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Mirrors > Home > MPE Home > Th. List > rpaddcld | Structured version Visualization version GIF version |
Description: Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpred.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
rpaddcld.1 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
Ref | Expression |
---|---|
rpaddcld | ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
2 | rpaddcld.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
3 | rpaddcl 11730 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 + 𝐵) ∈ ℝ+) | |
4 | 1, 2, 3 | syl2anc 691 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 (class class class)co 6549 + caddc 9818 ℝ+crp 11708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-rp 11709 |
This theorem is referenced by: xov1plusxeqvd 12189 sqrlem7 13837 rpcoshcl 14726 isosctrlem2 24349 lgamucov 24564 relgamcl 24588 pntrlog2bndlem2 25067 pntrlog2bndlem3 25068 pntrlog2bndlem4 25069 pntibndlem3 25081 pntlema 25085 pntlemb 25086 padicabv 25119 ubthlem2 27111 2sqmod 28979 iprodgam 30881 faclimlem1 30882 faclimlem3 30884 faclim 30885 iprodfac 30886 heicant 32614 ftc1anclem6 32660 heiborlem6 32785 pell1qrgaplem 36455 pell14qrgapw 36458 wallispilem4 38961 stirlinglem1 38967 stirlinglem5 38971 fourierdlem30 39030 |
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