Theorem ecopoverOLD 7852

Description: Obsolete proof of ecopover 7851 as of 1-May-2021. 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.) (Proof modification is discouraged.) (New usage is discouraged.)

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
ecopoverOLD	⊢ ∼ 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 ecopoverOLD

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 5245	. . . 4 ⊢ Rel ∼
3	2	a1i 11	. . 3 ⊢ (⊤ → Rel ∼ )
4		ecopopr.com	. . . . 5 ⊢ (𝑥 + 𝑦) = (𝑦 + 𝑥)
5	1, 4	ecopovsym 7849	. . . 4 ⊢ (𝑓 ∼ 𝑔 → 𝑔 ∼ 𝑓)
6	5	adantl 482	. . 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 7850	. . . 4 ⊢ ((𝑓 ∼ 𝑔 ∧ 𝑔 ∼ ℎ) → 𝑓 ∼ ℎ)
11	10	adantl 482	. . 3 ⊢ ((⊤ ∧ (𝑓 ∼ 𝑔 ∧ 𝑔 ∼ ℎ)) → 𝑓 ∼ ℎ)
12		vex 3203	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
13		vex 3203	. . . . . . . . . . 11 ⊢ ℎ ∈ V
14	12, 13, 4	caovcom 6831	. . . . . . . . . 10 ⊢ (𝑔 + ℎ) = (ℎ + 𝑔)
15	1	ecopoveq 7848	. . . . . . . . . 10 ⊢ (((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → (⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩ ↔ (𝑔 + ℎ) = (ℎ + 𝑔)))
16	14, 15	mpbiri 248	. . . . . . . . 9 ⊢ (((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → ⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩)
17	16	anidms 677	. . . . . . . 8 ⊢ ((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) → ⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩)
18	17	rgen2a 2977	. . . . . . 7 ⊢ ∀𝑔 ∈ 𝑆 ∀ℎ ∈ 𝑆 ⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩
19		breq12 4658	. . . . . . . . 9 ⊢ ((𝑓 = ⟨𝑔, ℎ⟩ ∧ 𝑓 = ⟨𝑔, ℎ⟩) → (𝑓 ∼ 𝑓 ↔ ⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩))
20	19	anidms 677	. . . . . . . 8 ⊢ (𝑓 = ⟨𝑔, ℎ⟩ → (𝑓 ∼ 𝑓 ↔ ⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩))
21	20	ralxp 5263	. . . . . . 7 ⊢ (∀𝑓 ∈ (𝑆 × 𝑆)𝑓 ∼ 𝑓 ↔ ∀𝑔 ∈ 𝑆 ∀ℎ ∈ 𝑆 ⟨𝑔, ℎ⟩ ∼ ⟨𝑔, ℎ⟩)
22	18, 21	mpbir 221	. . . . . 6 ⊢ ∀𝑓 ∈ (𝑆 × 𝑆)𝑓 ∼ 𝑓
23	22	rspec 2931	. . . . 5 ⊢ (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 ∼ 𝑓)
24	23	a1i 11	. . . 4 ⊢ (⊤ → (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 ∼ 𝑓))
25		opabssxp 5193	. . . . . . 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 3635	. . . . . 6 ⊢ ∼ ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
27	26	ssbri 4697	. . . . 5 ⊢ (𝑓 ∼ 𝑓 → 𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓)
28		brxp 5147	. . . . . 6 ⊢ (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓 ↔ (𝑓 ∈ (𝑆 × 𝑆) ∧ 𝑓 ∈ (𝑆 × 𝑆)))
29	28	simplbi 476	. . . . 5 ⊢ (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓 → 𝑓 ∈ (𝑆 × 𝑆))
30	27, 29	syl 17	. . . 4 ⊢ (𝑓 ∼ 𝑓 → 𝑓 ∈ (𝑆 × 𝑆))
31	24, 30	impbid1 215	. . 3 ⊢ (⊤ → (𝑓 ∈ (𝑆 × 𝑆) ↔ 𝑓 ∼ 𝑓))
32	3, 6, 11, 31	iserd 7768	. 2 ⊢ (⊤ → ∼ Er (𝑆 × 𝑆))
33	32	trud 1493	1 ⊢ ∼ Er (𝑆 × 𝑆)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ⊤wtru 1484  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ⟨cop 4183   class class class wbr 4653  {copab 4712   × cxp 5112  Rel wrel 5119  (class class class)co 6650   Er wer 7739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653  df-er 7742

This theorem is referenced by: (None)