Metamath Proof Explorer


Theorem ecopovtrn

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 NM, 11-Feb-1996) (Revised by Mario Carneiro, 26-Apr-2015)

Ref Expression
Hypotheses ecopopr.1 ˙ = x y | x S × S y S × S z w v u x = z w y = v u z + ˙ u = w + ˙ v
ecopopr.com x + ˙ y = y + ˙ x
ecopopr.cl x S y S x + ˙ y S
ecopopr.ass x + ˙ y + ˙ z = x + ˙ y + ˙ z
ecopopr.can x S y S x + ˙ y = x + ˙ z y = z
Assertion ecopovtrn A ˙ B B ˙ C A ˙ C

Proof

Step Hyp Ref Expression
1 ecopopr.1 ˙ = x y | x S × S y S × S z w v u x = z w y = v u z + ˙ u = w + ˙ v
2 ecopopr.com x + ˙ y = y + ˙ x
3 ecopopr.cl x S y S x + ˙ y S
4 ecopopr.ass x + ˙ y + ˙ z = x + ˙ y + ˙ z
5 ecopopr.can x S y S x + ˙ y = x + ˙ z y = z
6 opabssxp x y | x S × S y S × S z w v u x = z w y = v u z + ˙ u = w + ˙ v S × S × S × S
7 1 6 eqsstri ˙ S × S × S × S
8 7 brel A ˙ B A S × S B S × S
9 8 simpld A ˙ B A S × S
10 7 brel B ˙ C B S × S C S × S
11 9 10 anim12i A ˙ B B ˙ C A S × S B S × S C S × S
12 3anass A S × S B S × S C S × S A S × S B S × S C S × S
13 11 12 sylibr A ˙ B B ˙ C A S × S B S × S C S × S
14 eqid S × S = S × S
15 breq1 f g = A f g ˙ h t A ˙ h t
16 15 anbi1d f g = A f g ˙ h t h t ˙ s r A ˙ h t h t ˙ s r
17 breq1 f g = A f g ˙ s r A ˙ s r
18 16 17 imbi12d f g = A f g ˙ h t h t ˙ s r f g ˙ s r A ˙ h t h t ˙ s r A ˙ s r
19 breq2 h t = B A ˙ h t A ˙ B
20 breq1 h t = B h t ˙ s r B ˙ s r
21 19 20 anbi12d h t = B A ˙ h t h t ˙ s r A ˙ B B ˙ s r
22 21 imbi1d h t = B A ˙ h t h t ˙ s r A ˙ s r A ˙ B B ˙ s r A ˙ s r
23 breq2 s r = C B ˙ s r B ˙ C
24 23 anbi2d s r = C A ˙ B B ˙ s r A ˙ B B ˙ C
25 breq2 s r = C A ˙ s r A ˙ C
26 24 25 imbi12d s r = C A ˙ B B ˙ s r A ˙ s r A ˙ B B ˙ C A ˙ C
27 1 ecopoveq f S g S h S t S f g ˙ h t f + ˙ t = g + ˙ h
28 27 3adant3 f S g S h S t S s S r S f g ˙ h t f + ˙ t = g + ˙ h
29 1 ecopoveq h S t S s S r S h t ˙ s r h + ˙ r = t + ˙ s
30 29 3adant1 f S g S h S t S s S r S h t ˙ s r h + ˙ r = t + ˙ s
31 28 30 anbi12d f S g S h S t S s S r S f g ˙ h t h t ˙ s r f + ˙ t = g + ˙ h h + ˙ r = t + ˙ s
32 oveq12 f + ˙ t = g + ˙ h h + ˙ r = t + ˙ s f + ˙ t + ˙ h + ˙ r = g + ˙ h + ˙ t + ˙ s
33 vex h V
34 vex t V
35 vex f V
36 vex r V
37 33 34 35 2 4 36 caov411 h + ˙ t + ˙ f + ˙ r = f + ˙ t + ˙ h + ˙ r
38 vex g V
39 vex s V
40 38 34 33 2 4 39 caov411 g + ˙ t + ˙ h + ˙ s = h + ˙ t + ˙ g + ˙ s
41 38 34 33 2 4 39 caov4 g + ˙ t + ˙ h + ˙ s = g + ˙ h + ˙ t + ˙ s
42 40 41 eqtr3i h + ˙ t + ˙ g + ˙ s = g + ˙ h + ˙ t + ˙ s
43 32 37 42 3eqtr4g f + ˙ t = g + ˙ h h + ˙ r = t + ˙ s h + ˙ t + ˙ f + ˙ r = h + ˙ t + ˙ g + ˙ s
44 31 43 syl6bi f S g S h S t S s S r S f g ˙ h t h t ˙ s r h + ˙ t + ˙ f + ˙ r = h + ˙ t + ˙ g + ˙ s
45 3 caovcl h S t S h + ˙ t S
46 3 caovcl f S r S f + ˙ r S
47 ovex g + ˙ s V
48 47 5 caovcan h + ˙ t S f + ˙ r S h + ˙ t + ˙ f + ˙ r = h + ˙ t + ˙ g + ˙ s f + ˙ r = g + ˙ s
49 45 46 48 syl2an h S t S f S r S h + ˙ t + ˙ f + ˙ r = h + ˙ t + ˙ g + ˙ s f + ˙ r = g + ˙ s
50 49 3impb h S t S f S r S h + ˙ t + ˙ f + ˙ r = h + ˙ t + ˙ g + ˙ s f + ˙ r = g + ˙ s
51 50 3com12 f S h S t S r S h + ˙ t + ˙ f + ˙ r = h + ˙ t + ˙ g + ˙ s f + ˙ r = g + ˙ s
52 51 3adant3l f S h S t S s S r S h + ˙ t + ˙ f + ˙ r = h + ˙ t + ˙ g + ˙ s f + ˙ r = g + ˙ s
53 52 3adant1r f S g S h S t S s S r S h + ˙ t + ˙ f + ˙ r = h + ˙ t + ˙ g + ˙ s f + ˙ r = g + ˙ s
54 44 53 syld f S g S h S t S s S r S f g ˙ h t h t ˙ s r f + ˙ r = g + ˙ s
55 1 ecopoveq f S g S s S r S f g ˙ s r f + ˙ r = g + ˙ s
56 55 3adant2 f S g S h S t S s S r S f g ˙ s r f + ˙ r = g + ˙ s
57 54 56 sylibrd f S g S h S t S s S r S f g ˙ h t h t ˙ s r f g ˙ s r
58 14 18 22 26 57 3optocl A S × S B S × S C S × S A ˙ B B ˙ C A ˙ C
59 13 58 mpcom A ˙ B B ˙ C A ˙ C