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Theorem plpvlu 6579
 Description: Value of addition on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
Assertion
Ref Expression
plpvlu ((𝐴P𝐵P) → (𝐴 +P 𝐵) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦 +Q 𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦 +Q 𝑧)}⟩)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧

Proof of Theorem plpvlu
Dummy variables 𝑓 𝑔 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iplp 6509 . 2 +P = (𝑤P, 𝑣P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑤) ∧ ∈ (1st𝑣) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑤) ∧ ∈ (2nd𝑣) ∧ 𝑓 = (𝑔 +Q ))}⟩)
2 addclnq 6416 . 2 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
31, 2genipv 6550 1 ((𝐴P𝐵P) → (𝐴 +P 𝐵) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦 +Q 𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦 +Q 𝑧)}⟩)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ∃wrex 2304  {crab 2307  ⟨cop 3375  ‘cfv 4863  (class class class)co 5473  1st c1st 5726  2nd c2nd 5727  Qcnq 6321   +Q cplq 6323  Pcnp 6332   +P cpp 6334 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3868  ax-sep 3871  ax-nul 3879  ax-pow 3923  ax-pr 3940  ax-un 4141  ax-setind 4230  ax-iinf 4272 This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3577  df-int 3612  df-iun 3655  df-br 3761  df-opab 3815  df-mpt 3816  df-tr 3851  df-id 4026  df-iord 4074  df-on 4076  df-suc 4079  df-iom 4275  df-xp 4312  df-rel 4313  df-cnv 4314  df-co 4315  df-dm 4316  df-rn 4317  df-res 4318  df-ima 4319  df-iota 4828  df-fun 4865  df-fn 4866  df-f 4867  df-f1 4868  df-fo 4869  df-f1o 4870  df-fv 4871  df-ov 5476  df-oprab 5477  df-mpt2 5478  df-1st 5728  df-2nd 5729  df-recs 5881  df-irdg 5918  df-oadd 5966  df-omul 5967  df-er 6065  df-ec 6067  df-qs 6071  df-ni 6345  df-pli 6346  df-mi 6347  df-plpq 6385  df-enq 6388  df-nqqs 6389  df-plqqs 6390  df-inp 6507  df-iplp 6509 This theorem is referenced by:  addcomprg  6619
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