addclpr - Intuitionistic Logic Explorer

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Theorem addclpr 7538

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 7469	. . . 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 7512	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +_P 𝐵) ∈ (𝒫 Q × 𝒫 Q))
3		addclnq 7376	. . . 4 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +_Q 𝑧) ∈ Q)
4	1, 3	genpml 7518	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)))
5	1, 3	genpmu 7519	. . 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 7401	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <_Q 𝑦 ↔ (𝑧 +_Q 𝑥) <_Q (𝑧 +_Q 𝑦)))
8		addcomnqg 7382	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 +_Q 𝑦) = (𝑦 +_Q 𝑥))
9		addnqprl 7530	. . . . 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 7522	. . . 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 7531	. . . . 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 7523	. . . 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 7524	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴 +_P 𝐵))))
15		addlocpr 7537	. . 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 1177	. 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 7477	. 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 708 ∧ w3a 978 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 𝒫 cpw 3577 class class class wbr 4005 × cxp 4626 ‘cfv 5218 (class class class)co 5877 1^st c1st 6141 2^nd c2nd 6142 Qcnq 7281 +_Q cplq 7283 <_Q cltq 7286 Pcnp 7292 +_P cpp 7294

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-eprel 4291 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-1o 6419 df-2o 6420 df-oadd 6423 df-omul 6424 df-er 6537 df-ec 6539 df-qs 6543 df-ni 7305 df-pli 7306 df-mi 7307 df-lti 7308 df-plpq 7345 df-mpq 7346 df-enq 7348 df-nqqs 7349 df-plqqs 7350 df-mqqs 7351 df-1nqqs 7352 df-rq 7353 df-ltnqqs 7354 df-enq0 7425 df-nq0 7426 df-0nq0 7427 df-plq0 7428 df-mq0 7429 df-inp 7467 df-iplp 7469

This theorem is referenced by: addnqprlemfl 7560 addnqprlemfu 7561 addnqpr 7562 addassprg 7580 distrlem1prl 7583 distrlem1pru 7584 distrlem4prl 7585 distrlem4pru 7586 distrprg 7589 ltaddpr 7598 ltexpri 7614 addcanprleml 7615 addcanprlemu 7616 ltaprlem 7619 ltaprg 7620 prplnqu 7621 addextpr 7622 caucvgprlemcanl 7645 cauappcvgprlemladdru 7657 cauappcvgprlemladdrl 7658 cauappcvgprlemladd 7659 cauappcvgprlem1 7660 caucvgprlemladdrl 7679 caucvgprlem1 7680 caucvgprprlemnbj 7694 caucvgprprlemopu 7700 caucvgprprlemloc 7704 caucvgprprlemexbt 7707 caucvgprprlemexb 7708 caucvgprprlemaddq 7709 caucvgprprlem2 7711 enrer 7736 addcmpblnr 7740 mulcmpblnrlemg 7741 mulcmpblnr 7742 ltsrprg 7748 1sr 7752 m1r 7753 addclsr 7754 mulclsr 7755 addasssrg 7757 mulasssrg 7759 distrsrg 7760 m1p1sr 7761 m1m1sr 7762 lttrsr 7763 ltsosr 7765 0lt1sr 7766 0idsr 7768 1idsr 7769 00sr 7770 ltasrg 7771 recexgt0sr 7774 mulgt0sr 7779 aptisr 7780 mulextsr1lem 7781 mulextsr1 7782 archsr 7783 srpospr 7784 prsrcl 7785 prsradd 7787 prsrlt 7788 caucvgsrlemcau 7794 caucvgsrlemgt1 7796 mappsrprg 7805 map2psrprg 7806 pitonnlem1p1 7847 pitonnlem2 7848 pitonn 7849 pitoregt0 7850 pitore 7851 recnnre 7852 recidpirqlemcalc 7858 recidpirq 7859

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