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Theorem th3q 6241
 Description: Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
th3q.1 ∈ V
th3q.2 Er (𝑆 × 𝑆)
th3q.4 ((((𝑤𝑆𝑣𝑆) ∧ (𝑢𝑆𝑡𝑆)) ∧ ((𝑠𝑆𝑓𝑆) ∧ (𝑔𝑆𝑆))) → ((⟨𝑤, 𝑣𝑢, 𝑡⟩ ∧ ⟨𝑠, 𝑓𝑔, ⟩) → (⟨𝑤, 𝑣+𝑠, 𝑓⟩) (⟨𝑢, 𝑡+𝑔, ⟩)))
th3q.5 𝐺 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((𝑆 × 𝑆) / ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / )) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] 𝑦 = [⟨𝑢, 𝑡⟩] ) ∧ 𝑧 = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ))}
Assertion
Ref Expression
th3q (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ([⟨𝐴, 𝐵⟩] 𝐺[⟨𝐶, 𝐷⟩] ) = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] )
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠,𝑓,𝑔,,   𝑥,𝑆,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠,𝑓,𝑔,   𝑥,𝐴,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠,𝑓   𝑥,𝐵,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠,𝑓   𝑥,𝐶,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡   𝑥,𝐷,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡   𝑥, + ,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠,𝑓,𝑔,
Allowed substitution hints:   𝐴(𝑔,)   𝐵(𝑔,)   𝐶(𝑓,𝑔,,𝑠)   𝐷(𝑓,𝑔,,𝑠)   𝐺(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑓,𝑔,,𝑠)

Proof of Theorem th3q
StepHypRef Expression
1 opelxpi 4403 . . . 4 ((𝐴𝑆𝐵𝑆) → ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
2 th3q.1 . . . . 5 ∈ V
32ecelqsi 6190 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) → [⟨𝐴, 𝐵⟩] ∈ ((𝑆 × 𝑆) / ))
41, 3syl 14 . . 3 ((𝐴𝑆𝐵𝑆) → [⟨𝐴, 𝐵⟩] ∈ ((𝑆 × 𝑆) / ))
5 opelxpi 4403 . . . 4 ((𝐶𝑆𝐷𝑆) → ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆))
62ecelqsi 6190 . . . 4 (⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆) → [⟨𝐶, 𝐷⟩] ∈ ((𝑆 × 𝑆) / ))
75, 6syl 14 . . 3 ((𝐶𝑆𝐷𝑆) → [⟨𝐶, 𝐷⟩] ∈ ((𝑆 × 𝑆) / ))
84, 7anim12i 325 . 2 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ([⟨𝐴, 𝐵⟩] ∈ ((𝑆 × 𝑆) / ) ∧ [⟨𝐶, 𝐷⟩] ∈ ((𝑆 × 𝑆) / )))
9 eqid 2056 . . . 4 [⟨𝐴, 𝐵⟩] = [⟨𝐴, 𝐵⟩]
10 eqid 2056 . . . 4 [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩]
119, 10pm3.2i 261 . . 3 ([⟨𝐴, 𝐵⟩] = [⟨𝐴, 𝐵⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] )
12 eqid 2056 . . 3 [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)]
13 opeq12 3578 . . . . . 6 ((𝑤 = 𝐴𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
14 eceq1 6171 . . . . . . . . 9 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [⟨𝑤, 𝑣⟩] = [⟨𝐴, 𝐵⟩] )
1514eqeq2d 2067 . . . . . . . 8 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ↔ [⟨𝐴, 𝐵⟩] = [⟨𝐴, 𝐵⟩] ))
1615anbi1d 446 . . . . . . 7 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → (([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) ↔ ([⟨𝐴, 𝐵⟩] = [⟨𝐴, 𝐵⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] )))
17 oveq1 5546 . . . . . . . . 9 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → (⟨𝑤, 𝑣+𝐶, 𝐷⟩) = (⟨𝐴, 𝐵+𝐶, 𝐷⟩))
1817eceq1d 6172 . . . . . . . 8 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] )
1918eqeq2d 2067 . . . . . . 7 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ([(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] ↔ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] ))
2016, 19anbi12d 450 . . . . . 6 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ((([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] ) ↔ (([⟨𝐴, 𝐵⟩] = [⟨𝐴, 𝐵⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] )))
2113, 20syl 14 . . . . 5 ((𝑤 = 𝐴𝑣 = 𝐵) → ((([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] ) ↔ (([⟨𝐴, 𝐵⟩] = [⟨𝐴, 𝐵⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] )))
2221spc2egv 2659 . . . 4 ((𝐴𝑆𝐵𝑆) → ((([⟨𝐴, 𝐵⟩] = [⟨𝐴, 𝐵⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] ) → ∃𝑤𝑣(([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] )))
23 opeq12 3578 . . . . . . 7 ((𝑢 = 𝐶𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
24 eceq1 6171 . . . . . . . . . 10 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨𝑢, 𝑡⟩] = [⟨𝐶, 𝐷⟩] )
2524eqeq2d 2067 . . . . . . . . 9 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ↔ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ))
2625anbi2d 445 . . . . . . . 8 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → (([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ) ↔ ([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] )))
27 oveq2 5547 . . . . . . . . . 10 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → (⟨𝑤, 𝑣+𝑢, 𝑡⟩) = (⟨𝑤, 𝑣+𝐶, 𝐷⟩))
2827eceq1d 6172 . . . . . . . . 9 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] = [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] )
2928eqeq2d 2067 . . . . . . . 8 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ↔ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] ))
3026, 29anbi12d 450 . . . . . . 7 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ((([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ) ↔ (([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] )))
3123, 30syl 14 . . . . . 6 ((𝑢 = 𝐶𝑡 = 𝐷) → ((([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ) ↔ (([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] )))
3231spc2egv 2659 . . . . 5 ((𝐶𝑆𝐷𝑆) → ((([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] ) → ∃𝑢𝑡(([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] )))
33322eximdv 1778 . . . 4 ((𝐶𝑆𝐷𝑆) → (∃𝑤𝑣(([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] ) → ∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] )))
3422, 33sylan9 395 . . 3 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ((([⟨𝐴, 𝐵⟩] = [⟨𝐴, 𝐵⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] ) → ∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] )))
3511, 12, 34mp2ani 416 . 2 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ))
36 ecexg 6140 . . . 4 ( ∈ V → [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] ∈ V)
372, 36ax-mp 7 . . 3 [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] ∈ V
38 eqeq1 2062 . . . . . . . 8 (𝑥 = [⟨𝐴, 𝐵⟩] → (𝑥 = [⟨𝑤, 𝑣⟩] ↔ [⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ))
39 eqeq1 2062 . . . . . . . 8 (𝑦 = [⟨𝐶, 𝐷⟩] → (𝑦 = [⟨𝑢, 𝑡⟩] ↔ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ))
4038, 39bi2anan9 548 . . . . . . 7 ((𝑥 = [⟨𝐴, 𝐵⟩] 𝑦 = [⟨𝐶, 𝐷⟩] ) → ((𝑥 = [⟨𝑤, 𝑣⟩] 𝑦 = [⟨𝑢, 𝑡⟩] ) ↔ ([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] )))
41 eqeq1 2062 . . . . . . 7 (𝑧 = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] → (𝑧 = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ↔ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ))
4240, 41bi2anan9 548 . . . . . 6 (((𝑥 = [⟨𝐴, 𝐵⟩] 𝑦 = [⟨𝐶, 𝐷⟩] ) ∧ 𝑧 = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] ) → (((𝑥 = [⟨𝑤, 𝑣⟩] 𝑦 = [⟨𝑢, 𝑡⟩] ) ∧ 𝑧 = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ) ↔ (([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] )))
43423impa 1110 . . . . 5 ((𝑥 = [⟨𝐴, 𝐵⟩] 𝑦 = [⟨𝐶, 𝐷⟩] 𝑧 = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] ) → (((𝑥 = [⟨𝑤, 𝑣⟩] 𝑦 = [⟨𝑢, 𝑡⟩] ) ∧ 𝑧 = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ) ↔ (([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] )))
44434exbidv 1766 . . . 4 ((𝑥 = [⟨𝐴, 𝐵⟩] 𝑦 = [⟨𝐶, 𝐷⟩] 𝑧 = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] ) → (∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] 𝑦 = [⟨𝑢, 𝑡⟩] ) ∧ 𝑧 = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ) ↔ ∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] )))
45 th3q.2 . . . . 5 Er (𝑆 × 𝑆)
46 th3q.4 . . . . 5 ((((𝑤𝑆𝑣𝑆) ∧ (𝑢𝑆𝑡𝑆)) ∧ ((𝑠𝑆𝑓𝑆) ∧ (𝑔𝑆𝑆))) → ((⟨𝑤, 𝑣𝑢, 𝑡⟩ ∧ ⟨𝑠, 𝑓𝑔, ⟩) → (⟨𝑤, 𝑣+𝑠, 𝑓⟩) (⟨𝑢, 𝑡+𝑔, ⟩)))
472, 45, 46th3qlem2 6239 . . . 4 ((𝑥 ∈ ((𝑆 × 𝑆) / ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] 𝑦 = [⟨𝑢, 𝑡⟩] ) ∧ 𝑧 = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ))
48 th3q.5 . . . 4 𝐺 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((𝑆 × 𝑆) / ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / )) ∧ ∃𝑤𝑣𝑢𝑡((𝑥 = [⟨𝑤, 𝑣⟩] 𝑦 = [⟨𝑢, 𝑡⟩] ) ∧ 𝑧 = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ))}
4944, 47, 48ovig 5649 . . 3 (([⟨𝐴, 𝐵⟩] ∈ ((𝑆 × 𝑆) / ) ∧ [⟨𝐶, 𝐷⟩] ∈ ((𝑆 × 𝑆) / ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] ∈ V) → (∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ) → ([⟨𝐴, 𝐵⟩] 𝐺[⟨𝐶, 𝐷⟩] ) = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] ))
5037, 49mp3an3 1232 . 2 (([⟨𝐴, 𝐵⟩] ∈ ((𝑆 × 𝑆) / ) ∧ [⟨𝐶, 𝐷⟩] ∈ ((𝑆 × 𝑆) / )) → (∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ) → ([⟨𝐴, 𝐵⟩] 𝐺[⟨𝐶, 𝐷⟩] ) = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] ))
518, 35, 50sylc 60 1 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ([⟨𝐴, 𝐵⟩] 𝐺[⟨𝐶, 𝐷⟩] ) = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] )
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   ∧ w3a 896   = wceq 1259  ∃wex 1397   ∈ wcel 1409  Vcvv 2574  ⟨cop 3405   class class class wbr 3791   × cxp 4370  (class class class)co 5539  {coprab 5540   Er wer 6133  [cec 6134   / cqs 6135 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fv 4937  df-ov 5542  df-oprab 5543  df-er 6136  df-ec 6138  df-qs 6142 This theorem is referenced by:  oviec  6242
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