Theorem addcnsr 7918

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

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		addclsr 7837	. . . 4 ⊢ ((𝐴 ∈ R ∧ 𝐶 ∈ R) → (𝐴 +R 𝐶) ∈ R)
2	1	ad2ant2r 509	. . 3 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (𝐴 +R 𝐶) ∈ R)
3		addclsr 7837	. . . 4 ⊢ ((𝐵 ∈ R ∧ 𝐷 ∈ R) → (𝐵 +R 𝐷) ∈ R)
4	3	ad2ant2l 508	. . 3 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (𝐵 +R 𝐷) ∈ R)
5		opelxpi 4696	. . 3 ⊢ (((𝐴 +R 𝐶) ∈ R ∧ (𝐵 +R 𝐷) ∈ R) → ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩ ∈ (R × R))
6	2, 4, 5	syl2anc 411	. 2 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩ ∈ (R × R))
7		simpll 527	. . . 4 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑤 = 𝐴)
8		simprl 529	. . . 4 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑢 = 𝐶)
9	7, 8	oveq12d 5943	. . 3 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → (𝑤 +R 𝑢) = (𝐴 +R 𝐶))
10		simplr 528	. . . 4 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑣 = 𝐵)
11		simprr 531	. . . 4 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑓 = 𝐷)
12	10, 11	oveq12d 5943	. . 3 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → (𝑣 +R 𝑓) = (𝐵 +R 𝐷))
13	9, 12	opeq12d 3817	. 2 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩ = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)
14		df-add 7907	. . 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)⟩))}
15		df-c 7902	. . . . . . 7 ⊢ ℂ = (R × R)
16	15	eleq2i 2263	. . . . . 6 ⊢ (𝑥 ∈ ℂ ↔ 𝑥 ∈ (R × R))
17	15	eleq2i 2263	. . . . . 6 ⊢ (𝑦 ∈ ℂ ↔ 𝑦 ∈ (R × R))
18	16, 17	anbi12i 460	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ↔ (𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)))
19	18	anbi1i 458	. . . 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)⟩)))
20	19	oprabbii 5981	. . 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)⟩))}
21	14, 20	eqtri 2217	. 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)⟩))}
22	6, 13, 21	ovi3 6064	1 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ⟨cop 3626   × cxp 4662  (class class class)co 5925  {coprab 5926  Rcnr 7381   +_R cplr 7385  ℂcc 7894   + caddc 7899

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-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-iplp 7552  df-enr 7810  df-nr 7811  df-plr 7812  df-c 7902  df-add 7907

This theorem is referenced by:  addresr  7921  addcnsrec  7926  axaddcl  7948  axaddcom  7954  ax0id  7962  axcnre  7965