addclpr - Intuitionistic Logic Explorer

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Theorem addclpr 7623

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 7554	. . . 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 7597	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +_P 𝐵) ∈ (𝒫 Q × 𝒫 Q))
3		addclnq 7461	. . . 4 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +_Q 𝑧) ∈ Q)
4	1, 3	genpml 7603	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)))
5	1, 3	genpmu 7604	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘(𝐴 +_P 𝐵)))
6	2, 4, 5	jca32 310	. 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 7486	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <_Q 𝑦 ↔ (𝑧 +_Q 𝑥) <_Q (𝑧 +_Q 𝑦)))
8		addcomnqg 7467	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 +_Q 𝑦) = (𝑦 +_Q 𝑥))
9		addnqprl 7615	. . . . 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 7607	. . . 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 7616	. . . . 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 7608	. . . 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 306	. . 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 7609	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴 +_P 𝐵))))
15		addlocpr 7622	. . 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 1179	. 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 7562	. 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 417	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +_P 𝐵) ∈ P)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 ∧ w3a 980 ∈ wcel 2167 ∀wral 2475 ∃wrex 2476 𝒫 cpw 3606 class class class wbr 4034 × cxp 4662 ‘cfv 5259 (class class class)co 5925 1^st c1st 6205 2^nd c2nd 6206 Qcnq 7366 +_Q cplq 7368 <_Q cltq 7371 Pcnp 7377 +_P cpp 7379

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-eprel 4325 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-irdg 6437 df-1o 6483 df-2o 6484 df-oadd 6487 df-omul 6488 df-er 6601 df-ec 6603 df-qs 6607 df-ni 7390 df-pli 7391 df-mi 7392 df-lti 7393 df-plpq 7430 df-mpq 7431 df-enq 7433 df-nqqs 7434 df-plqqs 7435 df-mqqs 7436 df-1nqqs 7437 df-rq 7438 df-ltnqqs 7439 df-enq0 7510 df-nq0 7511 df-0nq0 7512 df-plq0 7513 df-mq0 7514 df-inp 7552 df-iplp 7554

This theorem is referenced by: addnqprlemfl 7645 addnqprlemfu 7646 addnqpr 7647 addassprg 7665 distrlem1prl 7668 distrlem1pru 7669 distrlem4prl 7670 distrlem4pru 7671 distrprg 7674 ltaddpr 7683 ltexpri 7699 addcanprleml 7700 addcanprlemu 7701 ltaprlem 7704 ltaprg 7705 prplnqu 7706 addextpr 7707 caucvgprlemcanl 7730 cauappcvgprlemladdru 7742 cauappcvgprlemladdrl 7743 cauappcvgprlemladd 7744 cauappcvgprlem1 7745 caucvgprlemladdrl 7764 caucvgprlem1 7765 caucvgprprlemnbj 7779 caucvgprprlemopu 7785 caucvgprprlemloc 7789 caucvgprprlemexbt 7792 caucvgprprlemexb 7793 caucvgprprlemaddq 7794 caucvgprprlem2 7796 enrer 7821 addcmpblnr 7825 mulcmpblnrlemg 7826 mulcmpblnr 7827 ltsrprg 7833 1sr 7837 m1r 7838 addclsr 7839 mulclsr 7840 addasssrg 7842 mulasssrg 7844 distrsrg 7845 m1p1sr 7846 m1m1sr 7847 lttrsr 7848 ltsosr 7850 0lt1sr 7851 0idsr 7853 1idsr 7854 00sr 7855 ltasrg 7856 recexgt0sr 7859 mulgt0sr 7864 aptisr 7865 mulextsr1lem 7866 mulextsr1 7867 archsr 7868 srpospr 7869 prsrcl 7870 prsradd 7872 prsrlt 7873 caucvgsrlemcau 7879 caucvgsrlemgt1 7881 mappsrprg 7890 map2psrprg 7891 pitonnlem1p1 7932 pitonnlem2 7933 pitonn 7934 pitoregt0 7935 pitore 7936 recnnre 7937 recidpirqlemcalc 7943 recidpirq 7944

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