rpaddcl - Intuitionistic Logic Explorer

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Theorem rpaddcl 9585

Description: Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

**Assertion**
Ref	Expression
rpaddcl	⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) → (𝐴 + 𝐵) ∈ ℝ⁺)

**Proof of Theorem rpaddcl**
Step	Hyp	Ref	Expression
1		rpre 9568	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		rpre 9568	. . 3 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ∈ ℝ)
3		readdcl 7859	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
4	1, 2, 3	syl2an 287	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) → (𝐴 + 𝐵) ∈ ℝ)
5		elrp 9563	. . 3 ⊢ (𝐴 ∈ ℝ⁺ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
6		elrp 9563	. . 3 ⊢ (𝐵 ∈ ℝ⁺ ↔ (𝐵 ∈ ℝ ∧ 0 < 𝐵))
7		addgt0 8324	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 < 𝐵)) → 0 < (𝐴 + 𝐵))
8	7	an4s 578	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 + 𝐵))
9	5, 6, 8	syl2anb 289	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) → 0 < (𝐴 + 𝐵))
10		elrp 9563	. 2 ⊢ ((𝐴 + 𝐵) ∈ ℝ⁺ ↔ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵)))
11	4, 9, 10	sylanbrc 414	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) → (𝐴 + 𝐵) ∈ ℝ⁺)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2128 class class class wbr 3966 (class class class)co 5825 ℝcr 7732 0cc0 7733 + caddc 7736 < clt 7913 ℝ⁺crp 9561

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-addcom 7833 ax-addass 7835 ax-i2m1 7838 ax-0id 7841 ax-rnegex 7842 ax-pre-lttrn 7847 ax-pre-ltadd 7849

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-xp 4593 df-iota 5136 df-fv 5179 df-ov 5828 df-pnf 7915 df-mnf 7916 df-ltxr 7918 df-rp 9562

This theorem is referenced by: rpaddcld 9620 fsumrpcl 11305 isumrpcl 11395 efgt1p2 11596

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