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Mirrors > Home > MPE Home > Th. List > plpv | Structured version Visualization version GIF version |
Description: Value of addition on positive reals. (Contributed by NM, 28-Feb-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
plpv | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +Q 𝑧)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-plp 10394 | . 2 ⊢ +P = (𝑢 ∈ P, 𝑣 ∈ P ↦ {𝑓 ∣ ∃𝑔 ∈ 𝑢 ∃ℎ ∈ 𝑣 𝑓 = (𝑔 +Q ℎ)}) | |
2 | addclnq 10356 | . 2 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q) | |
3 | 1, 2 | genpv 10410 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +Q 𝑧)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {cab 2776 ∃wrex 3107 (class class class)co 7135 +Q cplq 10266 Pcnp 10270 +P cpp 10272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-omul 8090 df-er 8272 df-ni 10283 df-pli 10284 df-mi 10285 df-lti 10286 df-plpq 10319 df-enq 10322 df-nq 10323 df-erq 10324 df-plq 10325 df-1nq 10327 df-np 10392 df-plp 10394 |
This theorem is referenced by: addcompr 10432 |
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