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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 7244 | . . . 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 7287 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ (𝒫 Q × 𝒫 Q)) |
3 | addclnq 7151 | . . . 4 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +Q 𝑧) ∈ Q) | |
4 | 1, 3 | genpml 7293 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))) |
5 | 1, 3 | genpmu 7294 | . . 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 7176 | . . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ (𝑧 +Q 𝑥) <Q (𝑧 +Q 𝑦))) | |
8 | addcomnqg 7157 | . . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥)) | |
9 | addnqprl 7305 | . . . . 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 7297 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘(𝐴 +P 𝐵))))) |
11 | addnqpru 7306 | . . . . 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 7298 | . . . 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 7299 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴 +P 𝐵)))) |
15 | addlocpr 7312 | . . 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 1146 | . 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 7252 | . 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 682 ∧ w3a 947 ∈ wcel 1465 ∀wral 2393 ∃wrex 2394 𝒫 cpw 3480 class class class wbr 3899 × cxp 4507 ‘cfv 5093 (class class class)co 5742 1st c1st 6004 2nd c2nd 6005 Qcnq 7056 +Q cplq 7058 <Q cltq 7061 Pcnp 7067 +P cpp 7069 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-eprel 4181 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-1o 6281 df-2o 6282 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-pli 7081 df-mi 7082 df-lti 7083 df-plpq 7120 df-mpq 7121 df-enq 7123 df-nqqs 7124 df-plqqs 7125 df-mqqs 7126 df-1nqqs 7127 df-rq 7128 df-ltnqqs 7129 df-enq0 7200 df-nq0 7201 df-0nq0 7202 df-plq0 7203 df-mq0 7204 df-inp 7242 df-iplp 7244 |
This theorem is referenced by: addnqprlemfl 7335 addnqprlemfu 7336 addnqpr 7337 addassprg 7355 distrlem1prl 7358 distrlem1pru 7359 distrlem4prl 7360 distrlem4pru 7361 distrprg 7364 ltaddpr 7373 ltexpri 7389 addcanprleml 7390 addcanprlemu 7391 ltaprlem 7394 ltaprg 7395 prplnqu 7396 addextpr 7397 caucvgprlemcanl 7420 cauappcvgprlemladdru 7432 cauappcvgprlemladdrl 7433 cauappcvgprlemladd 7434 cauappcvgprlem1 7435 caucvgprlemladdrl 7454 caucvgprlem1 7455 caucvgprprlemnbj 7469 caucvgprprlemopu 7475 caucvgprprlemloc 7479 caucvgprprlemexbt 7482 caucvgprprlemexb 7483 caucvgprprlemaddq 7484 caucvgprprlem2 7486 enrer 7511 addcmpblnr 7515 mulcmpblnrlemg 7516 mulcmpblnr 7517 ltsrprg 7523 1sr 7527 m1r 7528 addclsr 7529 mulclsr 7530 addasssrg 7532 mulasssrg 7534 distrsrg 7535 m1p1sr 7536 m1m1sr 7537 lttrsr 7538 ltsosr 7540 0lt1sr 7541 0idsr 7543 1idsr 7544 00sr 7545 ltasrg 7546 recexgt0sr 7549 mulgt0sr 7554 aptisr 7555 mulextsr1lem 7556 mulextsr1 7557 archsr 7558 srpospr 7559 prsrcl 7560 prsradd 7562 prsrlt 7563 caucvgsrlemcau 7569 caucvgsrlemgt1 7571 mappsrprg 7580 map2psrprg 7581 pitonnlem1p1 7622 pitonnlem2 7623 pitonn 7624 pitoregt0 7625 pitore 7626 recnnre 7627 recidpirqlemcalc 7633 recidpirq 7634 |
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