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Mirrors > Home > ILE Home > Th. List > addclpr | GIF version |
Description: Closure of addition on positive reals. First statement of Proposition 9-3.5 of [Gleason] p. 123. Combination of Lemma 11.13 and Lemma 11.16 in [BauerTaylor], p. 53. (Contributed by NM, 13-Mar-1996.) |
Ref | Expression |
---|---|
addclpr | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iplp 7276 | . . . 4 ⊢ +P = (𝑤 ∈ P, 𝑣 ∈ P ↦ 〈{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦 +Q 𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦 +Q 𝑧))}〉) | |
2 | 1 | genpelxp 7319 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ (𝒫 Q × 𝒫 Q)) |
3 | addclnq 7183 | . . . 4 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +Q 𝑧) ∈ Q) | |
4 | 1, 3 | genpml 7325 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))) |
5 | 1, 3 | genpmu 7326 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))) |
6 | 2, 4, 5 | jca32 308 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 𝐵) ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))))) |
7 | ltanqg 7208 | . . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ (𝑧 +Q 𝑥) <Q (𝑧 +Q 𝑦))) | |
8 | addcomnqg 7189 | . . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥)) | |
9 | addnqprl 7337 | . . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (1st ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (1st ‘𝐵))) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → 𝑥 ∈ (1st ‘(𝐴 +P 𝐵)))) | |
10 | 1, 3, 7, 8, 9 | genprndl 7329 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘(𝐴 +P 𝐵))))) |
11 | addnqpru 7338 | . . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2nd ‘𝐵))) ∧ 𝑥 ∈ Q) → ((𝑔 +Q ℎ) <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴 +P 𝐵)))) | |
12 | 1, 3, 7, 8, 11 | genprndu 7330 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴 +P 𝐵))))) |
13 | 10, 12 | jca 304 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∀𝑞 ∈ Q (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘(𝐴 +P 𝐵)))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴 +P 𝐵)))))) |
14 | 1, 3, 7, 8 | genpdisj 7331 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴 +P 𝐵)))) |
15 | addlocpr 7344 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))))) | |
16 | 13, 14, 15 | 3jca 1161 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((∀𝑞 ∈ Q (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘(𝐴 +P 𝐵)))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴 +P 𝐵))))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴 +P 𝐵))) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))) |
17 | elnp1st2nd 7284 | . 2 ⊢ ((𝐴 +P 𝐵) ∈ P ↔ (((𝐴 +P 𝐵) ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘(𝐴 +P 𝐵)))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴 +P 𝐵))))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴 +P 𝐵))) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))))))) | |
18 | 6, 16, 17 | sylanbrc 413 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 697 ∧ w3a 962 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 𝒫 cpw 3510 class class class wbr 3929 × cxp 4537 ‘cfv 5123 (class class class)co 5774 1st c1st 6036 2nd c2nd 6037 Qcnq 7088 +Q cplq 7090 <Q cltq 7093 Pcnp 7099 +P cpp 7101 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-2o 6314 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 df-enq0 7232 df-nq0 7233 df-0nq0 7234 df-plq0 7235 df-mq0 7236 df-inp 7274 df-iplp 7276 |
This theorem is referenced by: addnqprlemfl 7367 addnqprlemfu 7368 addnqpr 7369 addassprg 7387 distrlem1prl 7390 distrlem1pru 7391 distrlem4prl 7392 distrlem4pru 7393 distrprg 7396 ltaddpr 7405 ltexpri 7421 addcanprleml 7422 addcanprlemu 7423 ltaprlem 7426 ltaprg 7427 prplnqu 7428 addextpr 7429 caucvgprlemcanl 7452 cauappcvgprlemladdru 7464 cauappcvgprlemladdrl 7465 cauappcvgprlemladd 7466 cauappcvgprlem1 7467 caucvgprlemladdrl 7486 caucvgprlem1 7487 caucvgprprlemnbj 7501 caucvgprprlemopu 7507 caucvgprprlemloc 7511 caucvgprprlemexbt 7514 caucvgprprlemexb 7515 caucvgprprlemaddq 7516 caucvgprprlem2 7518 enrer 7543 addcmpblnr 7547 mulcmpblnrlemg 7548 mulcmpblnr 7549 ltsrprg 7555 1sr 7559 m1r 7560 addclsr 7561 mulclsr 7562 addasssrg 7564 mulasssrg 7566 distrsrg 7567 m1p1sr 7568 m1m1sr 7569 lttrsr 7570 ltsosr 7572 0lt1sr 7573 0idsr 7575 1idsr 7576 00sr 7577 ltasrg 7578 recexgt0sr 7581 mulgt0sr 7586 aptisr 7587 mulextsr1lem 7588 mulextsr1 7589 archsr 7590 srpospr 7591 prsrcl 7592 prsradd 7594 prsrlt 7595 caucvgsrlemcau 7601 caucvgsrlemgt1 7603 mappsrprg 7612 map2psrprg 7613 pitonnlem1p1 7654 pitonnlem2 7655 pitonn 7656 pitoregt0 7657 pitore 7658 recnnre 7659 recidpirqlemcalc 7665 recidpirq 7666 |
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