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Mirrors > Home > MPE Home > Th. List > rpssre | Structured version Visualization version GIF version |
Description: The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.) |
Ref | Expression |
---|---|
rpssre | ⊢ ℝ+ ⊆ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 11715 | . 2 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ) | |
2 | 1 | ssriv 3572 | 1 ⊢ ℝ+ ⊆ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3540 ℝcr 9814 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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-in 3547 df-ss 3554 df-rp 11709 |
This theorem is referenced by: rpred 11748 rpexpcl 12741 sqrlem3 13833 fsumrpcl 14315 o1fsum 14386 divrcnv 14423 fprodrpcl 14525 rprisefaccl 14593 lebnumlem2 22569 bcthlem1 22929 bcthlem5 22933 aalioulem2 23892 efcvx 24007 pilem2 24010 pilem3 24011 dvrelog 24183 relogcn 24184 logcn 24193 advlog 24200 advlogexp 24201 loglesqrt 24299 rlimcnp 24492 rlimcnp3 24494 cxplim 24498 cxp2lim 24503 cxploglim 24504 divsqrtsumo1 24510 amgmlem 24516 logexprlim 24750 chto1ub 24965 chpo1ub 24969 chpo1ubb 24970 vmadivsum 24971 vmadivsumb 24972 rpvmasumlem 24976 dchrmusum2 24983 dchrvmasumlem2 24987 dchrvmasumiflem2 24991 dchrisum0fno1 25000 rpvmasum2 25001 dchrisum0lem1 25005 dchrisum0lem2a 25006 dchrisum0lem2 25007 dchrisum0 25009 dchrmusumlem 25011 rplogsum 25016 dirith2 25017 mudivsum 25019 mulogsumlem 25020 mulogsum 25021 mulog2sumlem2 25024 mulog2sumlem3 25025 log2sumbnd 25033 selberglem1 25034 selberglem2 25035 selberg2lem 25039 selberg2 25040 pntrmax 25053 pntrsumo1 25054 selbergr 25057 pntlem3 25098 pnt2 25102 xrge0iifhom 29311 omssubadd 29689 signsplypnf 29953 signsply0 29954 taupilem2 32345 taupi 32346 ptrecube 32579 heicant 32614 totbndbnd 32758 rpexpmord 36531 seff 37530 ioorrnopnlem 39200 vonioolem1 39571 elbigolo1 42149 amgmwlem 42357 amgmlemALT 42358 |
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