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Theorem th3q 6377
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 4459 . . . 4 ((𝐴𝑆𝐵𝑆) → ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
2 th3q.1 . . . . 5 ∈ V
32ecelqsi 6326 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) → [⟨𝐴, 𝐵⟩] ∈ ((𝑆 × 𝑆) / ))
41, 3syl 14 . . 3 ((𝐴𝑆𝐵𝑆) → [⟨𝐴, 𝐵⟩] ∈ ((𝑆 × 𝑆) / ))
5 opelxpi 4459 . . . 4 ((𝐶𝑆𝐷𝑆) → ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆))
62ecelqsi 6326 . . . 4 (⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆) → [⟨𝐶, 𝐷⟩] ∈ ((𝑆 × 𝑆) / ))
75, 6syl 14 . . 3 ((𝐶𝑆𝐷𝑆) → [⟨𝐶, 𝐷⟩] ∈ ((𝑆 × 𝑆) / ))
84, 7anim12i 331 . 2 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ([⟨𝐴, 𝐵⟩] ∈ ((𝑆 × 𝑆) / ) ∧ [⟨𝐶, 𝐷⟩] ∈ ((𝑆 × 𝑆) / )))
9 eqid 2088 . . . 4 [⟨𝐴, 𝐵⟩] = [⟨𝐴, 𝐵⟩]
10 eqid 2088 . . . 4 [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩]
119, 10pm3.2i 266 . . 3 ([⟨𝐴, 𝐵⟩] = [⟨𝐴, 𝐵⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] )
12 eqid 2088 . . 3 [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)]
13 opeq12 3619 . . . . . 6 ((𝑤 = 𝐴𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
14 eceq1 6307 . . . . . . . . 9 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [⟨𝑤, 𝑣⟩] = [⟨𝐴, 𝐵⟩] )
1514eqeq2d 2099 . . . . . . . 8 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ↔ [⟨𝐴, 𝐵⟩] = [⟨𝐴, 𝐵⟩] ))
1615anbi1d 453 . . . . . . 7 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → (([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) ↔ ([⟨𝐴, 𝐵⟩] = [⟨𝐴, 𝐵⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] )))
17 oveq1 5641 . . . . . . . . 9 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → (⟨𝑤, 𝑣+𝐶, 𝐷⟩) = (⟨𝐴, 𝐵+𝐶, 𝐷⟩))
1817eceq1d 6308 . . . . . . . 8 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] )
1918eqeq2d 2099 . . . . . . 7 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ([(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] ↔ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] ))
2016, 19anbi12d 457 . . . . . 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 2708 . . . 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 3619 . . . . . . 7 ((𝑢 = 𝐶𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
24 eceq1 6307 . . . . . . . . . 10 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨𝑢, 𝑡⟩] = [⟨𝐶, 𝐷⟩] )
2524eqeq2d 2099 . . . . . . . . 9 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ↔ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ))
2625anbi2d 452 . . . . . . . 8 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → (([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ) ↔ ([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] )))
27 oveq2 5642 . . . . . . . . . 10 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → (⟨𝑤, 𝑣+𝑢, 𝑡⟩) = (⟨𝑤, 𝑣+𝐶, 𝐷⟩))
2827eceq1d 6308 . . . . . . . . 9 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] = [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] )
2928eqeq2d 2099 . . . . . . . 8 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ↔ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] ))
3026, 29anbi12d 457 . . . . . . 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 2708 . . . . 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 1810 . . . 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 401 . . 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 423 . 2 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ))
36 ecexg 6276 . . . 4 ( ∈ V → [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] ∈ V)
372, 36ax-mp 7 . . 3 [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] ∈ V
38 eqeq1 2094 . . . . . . . 8 (𝑥 = [⟨𝐴, 𝐵⟩] → (𝑥 = [⟨𝑤, 𝑣⟩] ↔ [⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ))
39 eqeq1 2094 . . . . . . . 8 (𝑦 = [⟨𝐶, 𝐷⟩] → (𝑦 = [⟨𝑢, 𝑡⟩] ↔ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ))
4038, 39bi2anan9 573 . . . . . . 7 ((𝑥 = [⟨𝐴, 𝐵⟩] 𝑦 = [⟨𝐶, 𝐷⟩] ) → ((𝑥 = [⟨𝑤, 𝑣⟩] 𝑦 = [⟨𝑢, 𝑡⟩] ) ↔ ([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] )))
41 eqeq1 2094 . . . . . . 7 (𝑧 = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] → (𝑧 = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ↔ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ))
4240, 41bi2anan9 573 . . . . . 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 1138 . . . . 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 1798 . . . 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 6375 . . . 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 5748 . . 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 1262 . 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 61 1 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ([⟨𝐴, 𝐵⟩] 𝐺[⟨𝐶, 𝐷⟩] ) = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] )
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 924   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619  cop 3444   class class class wbr 3837   × cxp 4426  (class class class)co 5634  {coprab 5635   Er wer 6269  [cec 6270   / cqs 6271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fv 5010  df-ov 5637  df-oprab 5638  df-er 6272  df-ec 6274  df-qs 6278
This theorem is referenced by:  oviec  6378
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