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Theorem th3q 6500
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 4539 . . . 4 ((𝐴𝑆𝐵𝑆) → ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
2 th3q.1 . . . . 5 ∈ V
32ecelqsi 6449 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) → [⟨𝐴, 𝐵⟩] ∈ ((𝑆 × 𝑆) / ))
41, 3syl 14 . . 3 ((𝐴𝑆𝐵𝑆) → [⟨𝐴, 𝐵⟩] ∈ ((𝑆 × 𝑆) / ))
5 opelxpi 4539 . . . 4 ((𝐶𝑆𝐷𝑆) → ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆))
62ecelqsi 6449 . . . 4 (⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆) → [⟨𝐶, 𝐷⟩] ∈ ((𝑆 × 𝑆) / ))
75, 6syl 14 . . 3 ((𝐶𝑆𝐷𝑆) → [⟨𝐶, 𝐷⟩] ∈ ((𝑆 × 𝑆) / ))
84, 7anim12i 334 . 2 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ([⟨𝐴, 𝐵⟩] ∈ ((𝑆 × 𝑆) / ) ∧ [⟨𝐶, 𝐷⟩] ∈ ((𝑆 × 𝑆) / )))
9 eqid 2115 . . . 4 [⟨𝐴, 𝐵⟩] = [⟨𝐴, 𝐵⟩]
10 eqid 2115 . . . 4 [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩]
119, 10pm3.2i 268 . . 3 ([⟨𝐴, 𝐵⟩] = [⟨𝐴, 𝐵⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] )
12 eqid 2115 . . 3 [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)]
13 opeq12 3675 . . . . . 6 ((𝑤 = 𝐴𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
14 eceq1 6430 . . . . . . . . 9 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [⟨𝑤, 𝑣⟩] = [⟨𝐴, 𝐵⟩] )
1514eqeq2d 2127 . . . . . . . 8 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ↔ [⟨𝐴, 𝐵⟩] = [⟨𝐴, 𝐵⟩] ))
1615anbi1d 458 . . . . . . 7 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → (([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) ↔ ([⟨𝐴, 𝐵⟩] = [⟨𝐴, 𝐵⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] )))
17 oveq1 5747 . . . . . . . . 9 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → (⟨𝑤, 𝑣+𝐶, 𝐷⟩) = (⟨𝐴, 𝐵+𝐶, 𝐷⟩))
1817eceq1d 6431 . . . . . . . 8 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] )
1918eqeq2d 2127 . . . . . . 7 (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ([(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] ↔ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] ))
2016, 19anbi12d 462 . . . . . 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 2747 . . . 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 3675 . . . . . . 7 ((𝑢 = 𝐶𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
24 eceq1 6430 . . . . . . . . . 10 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨𝑢, 𝑡⟩] = [⟨𝐶, 𝐷⟩] )
2524eqeq2d 2127 . . . . . . . . 9 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ↔ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ))
2625anbi2d 457 . . . . . . . 8 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → (([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ) ↔ ([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] )))
27 oveq2 5748 . . . . . . . . . 10 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → (⟨𝑤, 𝑣+𝑢, 𝑡⟩) = (⟨𝑤, 𝑣+𝐶, 𝐷⟩))
2827eceq1d 6431 . . . . . . . . 9 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] = [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] )
2928eqeq2d 2127 . . . . . . . 8 (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ↔ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝐶, 𝐷⟩)] ))
3026, 29anbi12d 462 . . . . . . 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 2747 . . . . 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 1836 . . . 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 404 . . 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 426 . 2 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ∃𝑤𝑣𝑢𝑡(([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ) ∧ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ))
36 ecexg 6399 . . . 4 ( ∈ V → [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] ∈ V)
372, 36ax-mp 5 . . 3 [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] ∈ V
38 eqeq1 2122 . . . . . . . 8 (𝑥 = [⟨𝐴, 𝐵⟩] → (𝑥 = [⟨𝑤, 𝑣⟩] ↔ [⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ))
39 eqeq1 2122 . . . . . . . 8 (𝑦 = [⟨𝐶, 𝐷⟩] → (𝑦 = [⟨𝑢, 𝑡⟩] ↔ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] ))
4038, 39bi2anan9 578 . . . . . . 7 ((𝑥 = [⟨𝐴, 𝐵⟩] 𝑦 = [⟨𝐶, 𝐷⟩] ) → ((𝑥 = [⟨𝑤, 𝑣⟩] 𝑦 = [⟨𝑢, 𝑡⟩] ) ↔ ([⟨𝐴, 𝐵⟩] = [⟨𝑤, 𝑣⟩] ∧ [⟨𝐶, 𝐷⟩] = [⟨𝑢, 𝑡⟩] )))
41 eqeq1 2122 . . . . . . 7 (𝑧 = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] → (𝑧 = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ↔ [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [(⟨𝑤, 𝑣+𝑢, 𝑡⟩)] ))
4240, 41bi2anan9 578 . . . . . 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 1159 . . . . 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 1824 . . . 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 6498 . . . 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 5858 . . 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 1287 . 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 62 1 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ([⟨𝐴, 𝐵⟩] 𝐺[⟨𝐶, 𝐷⟩] ) = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] )
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 945   = wceq 1314  wex 1451  wcel 1463  Vcvv 2658  cop 3498   class class class wbr 3897   × cxp 4505  (class class class)co 5740  {coprab 5741   Er wer 6392  [cec 6393   / cqs 6394
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fv 5099  df-ov 5743  df-oprab 5744  df-er 6395  df-ec 6397  df-qs 6401
This theorem is referenced by:  oviec  6501
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