Theorem addcnsr 11147

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 5439	. 2 ⊢ ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩ ∈ V
2		oveq1 7421	. . . 4 ⊢ (𝑤 = 𝐴 → (𝑤 +R 𝑢) = (𝐴 +R 𝑢))
3		oveq1 7421	. . . 4 ⊢ (𝑣 = 𝐵 → (𝑣 +R 𝑓) = (𝐵 +R 𝑓))
4		opeq12 4835	. . . 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 608	. . 3 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩ = ⟨(𝐴 +R 𝑢), (𝐵 +R 𝑓)⟩)
6		oveq2 7422	. . . 4 ⊢ (𝑢 = 𝐶 → (𝐴 +R 𝑢) = (𝐴 +R 𝐶))
7		oveq2 7422	. . . 4 ⊢ (𝑓 = 𝐷 → (𝐵 +R 𝑓) = (𝐵 +R 𝐷))
8		opeq12 4835	. . . 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 608	. . 3 ⊢ ((𝑢 = 𝐶 ∧ 𝑓 = 𝐷) → ⟨(𝐴 +R 𝑢), (𝐵 +R 𝑓)⟩ = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)
10	5, 9	sylan9eq 2815	. 2 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩ = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)
11		df-add 11138	. . 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 11133	. . . . . . 7 ⊢ ℂ = (R × R)
13	12	eleq2i 2852	. . . . . 6 ⊢ (𝑥 ∈ ℂ ↔ 𝑥 ∈ (R × R))
14	12	eleq2i 2852	. . . . . 6 ⊢ (𝑦 ∈ ℂ ↔ 𝑦 ∈ (R × R))
15	13, 14	anbi12i 640	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ↔ (𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)))
16	15	anbi1i 636	. . . 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 7481	. . 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 2783	. 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 7577	1 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ⟨cop 4590   × cxp 5653  (class class class)co 7414  {coprab 7415  Rcnr 10877   +_R cplr 10881  ℂcc 11125   + caddc 11130

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7417  df-oprab 7418  df-c 11133  df-add 11138

This theorem is used by:  addresr  11150  addcnsrec  11155  axaddf  11157  axcnre  11176