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Theorem addvalex 7616
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 7709. (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 5743 . 2 (𝐴 + 𝐵) = ( + ‘⟨𝐴, 𝐵⟩)
2 df-nr 7499 . . . . 5 R = ((P × P) / ~R )
3 npex 7245 . . . . . . 7 P ∈ V
43, 3xpex 4622 . . . . . 6 (P × P) ∈ V
54qsex 6452 . . . . 5 ((P × P) / ~R ) ∈ V
62, 5eqeltri 2188 . . . 4 R ∈ V
7 df-add 7595 . . . . 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 7590 . . . . . . . . 9 ℂ = (R × R)
98eleq2i 2182 . . . . . . . 8 (𝑥 ∈ ℂ ↔ 𝑥 ∈ (R × R))
108eleq2i 2182 . . . . . . . 8 (𝑦 ∈ ℂ ↔ 𝑦 ∈ (R × R))
119, 10anbi12i 453 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ↔ (𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)))
1211anbi1i 451 . . . . . 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 5792 . . . . 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 2136 . . . 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 5993 . . 3 + ∈ V
16 opexg 4118 . . 3 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ V)
17 fvexg 5406 . . 3 (( + ∈ V ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ( + ‘⟨𝐴, 𝐵⟩) ∈ V)
1815, 16, 17sylancr 408 . 2 ((𝐴𝑉𝐵𝑊) → ( + ‘⟨𝐴, 𝐵⟩) ∈ V)
191, 18eqeltrid 2202 1 ((𝐴𝑉𝐵𝑊) → (𝐴 + 𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wex 1451  wcel 1463  Vcvv 2658  cop 3498   × cxp 4505  cfv 5091  (class class class)co 5740  {coprab 5741   / cqs 6394  Pcnp 7063   ~R cer 7068  Rcnr 7069   +R cplr 7073  cc 7582   + caddc 7587
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-qs 6401  df-ni 7076  df-nqqs 7120  df-inp 7238  df-nr 7499  df-c 7590  df-add 7595
This theorem is referenced by:  peano2nnnn  7625
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