Theorem plpvlu 7666

Description: Value of addition on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)

Assertion

Ref	Expression
plpvlu	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑥 = (𝑦 +Q 𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑥 = (𝑦 +Q 𝑧)}⟩)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧

Proof of Theorem plpvlu

Dummy variables 𝑓 𝑔 ℎ 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iplp 7596	. 2 ⊢ +P = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1st ‘𝑤) ∧ ℎ ∈ (1st ‘𝑣) ∧ 𝑓 = (𝑔 +Q ℎ))}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2nd ‘𝑤) ∧ ℎ ∈ (2nd ‘𝑣) ∧ 𝑓 = (𝑔 +Q ℎ))}⟩)
2		addclnq 7503	. 2 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
3	1, 2	genipv 7637	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑥 = (𝑦 +Q 𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑥 = (𝑦 +Q 𝑧)}⟩)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  ∃wrex 2486  {crab 2489  ⟨cop 3640  ‘cfv 5279  (class class class)co 5956  1^st c1st 6236  2^nd c2nd 6237  Qcnq 7408   +_Q cplq 7410  Pcnp 7419   +_P cpp 7421

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-id 4347  df-iord 4420  df-on 4422  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-ov 5959  df-oprab 5960  df-mpo 5961  df-1st 6238  df-2nd 6239  df-recs 6403  df-irdg 6468  df-oadd 6518  df-omul 6519  df-er 6632  df-ec 6634  df-qs 6638  df-ni 7432  df-pli 7433  df-mi 7434  df-plpq 7472  df-enq 7475  df-nqqs 7476  df-plqqs 7477  df-inp 7594  df-iplp 7596

This theorem is referenced by:  addcomprg  7706