑆) (B (𝑆 × 𝑆) 𝐶 (𝑆 × 𝑆))))
97, 8sylibr 137 . 2 ((A B B 𝐶) → (A (𝑆 × 𝑆) B (𝑆 × 𝑆) 𝐶 (𝑆 × 𝑆)))
10 eqid 2037 . . 3 (𝑆 × 𝑆) = (𝑆 × 𝑆)
11 breq1 3758 . . . . 5 (⟨f, g⟩ = A → (⟨f, g, 𝑡⟩ ↔ A , 𝑡⟩))
1211anbi1d 438 . . . 4 (⟨f, g⟩ = A → ((⟨f, g, 𝑡, 𝑡𝑠, 𝑟⟩) ↔ (A , 𝑡, 𝑡𝑠, 𝑟⟩)))
13 breq1 3758 . . . 4 (⟨f, g⟩ = A → (⟨f, g𝑠, 𝑟⟩ ↔ A 𝑠, 𝑟⟩))
1412, 13imbi12d 223 . . 3 (⟨f, g⟩ = A → (((⟨f, g, 𝑡, 𝑡𝑠, 𝑟⟩) → ⟨f, g𝑠, 𝑟⟩) ↔ ((A , 𝑡, 𝑡𝑠, 𝑟⟩) → A 𝑠, 𝑟⟩)))
15 breq2 3759 . . . . 5 (⟨, 𝑡⟩ = B → (A , 𝑡⟩ ↔ A B))
16 breq1 3758 . . . . 5 (⟨, 𝑡⟩ = B → (⟨, 𝑡𝑠, 𝑟⟩ ↔ B 𝑠, 𝑟⟩))
1715, 16anbi12d 442 . . . 4 (⟨, 𝑡⟩ = B → ((A , 𝑡, 𝑡𝑠, 𝑟⟩) ↔ (A B B 𝑠, 𝑟⟩)))
1817imbi1d 220 . . 3 (⟨, 𝑡⟩ = B → (((A , 𝑡, 𝑡𝑠, 𝑟⟩) → A 𝑠, 𝑟⟩) ↔ ((A B B 𝑠, 𝑟⟩) → A 𝑠, 𝑟⟩)))
19 breq2 3759 . . . . 5 (⟨𝑠, 𝑟⟩ = 𝐶 → (B 𝑠, 𝑟⟩ ↔ B 𝐶))
2019anbi2d 437 . . . 4 (⟨𝑠, 𝑟⟩ = 𝐶 → ((A B B 𝑠, 𝑟⟩) ↔ (A B B 𝐶)))
21 breq2 3759 . . . 4 (⟨𝑠, 𝑟⟩ = 𝐶 → (A 𝑠, 𝑟⟩ ↔ A 𝐶))
2220, 21imbi12d 223 . . 3 (⟨𝑠, 𝑟⟩ = 𝐶 → (((A B B 𝑠, 𝑟⟩) → A 𝑠, 𝑟⟩) ↔ ((A B B 𝐶) → A 𝐶)))
231ecopoveq 6137 . . . . . . . 8 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆)) → (⟨f, g, 𝑡⟩ ↔ (f + 𝑡) = (g + )))
24233adant3 923 . . . . . . 7 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → (⟨f, g, 𝑡⟩ ↔ (f + 𝑡) = (g + )))
251ecopoveq 6137 . . . . . . . 8 ((( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → (⟨, 𝑡𝑠, 𝑟⟩ ↔ ( + 𝑟) = (𝑡 + 𝑠)))
26253adant1 921 . . . . . . 7 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → (⟨, 𝑡𝑠, 𝑟⟩ ↔ ( + 𝑟) = (𝑡 + 𝑠)))
2724, 26anbi12d 442 . . . . . 6 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → ((⟨f, g, 𝑡, 𝑡𝑠, 𝑟⟩) ↔ ((f + 𝑡) = (g + ) ( + 𝑟) = (𝑡 + 𝑠))))
28 oveq12 5464 . . . . . . 7 (((f + 𝑡) = (g + ) ( + 𝑟) = (𝑡 + 𝑠)) → ((f + 𝑡) + ( + 𝑟)) = ((g + ) + (𝑡 + 𝑠)))
29 simp2l 929 . . . . . . . . 9 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → 𝑆)
30 simp2r 930 . . . . . . . . 9 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → 𝑡 𝑆)
31 simp1l 927 . . . . . . . . 9 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → f 𝑆)
32 ecopoprg.com . . . . . . . . . 10 ((x 𝑆 y 𝑆) → (x + y) = (y + x))
3332adantl 262 . . . . . . . . 9 ((((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) (x 𝑆 y 𝑆)) → (x + y) = (y + x))
34 ecopoprg.ass . . . . . . . . . 10 ((x 𝑆 y 𝑆 z 𝑆) → ((x + y) + z) = (x + (y + z)))
3534adantl 262 . . . . . . . . 9 ((((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) (x 𝑆 y 𝑆 z 𝑆)) → ((x + y) + z) = (x + (y + z)))
36 simp3r 932 . . . . . . . . 9 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → 𝑟 𝑆)
37 ecopoprg.cl . . . . . . . . . 10 ((x 𝑆 y 𝑆) → (x + y) 𝑆)
3837adantl 262 . . . . . . . . 9 ((((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) (x 𝑆 y 𝑆)) → (x + y) 𝑆)
3929, 30, 31, 33, 35, 36, 38caov411d 5628 . . . . . . . 8 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → (( + 𝑡) + (f + 𝑟)) = ((f + 𝑡) + ( + 𝑟)))
40 simp1r 928 . . . . . . . . . 10 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → g 𝑆)
41 simp3l 931 . . . . . . . . . 10 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → 𝑠 𝑆)
4240, 30, 29, 33, 35, 41, 38caov411d 5628 . . . . . . . . 9 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → ((g + 𝑡) + ( + 𝑠)) = (( + 𝑡) + (g + 𝑠)))
4340, 30, 29, 33, 35, 41, 38caov4d 5627 . . . . . . . . 9 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → ((g + 𝑡) + ( + 𝑠)) = ((g + ) + (𝑡 + 𝑠)))
4442, 43eqtr3d 2071 . . . . . . . 8 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → (( + 𝑡) + (g + 𝑠)) = ((g + ) + (𝑡 + 𝑠)))
4539, 44eqeq12d 2051 . . . . . . 7 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → ((( + 𝑡) + (f + 𝑟)) = (( + 𝑡) + (g + 𝑠)) ↔ ((f + 𝑡) + ( + 𝑟)) = ((g + ) + (𝑡 + 𝑠))))
4628, 45syl5ibr 145 . . . . . 6 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → (((f + 𝑡) = (g + ) ( + 𝑟) = (𝑡 + 𝑠)) → (( + 𝑡) + (f + 𝑟)) = (( + 𝑡) + (g + 𝑠))))
4727, 46sylbid 139 . . . . 5 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → ((⟨f, g, 𝑡, 𝑡𝑠, 𝑟⟩) → (( + 𝑡) + (f + 𝑟)) = (( + 𝑡) + (g + 𝑠))))
48 ecopoprg.can . . . . . . . 8 ((x 𝑆 y 𝑆 z 𝑆) → ((x + y) = (x + z) → y = z))
49 oveq2 5463 . . . . . . . 8 (y = z → (x + y) = (x + z))
5048, 49impbid1 130 . . . . . . 7 ((x 𝑆 y 𝑆 z 𝑆) → ((x + y) = (x + z) ↔ y = z))
5150adantl 262 . . . . . 6 ((((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) (x 𝑆 y 𝑆 z 𝑆)) → ((x + y) = (x + z) ↔ y = z))
5237caovcl 5597 . . . . . . 7 (( 𝑆 𝑡 𝑆) → ( + 𝑡) 𝑆)
5329, 30, 52syl2anc 391 . . . . . 6 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → ( + 𝑡) 𝑆)
5437caovcl 5597 . . . . . . 7 ((f 𝑆 𝑟 𝑆) → (f + 𝑟) 𝑆)
5531, 36, 54syl2anc 391 . . . . . 6 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → (f + 𝑟) 𝑆)
5638, 40, 41caovcld 5596 . . . . . 6 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → (g + 𝑠) 𝑆)
5751, 53, 55, 56caovcand 5605 . . . . 5 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → ((( + 𝑡) + (f + 𝑟)) = (( + 𝑡) + (g + 𝑠)) ↔ (f + 𝑟) = (g + 𝑠)))
5847, 57sylibd 138 . . . 4 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → ((⟨f, g, 𝑡, 𝑡𝑠, 𝑟⟩) → (f + 𝑟) = (g + 𝑠)))
591ecopoveq 6137 . . . . 5 (((f 𝑆 g 𝑆) (𝑠 𝑆 𝑟 𝑆)) → (⟨f, g𝑠, 𝑟⟩ ↔ (f + 𝑟) = (g + 𝑠)))
60593adant2 922 . . . 4 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → (⟨f, g𝑠, 𝑟⟩ ↔ (f + 𝑟) = (g + 𝑠)))
6158, 60sylibrd 158 . . 3 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → ((⟨f, g, 𝑡, 𝑡𝑠, 𝑟⟩) → ⟨f, g𝑠, 𝑟⟩))
6210, 14, 18, 22, 613optocl 4361 . 2 ((A (𝑆 × 𝑆) B (𝑆 × 𝑆) 𝐶 (𝑆 × 𝑆)) → ((A B B 𝐶) → A 𝐶))
639, 62mpcom 32 1 ((A B B 𝐶) → A 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242  wex 1378   wcel 1390  cop 3370   class class class wbr 3755  {copab 3808   × cxp 4286  (class class class)co 5455
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-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-iota 4810  df-fv 4853  df-ov 5458
This theorem is referenced by:  ecopoverg  6143
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