Theorem dfplq0qs 7431

Description: Addition on nonnegative fractions. This definition is similar to df-plq0 7428 but expands Q₀. (Contributed by Jim Kingdon, 24-Nov-2019.)

Assertion

Ref	Expression
dfplq0qs	⊢ +Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·o 𝑓) +o (𝑣 ·o 𝑢)), (𝑣 ·o 𝑓)⟩] ~Q0 ))}

Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑓

Proof of Theorem dfplq0qs

Step	Hyp	Ref	Expression
1		df-plq0 7428	. 2 ⊢ +Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ Q0 ∧ 𝑦 ∈ Q0) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·o 𝑓) +o (𝑣 ·o 𝑢)), (𝑣 ·o 𝑓)⟩] ~Q0 ))}
2		df-nq0 7426	. . . . . 6 ⊢ Q0 = ((ω × N) / ~Q0 )
3	2	eleq2i 2244	. . . . 5 ⊢ (𝑥 ∈ Q0 ↔ 𝑥 ∈ ((ω × N) / ~Q0 ))
4	2	eleq2i 2244	. . . . 5 ⊢ (𝑦 ∈ Q0 ↔ 𝑦 ∈ ((ω × N) / ~Q0 ))
5	3, 4	anbi12i 460	. . . 4 ⊢ ((𝑥 ∈ Q0 ∧ 𝑦 ∈ Q0) ↔ (𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )))
6	5	anbi1i 458	. . 3 ⊢ (((𝑥 ∈ Q0 ∧ 𝑦 ∈ Q0) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·o 𝑓) +o (𝑣 ·o 𝑢)), (𝑣 ·o 𝑓)⟩] ~Q0 )) ↔ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·o 𝑓) +o (𝑣 ·o 𝑢)), (𝑣 ·o 𝑓)⟩] ~Q0 )))
7	6	oprabbii 5932	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ Q0 ∧ 𝑦 ∈ Q0) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·o 𝑓) +o (𝑣 ·o 𝑢)), (𝑣 ·o 𝑓)⟩] ~Q0 ))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·o 𝑓) +o (𝑣 ·o 𝑢)), (𝑣 ·o 𝑓)⟩] ~Q0 ))}
8	1, 7	eqtri 2198	1 ⊢ +Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·o 𝑓) +o (𝑣 ·o 𝑢)), (𝑣 ·o 𝑓)⟩] ~Q0 ))}

Syntax hints:   ∧ wa 104   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ⟨cop 3597  ωcom 4591   × cxp 4626  (class class class)co 5877  {coprab 5878   +_o coa 6416   ·_o comu 6417  [cec 6535   / cqs 6536  Ncnpi 7273   ~_Q0 ceq0 7287  Q₀cnq0 7288   +_Q0 cplq0 7290

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-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-oprab 5881  df-nq0 7426  df-plq0 7428

This theorem is referenced by:  addnnnq0  7450