Theorem addcnsr 11049

Description: Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
addcnsr	⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)

Proof of Theorem addcnsr

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opex 5403	. 2 ⊢ ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩ ∈ V
2		oveq1 7363	. . . 4 ⊢ (𝑤 = 𝐴 → (𝑤 +R 𝑢) = (𝐴 +R 𝑢))
3		oveq1 7363	. . . 4 ⊢ (𝑣 = 𝐵 → (𝑣 +R 𝑓) = (𝐵 +R 𝑓))
4		opeq12 4806	. . . 4 ⊢ (((𝑤 +R 𝑢) = (𝐴 +R 𝑢) ∧ (𝑣 +R 𝑓) = (𝐵 +R 𝑓)) → ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩ = ⟨(𝐴 +R 𝑢), (𝐵 +R 𝑓)⟩)
5	2, 3, 4	syl2an 602	. . 3 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩ = ⟨(𝐴 +R 𝑢), (𝐵 +R 𝑓)⟩)
6		oveq2 7364	. . . 4 ⊢ (𝑢 = 𝐶 → (𝐴 +R 𝑢) = (𝐴 +R 𝐶))
7		oveq2 7364	. . . 4 ⊢ (𝑓 = 𝐷 → (𝐵 +R 𝑓) = (𝐵 +R 𝐷))
8		opeq12 4806	. . . 4 ⊢ (((𝐴 +R 𝑢) = (𝐴 +R 𝐶) ∧ (𝐵 +R 𝑓) = (𝐵 +R 𝐷)) → ⟨(𝐴 +R 𝑢), (𝐵 +R 𝑓)⟩ = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)
9	6, 7, 8	syl2an 602	. . 3 ⊢ ((𝑢 = 𝐶 ∧ 𝑓 = 𝐷) → ⟨(𝐴 +R 𝑢), (𝐵 +R 𝑓)⟩ = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)
10	5, 9	sylan9eq 2794	. 2 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩ = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)
11		df-add 11040	. . 3 ⊢ + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
12		df-c 11035	. . . . . . 7 ⊢ ℂ = (R × R)
13	12	eleq2i 2831	. . . . . 6 ⊢ (𝑥 ∈ ℂ ↔ 𝑥 ∈ (R × R))
14	12	eleq2i 2831	. . . . . 6 ⊢ (𝑦 ∈ ℂ ↔ 𝑦 ∈ (R × R))
15	13, 14	anbi12i 634	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ↔ (𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)))
16	15	anbi1i 630	. . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)) ↔ ((𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)))
17	16	oprabbii 7423	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
18	11, 17	eqtri 2762	. 2 ⊢ + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
19	1, 10, 18	ov3 7519	1 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ⟨cop 4561   × cxp 5616  (class class class)co 7356  {coprab 7357  Rcnr 10779   +_R cplr 10783  ℂcc 11027   + caddc 11032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359  df-oprab 7360  df-c 11035  df-add 11040

This theorem is referenced by:  addresr  11052  addcnsrec  11057  axaddf  11059  axcnre  11078