Theorem oviec 6805

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 4803	. . . . 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 4803	. . . . 5 ⊢ (((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑡 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆)) → (⟨𝑔, ℎ⟩ ∼ ⟨𝑡, 𝑠⟩ ↔ 𝜒))
9	6, 8	bi2anan9 608	. . . 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 6154	. . . . . 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 6154	. . . . . 6 ⊢ (((𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆) ∧ (𝑡 ∈ 𝑆 ∧ 𝑠 ∈ 𝑆)) → (⟨𝑐, 𝑑⟩ + ⟨𝑡, 𝑠⟩) = 𝐿)
17	13, 16	breqan12d 4102	. . . . 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 590	. . . 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 2296	. . . . . . 7 ⊢ (𝑥 ∈ 𝑄 ↔ 𝑥 ∈ ((𝑆 × 𝑆) / ∼ ))
23	21	eleq2i 2296	. . . . . . 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 6071	. . . 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 2250	. . 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 6804	. 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 6154	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = 𝐻)
32	31	eceq1d 6733	. 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → [(⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩)] ∼ = [𝐻] ∼ )
33	28, 32	eqtrd 2262	1 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ([⟨𝐴, 𝐵⟩] ∼ ⨣ [⟨𝐶, 𝐷⟩] ∼ ) = [𝐻] ∼ )

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2800  ⟨cop 3670   class class class wbr 4086  {copab 4147   × cxp 4721  (class class class)co 6013  {coprab 6014   Er wer 6694  [cec 6695   / cqs 6696

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fv 5332  df-ov 6016  df-oprab 6017  df-er 6697  df-ec 6699  df-qs 6703

This theorem is referenced by:  addpipqqs  7580  mulpipqqs  7583