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Mirrors > Home > MPE Home > Th. List > addclpr | Structured version Visualization version GIF version |
Description: Closure of addition on positive reals. First statement of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
addclpr | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-plp 11020 | . 2 ⊢ +P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦 +Q 𝑧)}) | |
2 | addclnq 10982 | . 2 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +Q 𝑧) ∈ Q) | |
3 | ltanq 11008 | . 2 ⊢ (ℎ ∈ Q → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔))) | |
4 | addcomnq 10988 | . 2 ⊢ (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥) | |
5 | addclprlem2 11054 | . 2 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → 𝑥 ∈ (𝐴 +P 𝐵))) | |
6 | 1, 2, 3, 4, 5 | genpcl 11045 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 (class class class)co 7430 +Q cplq 10892 Pcnp 10896 +P cpp 10898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-omul 8509 df-er 8743 df-ni 10909 df-pli 10910 df-mi 10911 df-lti 10912 df-plpq 10945 df-mpq 10946 df-ltpq 10947 df-enq 10948 df-nq 10949 df-erq 10950 df-plq 10951 df-mq 10952 df-1nq 10953 df-rq 10954 df-ltnq 10955 df-np 11018 df-plp 11020 |
This theorem is referenced by: addasspr 11059 distrlem1pr 11062 distrlem4pr 11063 ltaddpr 11071 ltexprlem7 11079 ltaprlem 11081 ltapr 11082 addcanpr 11083 enrer 11100 addcmpblnr 11106 mulcmpblnr 11108 ltsrpr 11114 1sr 11118 m1r 11119 addclsr 11120 mulclsr 11121 addasssr 11125 mulasssr 11127 distrsr 11128 m1p1sr 11129 m1m1sr 11130 ltsosr 11131 0lt1sr 11132 0idsr 11134 1idsr 11135 00sr 11136 ltasr 11137 recexsrlem 11140 mulgt0sr 11142 mappsrpr 11145 |
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