addclpr - Intuitionistic Logic Explorer

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Theorem addclpr 6727

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.)

**Assertion**
Ref	Expression
addclpr	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +_P 𝐵) ∈ P)

Proof of Theorem addclpr Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑔 ℎ 𝑞 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iplp 6658	. . . 4 ⊢ +_P = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦 +_Q 𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦 +_Q 𝑧))}⟩)
2	1	genpelxp 6701	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +_P 𝐵) ∈ (𝒫 Q × 𝒫 Q))
3		addclnq 6565	. . . 4 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +_Q 𝑧) ∈ Q)
4	1, 3	genpml 6707	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)))
5	1, 3	genpmu 6708	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)))
6	2, 4, 5	jca32 303	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +_P 𝐵) ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)))))
7		ltanqg 6590	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <_Q 𝑦 ↔ (𝑧 +_Q 𝑥) <_Q (𝑧 +_Q 𝑦)))
8		addcomnqg 6571	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 +_Q 𝑦) = (𝑦 +_Q 𝑥))
9		addnqprl 6719	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (1^st ‘𝐵))) ∧ 𝑥 ∈ Q) → (𝑥 <_Q (𝑔 +_Q ℎ) → 𝑥 ∈ (1^st ‘(𝐴 +_P 𝐵))))
10	1, 3, 7, 8, 9	genprndl 6711	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴 +_P 𝐵)))))
11		addnqpru 6720	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (2^nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2^nd ‘𝐵))) ∧ 𝑥 ∈ Q) → ((𝑔 +_Q ℎ) <_Q 𝑥 → 𝑥 ∈ (2^nd ‘(𝐴 +_P 𝐵))))
12	1, 3, 7, 8, 11	genprndu 6712	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘(𝐴 +_P 𝐵)))))
13	10, 12	jca 300	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴 +_P 𝐵)))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘(𝐴 +_P 𝐵))))))
14	1, 3, 7, 8	genpdisj 6713	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴 +_P 𝐵))))
15		addlocpr 6726	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∨ 𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)))))
16	13, 14, 15	3jca 1118	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴 +_P 𝐵)))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘(𝐴 +_P 𝐵))))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴 +_P 𝐵))) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∨ 𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵))))))
17		elnp1st2nd 6666	. 2 ⊢ ((𝐴 +_P 𝐵) ∈ P ↔ (((𝐴 +_P 𝐵) ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴 +_P 𝐵)))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘(𝐴 +_P 𝐵))))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴 +_P 𝐵))) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∨ 𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)))))))
18	6, 16, 17	sylanbrc 408	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +_P 𝐵) ∈ P)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 661 ∧ w3a 919 ∈ wcel 1433 ∀wral 2348 ∃wrex 2349 𝒫 cpw 3382 class class class wbr 3785 × cxp 4361 ‘cfv 4922 (class class class)co 5532 1^st c1st 5785 2^nd c2nd 5786 Qcnq 6470 +_Q cplq 6472 <_Q cltq 6475 Pcnp 6481 +_P cpp 6483

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329

This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-iplp 6658

This theorem is referenced by: addnqprlemfl 6749 addnqprlemfu 6750 addnqpr 6751 addassprg 6769 distrlem1prl 6772 distrlem1pru 6773 distrlem4prl 6774 distrlem4pru 6775 distrprg 6778 ltaddpr 6787 ltexpri 6803 addcanprleml 6804 addcanprlemu 6805 ltaprlem 6808 ltaprg 6809 prplnqu 6810 addextpr 6811 caucvgprlemcanl 6834 cauappcvgprlemladdru 6846 cauappcvgprlemladdrl 6847 cauappcvgprlemladd 6848 cauappcvgprlem1 6849 caucvgprlemladdrl 6868 caucvgprlem1 6869 caucvgprprlemnbj 6883 caucvgprprlemopu 6889 caucvgprprlemloc 6893 caucvgprprlemexbt 6896 caucvgprprlemexb 6897 caucvgprprlemaddq 6898 caucvgprprlem2 6900 enrer 6912 addcmpblnr 6916 mulcmpblnrlemg 6917 mulcmpblnr 6918 ltsrprg 6924 1sr 6928 m1r 6929 addclsr 6930 mulclsr 6931 addasssrg 6933 mulasssrg 6935 distrsrg 6936 m1p1sr 6937 m1m1sr 6938 lttrsr 6939 ltsosr 6941 0lt1sr 6942 0idsr 6944 1idsr 6945 00sr 6946 ltasrg 6947 recexgt0sr 6950 mulgt0sr 6954 aptisr 6955 mulextsr1lem 6956 mulextsr1 6957 archsr 6958 srpospr 6959 prsrcl 6960 prsradd 6962 prsrlt 6963 caucvgsrlemcau 6969 caucvgsrlemgt1 6971 pitonnlem1p1 7014 pitonnlem2 7015 pitonn 7016 pitoregt0 7017 pitore 7018 recnnre 7019 recidpirqlemcalc 7025 recidpirq 7026

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