Theorem ecopover 6628

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 an equivalence relation. (Contributed by NM, 16-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
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

Ref	Expression
ecopover	⊢ ∼ Er (𝑆 × 𝑆)

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢, +   𝑥,𝑆,𝑦,𝑧,𝑤,𝑣,𝑢

Allowed substitution hints:   ∼ (𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem ecopover

Dummy variables 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ecopopr.1	. . . . 5 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}
2	1	relopabi 4750	. . . 4 ⊢ Rel ∼
3	2	a1i 9	. . 3 ⊢ (⊤ → Rel ∼ )
4		ecopopr.com	. . . . 5 ⊢ (𝑥 + 𝑦) = (𝑦 + 𝑥)
5	1, 4	ecopovsym 6626	. . . 4 ⊢ (𝑓 ∼ 𝑔 → 𝑔 ∼ 𝑓)
6	5	adantl 277	. . 3 ⊢ ((⊤ ∧ 𝑓 ∼ 𝑔) → 𝑔 ∼ 𝑓)
7		ecopopr.cl	. . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
8		ecopopr.ass	. . . . 5 ⊢ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))
9		ecopopr.can	. . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))
10	1, 4, 7, 8, 9	ecopovtrn 6627	. . . 4 ⊢ ((𝑓 ∼ 𝑔 ∧ 𝑔 ∼ ℎ) → 𝑓 ∼ ℎ)
11	10	adantl 277	. . 3 ⊢ ((⊤ ∧ (𝑓 ∼ 𝑔 ∧ 𝑔 ∼ ℎ)) → 𝑓 ∼ ℎ)
12		vex 2740	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
13		vex 2740	. . . . . . . . . . 11 ⊢ ℎ ∈ V
14	12, 13, 4	caovcom 6027	. . . . . . . . . 10 ⊢ (𝑔 + ℎ) = (ℎ + 𝑔)
15	1	ecopoveq 6625	. . . . . . . . . 10 ⊢ (((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → (⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩ ↔ (𝑔 + ℎ) = (ℎ + 𝑔)))
16	14, 15	mpbiri 168	. . . . . . . . 9 ⊢ (((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → ⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩)
17	16	anidms 397	. . . . . . . 8 ⊢ ((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) → ⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩)
18	17	rgen2a 2531	. . . . . . 7 ⊢ ∀𝑔 ∈ 𝑆 ∀ℎ ∈ 𝑆 ⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩
19		breq12 4006	. . . . . . . . 9 ⊢ ((𝑓 = ⟨𝑔, ℎ⟩ ∧ 𝑓 = ⟨𝑔, ℎ⟩) → (𝑓 ∼ 𝑓 ↔ ⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩))
20	19	anidms 397	. . . . . . . 8 ⊢ (𝑓 = ⟨𝑔, ℎ⟩ → (𝑓 ∼ 𝑓 ↔ ⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩))
21	20	ralxp 4767	. . . . . . 7 ⊢ (∀𝑓 ∈ (𝑆 × 𝑆)𝑓 ∼ 𝑓 ↔ ∀𝑔 ∈ 𝑆 ∀ℎ ∈ 𝑆 ⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩)
22	18, 21	mpbir 146	. . . . . 6 ⊢ ∀𝑓 ∈ (𝑆 × 𝑆)𝑓 ∼ 𝑓
23	22	rspec 2529	. . . . 5 ⊢ (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 ∼ 𝑓)
24	23	a1i 9	. . . 4 ⊢ (⊤ → (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 ∼ 𝑓))
25		opabssxp 4698	. . . . . . 7 ⊢ {⟨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))
26	1, 25	eqsstri 3187	. . . . . 6 ⊢ ∼ ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
27	26	ssbri 4045	. . . . 5 ⊢ (𝑓 ∼ 𝑓 → 𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓)
28		brxp 4655	. . . . . 6 ⊢ (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓 ↔ (𝑓 ∈ (𝑆 × 𝑆) ∧ 𝑓 ∈ (𝑆 × 𝑆)))
29	28	simplbi 274	. . . . 5 ⊢ (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓 → 𝑓 ∈ (𝑆 × 𝑆))
30	27, 29	syl 14	. . . 4 ⊢ (𝑓 ∼ 𝑓 → 𝑓 ∈ (𝑆 × 𝑆))
31	24, 30	impbid1 142	. . 3 ⊢ (⊤ → (𝑓 ∈ (𝑆 × 𝑆) ↔ 𝑓 ∼ 𝑓))
32	3, 6, 11, 31	iserd 6556	. 2 ⊢ (⊤ → ∼ Er (𝑆 × 𝑆))
33	32	mptru 1362	1 ⊢ ∼ Er (𝑆 × 𝑆)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  ⊤wtru 1354  ∃wex 1492   ∈ wcel 2148  ∀wral 2455  ⟨cop 3595   class class class wbr 4001  {copab 4061   × cxp 4622  Rel wrel 4629  (class class class)co 5870   Er wer 6527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4119  ax-pow 4172  ax-pr 4207

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-iun 3887  df-br 4002  df-opab 4063  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-iota 5175  df-fv 5221  df-ov 5873  df-er 6530

This theorem is referenced by: (None)