rpaddcl - Intuitionistic Logic Explorer

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Theorem rpaddcl 9613

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 9596	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		rpre 9596	. . 3 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ∈ ℝ)
3		readdcl 7879	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
4	1, 2, 3	syl2an 287	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) → (𝐴 + 𝐵) ∈ ℝ)
5		elrp 9591	. . 3 ⊢ (𝐴 ∈ ℝ⁺ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
6		elrp 9591	. . 3 ⊢ (𝐵 ∈ ℝ⁺ ↔ (𝐵 ∈ ℝ ∧ 0 < 𝐵))
7		addgt0 8346	. . . 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 9591	. 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 2136 class class class wbr 3982 (class class class)co 5842 ℝcr 7752 0cc0 7753 + caddc 7756 < clt 7933 ℝ⁺crp 9589

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-addcom 7853 ax-addass 7855 ax-i2m1 7858 ax-0id 7861 ax-rnegex 7862 ax-pre-lttrn 7867 ax-pre-ltadd 7869

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-rab 2453 df-v 2728 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-xp 4610 df-iota 5153 df-fv 5196 df-ov 5845 df-pnf 7935 df-mnf 7936 df-ltxr 7938 df-rp 9590

This theorem is referenced by: rpaddcld 9648 fsumrpcl 11345 isumrpcl 11435 efgt1p2 11636

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