Theorem addvalex 7785

Description: Existence of a sum. This is dependent on how we define + so once we proceed to real number axioms we will replace it with theorems such as addcl 7878. (Contributed by Jim Kingdon, 14-Jul-2021.)

Assertion

Ref	Expression
addvalex	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 + 𝐵) ∈ V)

Proof of Theorem addvalex

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

Step	Hyp	Ref	Expression
1		df-ov 5845	. 2 ⊢ (𝐴 + 𝐵) = ( + ‘⟨𝐴, 𝐵⟩)
2		df-nr 7668	. . . . 5 ⊢ R = ((P × P) / ~R )
3		npex 7414	. . . . . . 7 ⊢ P ∈ V
4	3, 3	xpex 4719	. . . . . 6 ⊢ (P × P) ∈ V
5	4	qsex 6558	. . . . 5 ⊢ ((P × P) / ~R ) ∈ V
6	2, 5	eqeltri 2239	. . . 4 ⊢ R ∈ V
7		df-add 7764	. . . . 5 ⊢ + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
8		df-c 7759	. . . . . . . . 9 ⊢ ℂ = (R × R)
9	8	eleq2i 2233	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ ↔ 𝑥 ∈ (R × R))
10	8	eleq2i 2233	. . . . . . . 8 ⊢ (𝑦 ∈ ℂ ↔ 𝑦 ∈ (R × R))
11	9, 10	anbi12i 456	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ↔ (𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)))
12	11	anbi1i 454	. . . . . 6 ⊢ (((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)⟩)))
13	12	oprabbii 5897	. . . . 5 ⊢ {⟨⟨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)⟩))}
14	7, 13	eqtri 2186	. . . 4 ⊢ + = {⟨⟨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)⟩))}
15	6, 14	oprabex3 6097	. . 3 ⊢ + ∈ V
16		opexg 4206	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ V)
17		fvexg 5505	. . 3 ⊢ (( + ∈ V ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ( + ‘⟨𝐴, 𝐵⟩) ∈ V)
18	15, 16, 17	sylancr 411	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ( + ‘⟨𝐴, 𝐵⟩) ∈ V)
19	1, 18	eqeltrid 2253	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 + 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136  Vcvv 2726  ⟨cop 3579   × cxp 4602  ‘cfv 5188  (class class class)co 5842  {coprab 5843   / cqs 6500  Pcnp 7232   ~_R cer 7237  Rcnr 7238   +_R cplr 7242  ℂcc 7751   + caddc 7756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-qs 6507  df-ni 7245  df-nqqs 7289  df-inp 7407  df-nr 7668  df-c 7759  df-add 7764

This theorem is referenced by:  peano2nnnn  7794