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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 6566 | . . . 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 6609 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ (𝒫 Q × 𝒫 Q)) |
3 | addclnq 6473 | . . . 4 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +Q 𝑧) ∈ Q) | |
4 | 1, 3 | genpml 6615 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))) |
5 | 1, 3 | genpmu 6616 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))) |
6 | 2, 4, 5 | jca32 293 | . 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 6498 | . . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ (𝑧 +Q 𝑥) <Q (𝑧 +Q 𝑦))) | |
8 | addcomnqg 6479 | . . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥)) | |
9 | addnqprl 6627 | . . . . 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 6619 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘(𝐴 +P 𝐵))))) |
11 | addnqpru 6628 | . . . . 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 6620 | . . . 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 290 | . . 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 6621 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴 +P 𝐵)))) |
15 | addlocpr 6634 | . . 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 1084 | . 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 6574 | . 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 394 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 ∨ wo 629 ∧ w3a 885 ∈ wcel 1393 ∀wral 2306 ∃wrex 2307 𝒫 cpw 3359 class class class wbr 3764 × cxp 4343 ‘cfv 4902 (class class class)co 5512 1st c1st 5765 2nd c2nd 5766 Qcnq 6378 +Q cplq 6380 <Q cltq 6383 Pcnp 6389 +P cpp 6391 |
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 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
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 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-iplp 6566 |
This theorem is referenced by: addnqprlemfl 6657 addnqprlemfu 6658 addnqpr 6659 addassprg 6677 distrlem1prl 6680 distrlem1pru 6681 distrlem4prl 6682 distrlem4pru 6683 distrprg 6686 ltaddpr 6695 ltexpri 6711 addcanprleml 6712 addcanprlemu 6713 ltaprlem 6716 ltaprg 6717 prplnqu 6718 addextpr 6719 caucvgprlemcanl 6742 cauappcvgprlemladdru 6754 cauappcvgprlemladdrl 6755 cauappcvgprlemladd 6756 cauappcvgprlem1 6757 caucvgprlemladdrl 6776 caucvgprlem1 6777 caucvgprprlemnbj 6791 caucvgprprlemopu 6797 caucvgprprlemloc 6801 caucvgprprlemexbt 6804 caucvgprprlemexb 6805 caucvgprprlemaddq 6806 caucvgprprlem2 6808 enrer 6820 addcmpblnr 6824 mulcmpblnrlemg 6825 mulcmpblnr 6826 ltsrprg 6832 1sr 6836 m1r 6837 addclsr 6838 mulclsr 6839 addasssrg 6841 mulasssrg 6843 distrsrg 6844 m1p1sr 6845 m1m1sr 6846 lttrsr 6847 ltsosr 6849 0lt1sr 6850 0idsr 6852 1idsr 6853 00sr 6854 ltasrg 6855 recexgt0sr 6858 mulgt0sr 6862 aptisr 6863 mulextsr1lem 6864 mulextsr1 6865 archsr 6866 srpospr 6867 prsrcl 6868 prsradd 6870 prsrlt 6871 caucvgsrlemcau 6877 caucvgsrlemgt1 6879 pitonnlem1p1 6922 pitonnlem2 6923 pitonn 6924 pitoregt0 6925 pitore 6926 recnnre 6927 recidpirqlemcalc 6933 recidpirq 6934 |
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