Theorem oviec 6619

Description: Express an operation on equivalence classes of ordered pairs in terms of equivalence class of operations on ordered pairs. See iset.mm for additional comments describing the hypotheses. (Unnecessary distinct variable restrictions were removed by David Abernethy, 4-Jun-2013.) (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 4-Jun-2013.)

Hypotheses

Ref	Expression
oviec.1	⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝐻 ∈ (𝑆 × 𝑆))
oviec.2	⊢ (((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → 𝐾 ∈ (𝑆 × 𝑆))
oviec.3	⊢ (((𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆) ∧ (𝑡 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆)) → 𝐿 ∈ (𝑆 × 𝑆))
oviec.4	⊢ ∼ ∈ V
oviec.5	⊢ ∼ Er (𝑆 × 𝑆)
oviec.7	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑))}
oviec.8	⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → (𝜑 ↔ 𝜓))
oviec.9	⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → (𝜑 ↔ 𝜒))
oviec.10	⊢ + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝐽))}
oviec.11	⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → 𝐽 = 𝐾)
oviec.12	⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → 𝐽 = 𝐿)
oviec.13	⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝐽 = 𝐻)
oviec.14	⊢ ⨣ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝑄) ∧ ∃𝑎∃𝑏∃𝑐∃𝑑((𝑥 = [⟨𝑎, 𝑏⟩] ∼ ∧ 𝑦 = [⟨𝑐, 𝑑⟩] ∼ ) ∧ 𝑧 = [(⟨𝑎, 𝑏⟩ + ⟨𝑐, 𝑑⟩)] ∼ ))}
oviec.15	⊢ 𝑄 = ((𝑆 × 𝑆) / ∼ )
oviec.16	⊢ ((((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) ∧ ((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑡 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆))) → ((𝜓 ∧ 𝜒) → 𝐾 ∼ 𝐿))

Assertion

Ref	Expression
oviec	⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ([⟨𝐴, 𝐵⟩] ∼ ⨣ [⟨𝐶, 𝐷⟩] ∼ ) = [𝐻] ∼ )

Distinct variable groups:   𝑎,𝑏,𝑐,𝑑,𝑓,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧,𝐶   𝐷,𝑎,𝑏,𝑐,𝑑,𝑓,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝑥,𝐽,𝑦,𝑧   𝑔,𝑎,ℎ,𝐴,𝑏,𝑐,𝑑,𝑓,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝜒,𝑢,𝑣,𝑤,𝑧   𝑓,𝐻,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝐵,𝑎,𝑏,𝑐,𝑑,𝑓,𝑔,ℎ,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝑓,𝐾,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝜓,𝑢,𝑣,𝑤,𝑧   𝑓,𝐿,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝜑,𝑥,𝑦   𝑠,𝑎,𝑡,𝑆,𝑏,𝑐,𝑑,𝑓,𝑔,ℎ,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   + ,𝑎,𝑏,𝑐,𝑑,𝑔,ℎ,𝑠,𝑡,𝑥,𝑦,𝑧   ∼ ,𝑎,𝑏,𝑐,𝑑,𝑔,ℎ,𝑠,𝑡,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑧,𝑤,𝑣,𝑢,𝑡,𝑓,𝑔,ℎ,𝑠,𝑎,𝑏,𝑐,𝑑)   𝜓(𝑥,𝑦,𝑡,𝑓,𝑔,ℎ,𝑠,𝑎,𝑏,𝑐,𝑑)   𝜒(𝑥,𝑦,𝑡,𝑓,𝑔,ℎ,𝑠,𝑎,𝑏,𝑐,𝑑)   𝐴(𝑡,𝑠)   𝐵(𝑡,𝑠)   𝐶(𝑡,𝑔,ℎ,𝑠)   𝐷(𝑡,𝑔,ℎ,𝑠)   + (𝑤,𝑣,𝑢,𝑓)   ⨣ (𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑓,𝑔,ℎ,𝑠,𝑎,𝑏,𝑐,𝑑)   𝑄(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑓,𝑔,ℎ,𝑠,𝑎,𝑏,𝑐,𝑑)   ∼ (𝑤,𝑣,𝑢,𝑓)   𝐻(𝑡,𝑔,ℎ,𝑠,𝑎,𝑏,𝑐,𝑑)   𝐽(𝑤,𝑣,𝑢,𝑡,𝑓,𝑔,ℎ,𝑠,𝑎,𝑏,𝑐,𝑑)   𝐾(𝑡,𝑔,ℎ,𝑠,𝑎,𝑏,𝑐,𝑑)   𝐿(𝑡,𝑔,ℎ,𝑠,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem oviec

Step	Hyp	Ref	Expression
1		oviec.4	. . 3 ⊢ ∼ ∈ V
2		oviec.5	. . 3 ⊢ ∼ Er (𝑆 × 𝑆)
3		oviec.16	. . . 4 ⊢ ((((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) ∧ ((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑡 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆))) → ((𝜓 ∧ 𝜒) → 𝐾 ∼ 𝐿))
4		oviec.8	. . . . . 6 ⊢ (((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) ∧ (𝑣 = 𝑐 ∧ 𝑢 = 𝑑)) → (𝜑 ↔ 𝜓))
5		oviec.7	. . . . . 6 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑))}
6	4, 5	opbrop 4690	. . . . 5 ⊢ (((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → (⟨𝑎, 𝑏⟩ ∼ ⟨𝑐, 𝑑⟩ ↔ 𝜓))
7		oviec.9	. . . . . 6 ⊢ (((𝑧 = 𝑔 ∧ 𝑤 = ℎ) ∧ (𝑣 = 𝑡 ∧ 𝑢 = 𝑠)) → (𝜑 ↔ 𝜒))
8	7, 5	opbrop 4690	. . . . 5 ⊢ (((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑡 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆)) → (⟨𝑔, ℎ⟩ ∼ ⟨𝑡, 𝑠⟩ ↔ 𝜒))
9	6, 8	bi2anan9 601	. . . 4 ⊢ ((((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) ∧ ((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑡 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆))) → ((⟨𝑎, 𝑏⟩ ∼ ⟨𝑐, 𝑑⟩ ∧ ⟨𝑔, ℎ⟩ ∼ ⟨𝑡, 𝑠⟩) ↔ (𝜓 ∧ 𝜒)))
10		oviec.2	. . . . . . 7 ⊢ (((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → 𝐾 ∈ (𝑆 × 𝑆))
11		oviec.11	. . . . . . 7 ⊢ (((𝑤 = 𝑎 ∧ 𝑣 = 𝑏) ∧ (𝑢 = 𝑔 ∧ 𝑓 = ℎ)) → 𝐽 = 𝐾)
12		oviec.10	. . . . . . 7 ⊢ + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝐽))}
13	10, 11, 12	ovi3 5989	. . . . . 6 ⊢ (((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → (⟨𝑎, 𝑏⟩ + ⟨𝑔, ℎ⟩) = 𝐾)
14		oviec.3	. . . . . . 7 ⊢ (((𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆) ∧ (𝑡 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆)) → 𝐿 ∈ (𝑆 × 𝑆))
15		oviec.12	. . . . . . 7 ⊢ (((𝑤 = 𝑐 ∧ 𝑣 = 𝑑) ∧ (𝑢 = 𝑡 ∧ 𝑓 = 𝑠)) → 𝐽 = 𝐿)
16	14, 15, 12	ovi3 5989	. . . . . 6 ⊢ (((𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆) ∧ (𝑡 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆)) → (⟨𝑐, 𝑑⟩ + ⟨𝑡, 𝑠⟩) = 𝐿)
17	13, 16	breqan12d 4005	. . . . 5 ⊢ ((((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) ∧ ((𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆) ∧ (𝑡 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆))) → ((⟨𝑎, 𝑏⟩ + ⟨𝑔, ℎ⟩) ∼ (⟨𝑐, 𝑑⟩ + ⟨𝑡, 𝑠⟩) ↔ 𝐾 ∼ 𝐿))
18	17	an4s 583	. . . 4 ⊢ ((((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) ∧ ((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑡 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆))) → ((⟨𝑎, 𝑏⟩ + ⟨𝑔, ℎ⟩) ∼ (⟨𝑐, 𝑑⟩ + ⟨𝑡, 𝑠⟩) ↔ 𝐾 ∼ 𝐿))
19	3, 9, 18	3imtr4d 202	. . 3 ⊢ ((((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) ∧ ((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑡 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆))) → ((⟨𝑎, 𝑏⟩ ∼ ⟨𝑐, 𝑑⟩ ∧ ⟨𝑔, ℎ⟩ ∼ ⟨𝑡, 𝑠⟩) → (⟨𝑎, 𝑏⟩ + ⟨𝑔, ℎ⟩) ∼ (⟨𝑐, 𝑑⟩ + ⟨𝑡, 𝑠⟩)))
20		oviec.14	. . . 4 ⊢ ⨣ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝑄) ∧ ∃𝑎∃𝑏∃𝑐∃𝑑((𝑥 = [⟨𝑎, 𝑏⟩] ∼ ∧ 𝑦 = [⟨𝑐, 𝑑⟩] ∼ ) ∧ 𝑧 = [(⟨𝑎, 𝑏⟩ + ⟨𝑐, 𝑑⟩)] ∼ ))}
21		oviec.15	. . . . . . . 8 ⊢ 𝑄 = ((𝑆 × 𝑆) / ∼ )
22	21	eleq2i 2237	. . . . . . 7 ⊢ (𝑥 ∈ 𝑄 ↔ 𝑥 ∈ ((𝑆 × 𝑆) / ∼ ))
23	21	eleq2i 2237	. . . . . . 7 ⊢ (𝑦 ∈ 𝑄 ↔ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ ))
24	22, 23	anbi12i 457	. . . . . 6 ⊢ ((𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝑄) ↔ (𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )))
25	24	anbi1i 455	. . . . 5 ⊢ (((𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝑄) ∧ ∃𝑎∃𝑏∃𝑐∃𝑑((𝑥 = [⟨𝑎, 𝑏⟩] ∼ ∧ 𝑦 = [⟨𝑐, 𝑑⟩] ∼ ) ∧ 𝑧 = [(⟨𝑎, 𝑏⟩ + ⟨𝑐, 𝑑⟩)] ∼ )) ↔ ((𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )) ∧ ∃𝑎∃𝑏∃𝑐∃𝑑((𝑥 = [⟨𝑎, 𝑏⟩] ∼ ∧ 𝑦 = [⟨𝑐, 𝑑⟩] ∼ ) ∧ 𝑧 = [(⟨𝑎, 𝑏⟩ + ⟨𝑐, 𝑑⟩)] ∼ )))
26	25	oprabbii 5908	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝑄) ∧ ∃𝑎∃𝑏∃𝑐∃𝑑((𝑥 = [⟨𝑎, 𝑏⟩] ∼ ∧ 𝑦 = [⟨𝑐, 𝑑⟩] ∼ ) ∧ 𝑧 = [(⟨𝑎, 𝑏⟩ + ⟨𝑐, 𝑑⟩)] ∼ ))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )) ∧ ∃𝑎∃𝑏∃𝑐∃𝑑((𝑥 = [⟨𝑎, 𝑏⟩] ∼ ∧ 𝑦 = [⟨𝑐, 𝑑⟩] ∼ ) ∧ 𝑧 = [(⟨𝑎, 𝑏⟩ + ⟨𝑐, 𝑑⟩)] ∼ ))}
27	20, 26	eqtri 2191	. . 3 ⊢ ⨣ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )) ∧ ∃𝑎∃𝑏∃𝑐∃𝑑((𝑥 = [⟨𝑎, 𝑏⟩] ∼ ∧ 𝑦 = [⟨𝑐, 𝑑⟩] ∼ ) ∧ 𝑧 = [(⟨𝑎, 𝑏⟩ + ⟨𝑐, 𝑑⟩)] ∼ ))}
28	1, 2, 19, 27	th3q 6618	. 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ([⟨𝐴, 𝐵⟩] ∼ ⨣ [⟨𝐶, 𝐷⟩] ∼ ) = [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ )
29		oviec.1	. . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝐻 ∈ (𝑆 × 𝑆))
30		oviec.13	. . . 4 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝐽 = 𝐻)
31	29, 30, 12	ovi3 5989	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = 𝐻)
32	31	eceq1d 6549	. 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [𝐻] ∼ )
33	28, 32	eqtrd 2203	1 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ([⟨𝐴, 𝐵⟩] ∼ ⨣ [⟨𝐶, 𝐷⟩] ∼ ) = [𝐻] ∼ )

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  Vcvv 2730  ⟨cop 3586   class class class wbr 3989  {copab 4049   × cxp 4609  (class class class)co 5853  {coprab 5854   Er wer 6510  [cec 6511   / cqs 6512

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-in1 609  ax-in2 610  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-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  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-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  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-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-er 6513  df-ec 6515  df-qs 6519

This theorem is referenced by:  addpipqqs  7332  mulpipqqs  7335