Theorem oviec 6875

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 4829	. . . . 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 4829	. . . . 5 ⊢ (((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑡 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆)) → (⟨𝑔, ℎ⟩ ∼ ⟨𝑡, 𝑠⟩ ↔ 𝜒))
9	6, 8	bi2anan9 610	. . . 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 6191	. . . . . 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 6191	. . . . . 6 ⊢ (((𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆) ∧ (𝑡 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆)) → (⟨𝑐, 𝑑⟩ + ⟨𝑡, 𝑠⟩) = 𝐿)
17	13, 16	breqan12d 4125	. . . . 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 592	. . . 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 203	. . 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 2299	. . . . . . 7 ⊢ (𝑥 ∈ 𝑄 ↔ 𝑥 ∈ ((𝑆 × 𝑆) / ∼ ))
23	21	eleq2i 2299	. . . . . . 7 ⊢ (𝑦 ∈ 𝑄 ↔ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ ))
24	22, 23	anbi12i 460	. . . . . 6 ⊢ ((𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝑄) ↔ (𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )))
25	24	anbi1i 458	. . . . 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 6108	. . . 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 2253	. . 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 6874	. 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 6191	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = 𝐻)
32	31	eceq1d 6803	. 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [𝐻] ∼ )
33	28, 32	eqtrd 2265	1 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ([⟨𝐴, 𝐵⟩] ∼ ⨣ [⟨𝐶, 𝐷⟩] ∼ ) = [𝐻] ∼ )

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2203  Vcvv 2813  ⟨cop 3692   class class class wbr 4109  {copab 4170   × cxp 4747  (class class class)co 6050  {coprab 6051   Er wer 6764  [cec 6765   / cqs 6766

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-oprab 6054  df-er 6767  df-ec 6769  df-qs 6773

This theorem is referenced by:  addpipqqs  7685  mulpipqqs  7688