Theorem ecopoprtrn 3247

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 R, specified by the first hypothesis, is transitive.

Hypotheses

Ref	Expression
ecopopr.1	⊢ R = {⟨x, y⟩∣((x ∈ (S × S) ∧ y ∈ (S × S)) ∧ ∃z∃w∃v∃u((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (zFu) = (wFv)))}
ecopopr.com	⊢ (xFy) = (yFx)
ecopopr.cl	⊢ ((x ∈ S ∧ y ∈ S) → (xFy) ∈ S)
ecopopr.ass	⊢ ((xFy)Fz) = (xF(yFz))
ecopopr.can	⊢ ((x ∈ S ∧ y ∈ S) → ((xFy) = (xFz) → y = z))
ecopopr.3	⊢ B ∈ V
ecopopr.4	⊢ C ∈ V

Assertion

Ref	Expression
ecopoprtrn	⊢ ((ARB ∧ BRC) → ARC)

Distinct variable group(s):   x,y,z,w,v,u,F   x,S,y,z,w,v,u

Proof of Theorem ecopoprtrn

Step	Hyp	Ref	Expression
1		ecopopr.3	. . . . . 6 ⊢ B ∈ V
2		ecopopr.1	. . . . . . 7 ⊢ R = {⟨x, y⟩∣((x ∈ (S × S) ∧ y ∈ (S × S)) ∧ ∃z∃w∃v∃u((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (zFu) = (wFv)))}
3		opabssxp 2468	. . . . . . 7 ⊢ {⟨x, y⟩∣((x ∈ (S × S) ∧ y ∈ (S × S)) ∧ ∃z∃w∃v∃u((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (zFu) = (wFv)))} ⊆ ((S × S) × (S × S))
4	2, 3	eqsstr 1530	. . . . . 6 ⊢ R ⊆ ((S × S) × (S × S))
5	1, 4	brel 2459	. . . . 5 ⊢ (ARB → (A ∈ (S × S) ∧ B ∈ (S × S)))
6	5	pm3.26d 258	. . . 4 ⊢ (ARB → A ∈ (S × S))
7		ecopopr.4	. . . . 5 ⊢ C ∈ V
8	7, 4	brel 2459	. . . 4 ⊢ (BRC → (B ∈ (S × S) ∧ C ∈ (S × S)))
9	6, 8	anim12i 268	. . 3 ⊢ ((ARB ∧ BRC) → (A ∈ (S × S) ∧ (B ∈ (S × S) ∧ C ∈ (S × S))))
10		3anass 585	. . 3 ⊢ ((A ∈ (S × S) ∧ B ∈ (S × S) ∧ C ∈ (S × S)) ↔ (A ∈ (S × S) ∧ (B ∈ (S × S) ∧ C ∈ (S × S))))
11	9, 10	sylibr 175	. 2 ⊢ ((ARB ∧ BRC) → (A ∈ (S × S) ∧ B ∈ (S × S) ∧ C ∈ (S × S)))
12		cleqid 1102	. . 3 ⊢ (S × S) = (S × S)
13		breq1 2065	. . . . 5 ⊢ (⟨f, g⟩ = A → (⟨f, g⟩R⟨h, t⟩ ↔ AR⟨h, t⟩))
14	13	anbi1d 469	. . . 4 ⊢ (⟨f, g⟩ = A → ((⟨f, g⟩R⟨h, t⟩ ∧ ⟨h, t⟩R⟨s, r⟩) ↔ (AR⟨h, t⟩ ∧ ⟨h, t⟩R⟨s, r⟩)))
15		breq1 2065	. . . 4 ⊢ (⟨f, g⟩ = A → (⟨f, g⟩R⟨s, r⟩ ↔ AR⟨s, r⟩))
16	14, 15	imbi12d 474	. . 3 ⊢ (⟨f, g⟩ = A → (((⟨f, g⟩R⟨h, t⟩ ∧ ⟨h, t⟩R⟨s, r⟩) → ⟨f, g⟩R⟨s, r⟩) ↔ ((AR⟨h, t⟩ ∧ ⟨h, t⟩R⟨s, r⟩) → AR⟨s, r⟩)))
17		breq2 2066	. . . . 5 ⊢ (⟨h, t⟩ = B → (AR⟨h, t⟩ ↔ ARB))
18		breq1 2065	. . . . 5 ⊢ (⟨h, t⟩ = B → (⟨h, t⟩R⟨s, r⟩ ↔ BR⟨s, r⟩))
19	17, 18	anbi12d 476	. . . 4 ⊢ (⟨h, t⟩ = B → ((AR⟨h, t⟩ ∧ ⟨h, t⟩R⟨s, r⟩) ↔ (ARB ∧ BR⟨s, r⟩)))
20	19	imbi1d 465	. . 3 ⊢ (⟨h, t⟩ = B → (((AR⟨h, t⟩ ∧ ⟨h, t⟩R⟨s, r⟩) → AR⟨s, r⟩) ↔ ((ARB ∧ BR⟨s, r⟩) → AR⟨s, r⟩)))
21		breq2 2066	. . . . 5 ⊢ (⟨s, r⟩ = C → (BR⟨s, r⟩ ↔ BRC))
22	21	anbi2d 468	. . . 4 ⊢ (⟨s, r⟩ = C → ((ARB ∧ BR⟨s, r⟩) ↔ (ARB ∧ BRC)))
23		breq2 2066	. . . 4 ⊢ (⟨s, r⟩ = C → (AR⟨s, r⟩ ↔ ARC))
24	22, 23	imbi12d 474	. . 3 ⊢ (⟨s, r⟩ = C → (((ARB ∧ BR⟨s, r⟩) → AR⟨s, r⟩) ↔ ((ARB ∧ BRC) → ARC)))
25	2	ecopopreq 3244	. . . . . . . 8 ⊢ (((f ∈ S ∧ g ∈ S) ∧ (h ∈ S ∧ t ∈ S)) → (⟨f, g⟩R⟨h, t⟩ ↔ (fFt) = (gFh)))
26	25	3adant3 599	. . . . . . 7 ⊢ (((f ∈ S ∧ g ∈ S) ∧ (h ∈ S ∧ t ∈ S) ∧ (s ∈ S ∧ r ∈ S)) → (⟨f, g⟩R⟨h, t⟩ ↔ (fFt) = (gFh)))
27	2	ecopopreq 3244	. . . . . . . 8 ⊢ (((h ∈ S ∧ t ∈ S) ∧ (s ∈ S ∧ r ∈ S)) → (⟨h, t⟩R⟨s, r⟩ ↔ (hFr) = (tFs)))
28	27	3adant1 597	. . . . . . 7 ⊢ (((f ∈ S ∧ g ∈ S) ∧ (h ∈ S ∧ t ∈ S) ∧ (s ∈ S ∧ r ∈ S)) → (⟨h, t⟩R⟨s, r⟩ ↔ (hFr) = (tFs)))
29	26, 28	anbi12d 476	. . . . . 6 ⊢ (((f ∈ S ∧ g ∈ S) ∧ (h ∈ S ∧ t ∈ S) ∧ (s ∈ S ∧ r ∈ S)) → ((⟨f, g⟩R⟨h, t⟩ ∧ ⟨h, t⟩R⟨s, r⟩) ↔ ((fFt) = (gFh) ∧ (hFr) = (tFs))))
30		opreq12 3008	. . . . . . 7 ⊢ (((fFt) = (gFh) ∧ (hFr) = (tFs)) → ((fFt)F(hFr)) = ((gFh)F(tFs)))
31		visset 1350	. . . . . . . 8 ⊢ h ∈ V
32		visset 1350	. . . . . . . 8 ⊢ t ∈ V
33		visset 1350	. . . . . . . 8 ⊢ f ∈ V
34		ecopopr.com	. . . . . . . 8 ⊢ (xFy) = (yFx)
35		ecopopr.ass	. . . . . . . 8 ⊢ ((xFy)Fz) = (xF(yFz))
36		visset 1350	. . . . . . . 8 ⊢ r ∈ V
37	31, 32, 33, 34, 35, 36	caopr411 3079	. . . . . . 7 ⊢ ((hFt)F(fFr)) = ((fFt)F(hFr))
38		visset 1350	. . . . . . . . 9 ⊢ g ∈ V
39		visset 1350	. . . . . . . . 9 ⊢ s ∈ V
40	38, 32, 31, 34, 35, 39	caopr411 3079	. . . . . . . 8 ⊢ ((gFt)F(hFs)) = ((hFt)F(gFs))
41	38, 32, 31, 34, 35, 39	caopr4 3078	. . . . . . . 8 ⊢ ((gFt)F(hFs)) = ((gFh)F(tFs))
42	40, 41	eqtr3 1121	. . . . . . 7 ⊢ ((hFt)F(gFs)) = ((gFh)F(tFs))
43	30, 37, 42	3eqtr4g 1147	. . . . . 6 ⊢ (((fFt) = (gFh) ∧ (hFr) = (tFs)) → ((hFt)F(fFr)) = ((hFt)F(gFs)))
44	29, 43	syl6bi 187	. . . . 5 ⊢ (((f ∈ S ∧ g ∈ S) ∧ (h ∈ S ∧ t ∈ S) ∧ (s ∈ S ∧ r ∈ S)) → ((⟨f, g⟩R⟨h, t⟩ ∧ ⟨h, t⟩R⟨s, r⟩) → ((hFt)F(fFr)) = ((hFt)F(gFs))))
45		oprex 3018	. . . . . . . . . . 11 ⊢ (gFs) ∈ V
46		ecopopr.can	. . . . . . . . . . 11 ⊢ ((x ∈ S ∧ y ∈ S) → ((xFy) = (xFz) → y = z))
47	45, 46	caoprcan 3069	. . . . . . . . . 10 ⊢ (((hFt) ∈ S ∧ (fFr) ∈ S) → (((hFt)F(fFr)) = ((hFt)F(gFs)) → (fFr) = (gFs)))
48		ecopopr.cl	. . . . . . . . . . 11 ⊢ ((x ∈ S ∧ y ∈ S) → (xFy) ∈ S)
49	48	caoprcl 3066	. . . . . . . . . 10 ⊢ ((h ∈ S ∧ t ∈ S) → (hFt) ∈ S)
50	48	caoprcl 3066	. . . . . . . . . 10 ⊢ ((f ∈ S ∧ r ∈ S) → (fFr) ∈ S)
51	47, 49, 50	syl2an 349	. . . . . . . . 9 ⊢ (((h ∈ S ∧ t ∈ S) ∧ (f ∈ S ∧ r ∈ S)) → (((hFt)F(fFr)) = ((hFt)F(gFs)) → (fFr) = (gFs)))
52	51	3impb 610	. . . . . . . 8 ⊢ (((h ∈ S ∧ t ∈ S) ∧ f ∈ S ∧ r ∈ S) → (((hFt)F(fFr)) = ((hFt)F(gFs)) → (fFr) = (gFs)))
53	52	3com12 614	. . . . . . 7 ⊢ ((f ∈ S ∧ (h ∈ S ∧ t ∈ S) ∧ r ∈ S) → (((hFt)F(fFr)) = ((hFt)F(gFs)) → (fFr) = (gFs)))
54		pm3.27 260	. . . . . . 7 ⊢ ((s ∈ S ∧ r ∈ S) → r ∈ S)
55	53, 54	syl3an3 621	. . . . . 6 ⊢ ((f ∈ S ∧ (h ∈ S ∧ t ∈ S) ∧ (s ∈ S ∧ r ∈ S)) → (((hFt)F(fFr)) = ((hFt)F(gFs)) → (fFr) = (gFs)))
56		pm3.26 256	. . . . . 6 ⊢ ((f ∈ S ∧ g ∈ S) → f ∈ S)
57	55, 56	syl3an1 619	. . . . 5 ⊢ (((f ∈ S ∧ g ∈ S) ∧ (h ∈ S ∧ t ∈ S) ∧ (s ∈ S ∧ r ∈ S)) → (((hFt)F(fFr)) = ((hFt)F(gFs)) → (fFr) = (gFs)))
58	44, 57	syld 27	. . . 4 ⊢ (((f ∈ S ∧ g ∈ S) ∧ (h ∈ S ∧ t ∈ S) ∧ (s ∈ S ∧ r ∈ S)) → ((⟨f, g⟩R⟨h, t⟩ ∧ ⟨h, t⟩R⟨s, r⟩) → (fFr) = (gFs)))
59	2	ecopopreq 3244	. . . . 5 ⊢ (((f ∈ S ∧ g ∈ S) ∧ (s ∈ S ∧ r ∈ S)) → (⟨f, g⟩R⟨s, r⟩ ↔ (fFr) = (gFs)))
60	59	3adant2 598	. . . 4 ⊢ (((f ∈ S ∧ g ∈ S) ∧ (h ∈ S ∧ t ∈ S) ∧ (s ∈ S ∧ r ∈ S)) → (⟨f, g⟩R⟨s, r⟩ ↔ (fFr) = (gFs)))
61	58, 60	sylibrd 179	. . 3 ⊢ (((f ∈ S ∧ g ∈ S) ∧ (h ∈ S ∧ t ∈ S) ∧ (s ∈ S ∧ r ∈ S)) → ((⟨f, g⟩R⟨h, t⟩ ∧ ⟨h, t⟩R⟨s, r⟩) → ⟨f, g⟩R⟨s, r⟩))
62	12, 16, 20, 24, 61	3optocl 2471	. 2 ⊢ ((A ∈ (S × S) ∧ B ∈ (S × S) ∧ C ∈ (S × S)) → ((ARB ∧ BRC) → ARC))
63	11, 62	mpcom 49	1 ⊢ ((ARB ∧ BRC) → ARC)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∧ w3a 581  ∃wex 678   = weq 797   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   class class class wbr 2054  {copab 2055   × cxp 2408  (class class class)co 3001

This theorem is referenced by:  ecopoprer 3248

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

metamath.org