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Theorem ecopovtrng 6113
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 4337 . . . . . . 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 2948 . . . . . 6 ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
43brel 4315 . . . . 5 (A B → (A (𝑆 × 𝑆) B (𝑆 × 𝑆)))
54simpld 105 . . . 4 (A BA (𝑆 × 𝑆))
63brel 4315 . . . 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 875 . . 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 2018 . . 3 (𝑆 × 𝑆) = (𝑆 × 𝑆)
11 breq1 3737 . . . . 5 (⟨f, g⟩ = A → (⟨f, g, 𝑡⟩ ↔ A , 𝑡⟩))
1211anbi1d 441 . . . 4 (⟨f, g⟩ = A → ((⟨f, g, 𝑡, 𝑡𝑠, 𝑟⟩) ↔ (A , 𝑡, 𝑡𝑠, 𝑟⟩)))
13 breq1 3737 . . . 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 3738 . . . . 5 (⟨, 𝑡⟩ = B → (A , 𝑡⟩ ↔ A B))
16 breq1 3737 . . . . 5 (⟨, 𝑡⟩ = B → (⟨, 𝑡𝑠, 𝑟⟩ ↔ B 𝑠, 𝑟⟩))
1715, 16anbi12d 445 . . . 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 3738 . . . . 5 (⟨𝑠, 𝑟⟩ = 𝐶 → (B 𝑠, 𝑟⟩ ↔ B 𝐶))
2019anbi2d 440 . . . 4 (⟨𝑠, 𝑟⟩ = 𝐶 → ((A B B 𝑠, 𝑟⟩) ↔ (A B B 𝐶)))
21 breq2 3738 . . . 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 6108 . . . . . . . 8 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆)) → (⟨f, g, 𝑡⟩ ↔ (f + 𝑡) = (g + )))
24233adant3 910 . . . . . . 7 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → (⟨f, g, 𝑡⟩ ↔ (f + 𝑡) = (g + )))
251ecopoveq 6108 . . . . . . . 8 ((( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → (⟨, 𝑡𝑠, 𝑟⟩ ↔ ( + 𝑟) = (𝑡 + 𝑠)))
26253adant1 908 . . . . . . 7 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → (⟨, 𝑡𝑠, 𝑟⟩ ↔ ( + 𝑟) = (𝑡 + 𝑠)))
2724, 26anbi12d 445 . . . . . 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 5441 . . . . . . 7 (((f + 𝑡) = (g + ) ( + 𝑟) = (𝑡 + 𝑠)) → ((f + 𝑡) + ( + 𝑟)) = ((g + ) + (𝑡 + 𝑠)))
29 simp2l 916 . . . . . . . . 9 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → 𝑆)
30 simp2r 917 . . . . . . . . 9 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → 𝑡 𝑆)
31 simp1l 914 . . . . . . . . 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 919 . . . . . . . . 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 5605 . . . . . . . 8 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → (( + 𝑡) + (f + 𝑟)) = ((f + 𝑡) + ( + 𝑟)))
40 simp1r 915 . . . . . . . . . 10 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → g 𝑆)
41 simp3l 918 . . . . . . . . . 10 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → 𝑠 𝑆)
4240, 30, 29, 33, 35, 41, 38caov411d 5605 . . . . . . . . 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 5604 . . . . . . . . 9 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → ((g + 𝑡) + ( + 𝑠)) = ((g + ) + (𝑡 + 𝑠)))
4442, 43eqtr3d 2052 . . . . . . . 8 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → (( + 𝑡) + (g + 𝑠)) = ((g + ) + (𝑡 + 𝑠)))
4539, 44eqeq12d 2032 . . . . . . 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 5440 . . . . . . . 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 5574 . . . . . . 7 (( 𝑆 𝑡 𝑆) → ( + 𝑡) 𝑆)
5329, 30, 52syl2anc 393 . . . . . 6 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → ( + 𝑡) 𝑆)
5437caovcl 5574 . . . . . . 7 ((f 𝑆 𝑟 𝑆) → (f + 𝑟) 𝑆)
5531, 36, 54syl2anc 393 . . . . . 6 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → (f + 𝑟) 𝑆)
5638, 40, 41caovcld 5573 . . . . . 6 (((f 𝑆 g 𝑆) ( 𝑆 𝑡 𝑆) (𝑠 𝑆 𝑟 𝑆)) → (g + 𝑠) 𝑆)
5751, 53, 55, 56caovcand 5582 . . . . 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 6108 . . . . 5 (((f 𝑆 g 𝑆) (𝑠 𝑆 𝑟 𝑆)) → (⟨f, g𝑠, 𝑟⟩ ↔ (f + 𝑟) = (g + 𝑠)))
60593adant2 909 . . . 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 4341 . 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 871   = 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:  ecopoverg  6114
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