Theorem addcnsr 11093

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 5431	. 2 ⊢ ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩ ∈ V
2		oveq1 7403	. . . 4 ⊢ (𝑤 = 𝐴 → (𝑤 +R 𝑢) = (𝐴 +R 𝑢))
3		oveq1 7403	. . . 4 ⊢ (𝑣 = 𝐵 → (𝑣 +R 𝑓) = (𝐵 +R 𝑓))
4		opeq12 4833	. . . 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 605	. . 3 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩ = ⟨(𝐴 +R 𝑢), (𝐵 +R 𝑓)⟩)
6		oveq2 7404	. . . 4 ⊢ (𝑢 = 𝐶 → (𝐴 +R 𝑢) = (𝐴 +R 𝐶))
7		oveq2 7404	. . . 4 ⊢ (𝑓 = 𝐷 → (𝐵 +R 𝑓) = (𝐵 +R 𝐷))
8		opeq12 4833	. . . 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 605	. . 3 ⊢ ((𝑢 = 𝐶 ∧ 𝑓 = 𝐷) → ⟨(𝐴 +R 𝑢), (𝐵 +R 𝑓)⟩ = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)
10	5, 9	sylan9eq 2817	. 2 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩ = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)
11		df-add 11084	. . 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 11079	. . . . . . 7 ⊢ ℂ = (R × R)
13	12	eleq2i 2854	. . . . . 6 ⊢ (𝑥 ∈ ℂ ↔ 𝑥 ∈ (R × R))
14	12	eleq2i 2854	. . . . . 6 ⊢ (𝑦 ∈ ℂ ↔ 𝑦 ∈ (R × R))
15	13, 14	anbi12i 637	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ↔ (𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)))
16	15	anbi1i 633	. . . 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 7463	. . 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 2785	. 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 7559	1 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560  ∃wex 1799   ∈ wcel 2142  ⟨cop 4588   × cxp 5645  (class class class)co 7396  {coprab 7397  Rcnr 10823   +_R cplr 10827  ℂcc 11071   + caddc 11076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-iota 6477  df-fun 6523  df-fv 6529  df-ov 7399  df-oprab 7400  df-c 11079  df-add 11084

This theorem is referenced by:  addresr  11096  addcnsrec  11101  axaddf  11103  axcnre  11122