addclpr - Intuitionistic Logic Explorer

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Theorem addclpr 6520

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 6451	. . . 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 6494	. . 3 ⊢ ((A ∈ P ∧ B ∈ P) → (A +_P B) ∈ (𝒫 Q × 𝒫 Q))
3		addclnq 6359	. . . 4 ⊢ ((y ∈ Q ∧ z ∈ Q) → (y +_Q z) ∈ Q)
4	1, 3	genpml 6500	. . 3 ⊢ ((A ∈ P ∧ B ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(A +_P B)))
5	1, 3	genpmu 6501	. . 3 ⊢ ((A ∈ P ∧ B ∈ P) → ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘(A +_P B)))
6	2, 4, 5	jca32 293	. 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 6384	. . . . 5 ⊢ ((x ∈ Q ∧ y ∈ Q ∧ z ∈ Q) → (x <_Q y ↔ (z +_Q x) <_Q (z +_Q y)))
8		addcomnqg 6365	. . . . 5 ⊢ ((x ∈ Q ∧ y ∈ Q) → (x +_Q y) = (y +_Q x))
9		addnqprl 6512	. . . . 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 6504	. . . 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 6513	. . . . 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 6505	. . . 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 290	. . 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 6506	. . 3 ⊢ ((A ∈ P ∧ B ∈ P) → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘(A +_P B)) ∧ 𝑞 ∈ (2^nd ‘(A +_P B))))
15		addlocpr 6519	. . 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 1083	. 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 6459	. 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 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 628 ∧ w3a 884 ∈ wcel 1390 ∀wral 2300 ∃wrex 2301 𝒫 cpw 3351 class class class wbr 3755 × cxp 4286 ‘cfv 4845 (class class class)co 5455 1^st c1st 5707 2^nd c2nd 5708 Qcnq 6264 +_Q cplq 6266 <_Q cltq 6269 Pcnp 6275 +_P cpp 6277

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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254

This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-1o 5940 df-2o 5941 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 df-ltnqqs 6337 df-enq0 6407 df-nq0 6408 df-0nq0 6409 df-plq0 6410 df-mq0 6411 df-inp 6449 df-iplp 6451

This theorem is referenced by: addnqprlemfl 6540 addnqprlemfu 6541 addnqpr 6542 addassprg 6555 distrlem1prl 6558 distrlem1pru 6559 distrlem4prl 6560 distrlem4pru 6561 distrprg 6564 ltaddpr 6571 ltexpri 6587 addcanprleml 6588 addcanprlemu 6589 ltaprlem 6591 ltaprg 6592 addextpr 6593 cauappcvgprlemcan 6616 cauappcvgprlemladdru 6628 cauappcvgprlemladdrl 6629 cauappcvgprlemladd 6630 cauappcvgprlem1 6631 caucvgprlemladdrl 6649 caucvgprlem1 6650 enrer 6663 addcmpblnr 6667 mulcmpblnrlemg 6668 mulcmpblnr 6669 ltsrprg 6675 1sr 6679 m1r 6680 addclsr 6681 mulclsr 6682 addasssrg 6684 mulasssrg 6686 distrsrg 6687 m1p1sr 6688 m1m1sr 6689 lttrsr 6690 ltsosr 6692 0lt1sr 6693 0idsr 6695 1idsr 6696 00sr 6697 ltasrg 6698 recexgt0sr 6701 mulgt0sr 6704 aptisr 6705 mulextsr1lem 6706 mulextsr1 6707 archsr 6708 pitonnlem1p1 6742 pitonnlem2 6743 pitonn 6744

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