Theorem ecopoverg 6614

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 Jim Kingdon, 1-Sep-2019.)

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)))}
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
ecopoverg	⊢ ∼ 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 ecopoverg

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 4737	. . . 4 ⊢ Rel ∼
3	2	a1i 9	. . 3 ⊢ (⊤ → Rel ∼ )
4		ecopoprg.com	. . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
5	1, 4	ecopovsymg 6612	. . . 4 ⊢ (𝑓 ∼ 𝑔 → 𝑔 ∼ 𝑓)
6	5	adantl 275	. . 3 ⊢ ((⊤ ∧ 𝑓 ∼ 𝑔) → 𝑔 ∼ 𝑓)
7		ecopoprg.cl	. . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
8		ecopoprg.ass	. . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
9		ecopoprg.can	. . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧))
10	1, 4, 7, 8, 9	ecopovtrng 6613	. . . 4 ⊢ ((𝑓 ∼ 𝑔 ∧ 𝑔 ∼ ℎ) → 𝑓 ∼ ℎ)
11	10	adantl 275	. . 3 ⊢ ((⊤ ∧ (𝑓 ∼ 𝑔 ∧ 𝑔 ∼ ℎ)) → 𝑓 ∼ ℎ)
12	4	adantl 275	. . . . . . . . . . 11 ⊢ ((((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
13		simpll 524	. . . . . . . . . . 11 ⊢ (((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → 𝑔 ∈ 𝑆)
14		simplr 525	. . . . . . . . . . 11 ⊢ (((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → ℎ ∈ 𝑆)
15	12, 13, 14	caovcomd 6009	. . . . . . . . . 10 ⊢ (((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → (𝑔 + ℎ) = (ℎ + 𝑔))
16	1	ecopoveq 6608	. . . . . . . . . 10 ⊢ (((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → (⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩ ↔ (𝑔 + ℎ) = (ℎ + 𝑔)))
17	15, 16	mpbird 166	. . . . . . . . 9 ⊢ (((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → ⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩)
18	17	anidms 395	. . . . . . . 8 ⊢ ((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) → ⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩)
19	18	rgen2a 2524	. . . . . . 7 ⊢ ∀𝑔 ∈ 𝑆 ∀ℎ ∈ 𝑆 ⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩
20		breq12 3994	. . . . . . . . 9 ⊢ ((𝑓 = ⟨𝑔, ℎ⟩ ∧ 𝑓 = ⟨𝑔, ℎ⟩) → (𝑓 ∼ 𝑓 ↔ ⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩))
21	20	anidms 395	. . . . . . . 8 ⊢ (𝑓 = ⟨𝑔, ℎ⟩ → (𝑓 ∼ 𝑓 ↔ ⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩))
22	21	ralxp 4754	. . . . . . 7 ⊢ (∀𝑓 ∈ (𝑆 × 𝑆)𝑓 ∼ 𝑓 ↔ ∀𝑔 ∈ 𝑆 ∀ℎ ∈ 𝑆 ⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩)
23	19, 22	mpbir 145	. . . . . 6 ⊢ ∀𝑓 ∈ (𝑆 × 𝑆)𝑓 ∼ 𝑓
24	23	rspec 2522	. . . . 5 ⊢ (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 ∼ 𝑓)
25	24	a1i 9	. . . 4 ⊢ (⊤ → (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 ∼ 𝑓))
26		opabssxp 4685	. . . . . . 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))
27	1, 26	eqsstri 3179	. . . . . 6 ⊢ ∼ ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
28	27	ssbri 4033	. . . . 5 ⊢ (𝑓 ∼ 𝑓 → 𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓)
29		brxp 4642	. . . . . 6 ⊢ (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓 ↔ (𝑓 ∈ (𝑆 × 𝑆) ∧ 𝑓 ∈ (𝑆 × 𝑆)))
30	29	simplbi 272	. . . . 5 ⊢ (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓 → 𝑓 ∈ (𝑆 × 𝑆))
31	28, 30	syl 14	. . . 4 ⊢ (𝑓 ∼ 𝑓 → 𝑓 ∈ (𝑆 × 𝑆))
32	25, 31	impbid1 141	. . 3 ⊢ (⊤ → (𝑓 ∈ (𝑆 × 𝑆) ↔ 𝑓 ∼ 𝑓))
33	3, 6, 11, 32	iserd 6539	. 2 ⊢ (⊤ → ∼ Er (𝑆 × 𝑆))
34	33	mptru 1357	1 ⊢ ∼ Er (𝑆 × 𝑆)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 973   = wceq 1348  ⊤wtru 1349  ∃wex 1485   ∈ wcel 2141  ∀wral 2448  ⟨cop 3586   class class class wbr 3989  {copab 4049   × cxp 4609  Rel wrel 4616  (class class class)co 5853   Er wer 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fv 5206  df-ov 5856  df-er 6513

This theorem is referenced by:  enqer  7320  enrer  7697