rpaddcl - Intuitionistic Logic Explorer

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Theorem rpaddcl 9494

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 9477	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		rpre 9477	. . 3 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ∈ ℝ)
3		readdcl 7770	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
4	1, 2, 3	syl2an 287	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) → (𝐴 + 𝐵) ∈ ℝ)
5		elrp 9472	. . 3 ⊢ (𝐴 ∈ ℝ⁺ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
6		elrp 9472	. . 3 ⊢ (𝐵 ∈ ℝ⁺ ↔ (𝐵 ∈ ℝ ∧ 0 < 𝐵))
7		addgt0 8234	. . . 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 9472	. 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 1481 class class class wbr 3937 (class class class)co 5782 ℝcr 7643 0cc0 7644 + caddc 7647 < clt 7824 ℝ⁺crp 9470

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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-i2m1 7749 ax-0id 7752 ax-rnegex 7753 ax-pre-lttrn 7758 ax-pre-ltadd 7760

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-iota 5096 df-fv 5139 df-ov 5785 df-pnf 7826 df-mnf 7827 df-ltxr 7829 df-rp 9471

This theorem is referenced by: rpaddcld 9529 fsumrpcl 11205 isumrpcl 11295 efgt1p2 11438

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