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Theorem addvalex 7360
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 7446. (Contributed by Jim Kingdon, 14-Jul-2021.)
Assertion
Ref Expression
addvalex ((𝐴𝑉𝐵𝑊) → (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.
StepHypRef Expression
1 df-ov 5637 . 2 (𝐴 + 𝐵) = ( + ‘⟨𝐴, 𝐵⟩)
2 df-nr 7252 . . . . 5 R = ((P × P) / ~R )
3 npex 7011 . . . . . . 7 P ∈ V
43, 3xpex 4541 . . . . . 6 (P × P) ∈ V
54qsex 6329 . . . . 5 ((P × P) / ~R ) ∈ V
62, 5eqeltri 2160 . . . 4 R ∈ V
7 df-add 7340 . . . . 5 + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
8 df-c 7335 . . . . . . . . 9 ℂ = (R × R)
98eleq2i 2154 . . . . . . . 8 (𝑥 ∈ ℂ ↔ 𝑥 ∈ (R × R))
108eleq2i 2154 . . . . . . . 8 (𝑦 ∈ ℂ ↔ 𝑦 ∈ (R × R))
119, 10anbi12i 448 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ↔ (𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)))
1211anbi1i 446 . . . . . 6 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)) ↔ ((𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)))
1312oprabbii 5686 . . . . 5 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
147, 13eqtri 2108 . . . 4 + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
156, 14oprabex3 5882 . . 3 + ∈ V
16 opexg 4046 . . 3 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ V)
17 fvexg 5308 . . 3 (( + ∈ V ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ( + ‘⟨𝐴, 𝐵⟩) ∈ V)
1815, 16, 17sylancr 405 . 2 ((𝐴𝑉𝐵𝑊) → ( + ‘⟨𝐴, 𝐵⟩) ∈ V)
191, 18syl5eqel 2174 1 ((𝐴𝑉𝐵𝑊) → (𝐴 + 𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619  cop 3444   × cxp 4426  cfv 5002  (class class class)co 5634  {coprab 5635   / cqs 6271  Pcnp 6829   ~R cer 6834  Rcnr 6835   +R cplr 6839  cc 7327   + caddc 7332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-qs 6278  df-ni 6842  df-nqqs 6886  df-inp 7004  df-nr 7252  df-c 7335  df-add 7340
This theorem is referenced by:  peano2nnnn  7369
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