addclpr - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > addclpr		GIF version

Theorem addclpr 7800

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 7731	. . . 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 7774	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +_P 𝐵) ∈ (𝒫 Q × 𝒫 Q))
3		addclnq 7638	. . . 4 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +_Q 𝑧) ∈ Q)
4	1, 3	genpml 7780	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)))
5	1, 3	genpmu 7781	. . 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 7663	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <_Q 𝑦 ↔ (𝑧 +_Q 𝑥) <_Q (𝑧 +_Q 𝑦)))
8		addcomnqg 7644	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 +_Q 𝑦) = (𝑦 +_Q 𝑥))
9		addnqprl 7792	. . . . 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 7784	. . . 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 7793	. . . . 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 7785	. . . 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 7786	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴 +_P 𝐵))))
15		addlocpr 7799	. . 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 1204	. 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 7739	. 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 716 ∧ w3a 1005 ∈ wcel 2202 ∀wral 2511 ∃wrex 2512 𝒫 cpw 3656 class class class wbr 4093 × cxp 4729 ‘cfv 5333 (class class class)co 6028 1^st c1st 6310 2^nd c2nd 6311 Qcnq 7543 +_Q cplq 7545 <_Q cltq 7548 Pcnp 7554 +_P cpp 7556

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-eprel 4392 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-1o 6625 df-2o 6626 df-oadd 6629 df-omul 6630 df-er 6745 df-ec 6747 df-qs 6751 df-ni 7567 df-pli 7568 df-mi 7569 df-lti 7570 df-plpq 7607 df-mpq 7608 df-enq 7610 df-nqqs 7611 df-plqqs 7612 df-mqqs 7613 df-1nqqs 7614 df-rq 7615 df-ltnqqs 7616 df-enq0 7687 df-nq0 7688 df-0nq0 7689 df-plq0 7690 df-mq0 7691 df-inp 7729 df-iplp 7731

This theorem is referenced by: addnqprlemfl 7822 addnqprlemfu 7823 addnqpr 7824 addassprg 7842 distrlem1prl 7845 distrlem1pru 7846 distrlem4prl 7847 distrlem4pru 7848 distrprg 7851 ltaddpr 7860 ltexpri 7876 addcanprleml 7877 addcanprlemu 7878 ltaprlem 7881 ltaprg 7882 prplnqu 7883 addextpr 7884 caucvgprlemcanl 7907 cauappcvgprlemladdru 7919 cauappcvgprlemladdrl 7920 cauappcvgprlemladd 7921 cauappcvgprlem1 7922 caucvgprlemladdrl 7941 caucvgprlem1 7942 caucvgprprlemnbj 7956 caucvgprprlemopu 7962 caucvgprprlemloc 7966 caucvgprprlemexbt 7969 caucvgprprlemexb 7970 caucvgprprlemaddq 7971 caucvgprprlem2 7973 enrer 7998 addcmpblnr 8002 mulcmpblnrlemg 8003 mulcmpblnr 8004 ltsrprg 8010 1sr 8014 m1r 8015 addclsr 8016 mulclsr 8017 addasssrg 8019 mulasssrg 8021 distrsrg 8022 m1p1sr 8023 m1m1sr 8024 lttrsr 8025 ltsosr 8027 0lt1sr 8028 0idsr 8030 1idsr 8031 00sr 8032 ltasrg 8033 recexgt0sr 8036 mulgt0sr 8041 aptisr 8042 mulextsr1lem 8043 mulextsr1 8044 archsr 8045 srpospr 8046 prsrcl 8047 prsradd 8049 prsrlt 8050 caucvgsrlemcau 8056 caucvgsrlemgt1 8058 mappsrprg 8067 map2psrprg 8068 pitonnlem1p1 8109 pitonnlem2 8110 pitonn 8111 pitoregt0 8112 pitore 8113 recnnre 8114 recidpirqlemcalc 8120 recidpirq 8121

W3C validator