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| Mirrors > Home > MPE Home > Th. List > 5re | Structured version Visualization version GIF version | ||
| Description: The number 5 is real. (Contributed by NM, 27-May-1999.) |
| Ref | Expression |
|---|---|
| 5re | ⊢ 5 ∈ ℝ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-5 12332 | . 2 ⊢ 5 = (4 + 1) | |
| 2 | 4re 12350 | . . 3 ⊢ 4 ∈ ℝ | |
| 3 | 1re 11261 | . . 3 ⊢ 1 ∈ ℝ | |
| 4 | 2, 3 | readdcli 11276 | . 2 ⊢ (4 + 1) ∈ ℝ |
| 5 | 1, 4 | eqeltri 2837 | 1 ⊢ 5 ∈ ℝ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 (class class class)co 7431 ℝcr 11154 1c1 11156 + caddc 11158 4c4 12323 5c5 12324 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-i2m1 11223 ax-1ne0 11224 ax-rrecex 11227 ax-cnre 11228 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-2 12329 df-3 12330 df-4 12331 df-5 12332 |
| This theorem is referenced by: 6re 12356 6pos 12376 3lt5 12444 2lt5 12445 1lt5 12446 5lt6 12447 4lt6 12448 5lt7 12453 4lt7 12454 5lt8 12460 4lt8 12461 5lt9 12468 4lt9 12469 5lt10 12868 4lt10 12869 5recm6rec 12877 5eluz3 12927 5rp 13041 fz0to5un2tp 13671 ef01bndlem 16220 prm23ge5 16853 prmlem1 17145 vscandxnscandx 17368 slotsdifipndx 17379 slotstnscsi 17404 plendxnscandx 17417 slotsdnscsi 17436 sralemOLD 21176 srascaOLD 21184 zlmlemOLD 21528 ppiublem1 27246 ppiub 27248 bposlem3 27330 bposlem4 27331 bposlem5 27332 bposlem6 27333 bposlem8 27335 bposlem9 27336 lgsdir2lem1 27369 gausslemma2dlem4 27413 2lgslem3 27448 cchhllemOLD 28902 ex-id 30453 ex-sqrt 30473 threehalves 32897 cyc3conja 33177 resvvscaOLD 33364 zlmdsOLD 33962 zlmtsetOLD 33964 hgt750lem2 34667 hgt750leme 34673 problem2 35671 12gcd5e1 42004 lcmineqlem23 42052 3lexlogpow2ineq1 42059 3lexlogpow2ineq2 42060 aks4d1p1p4 42072 aks4d1p1p6 42074 aks4d1p1p7 42075 aks4d1p1p5 42076 stoweidlem13 46028 ceil5half3 47342 31prm 47584 gbegt5 47748 gbowgt5 47749 sbgoldbo 47774 nnsum3primesle9 47781 nnsum4primesodd 47783 evengpop3 47785 usgrexmpl1lem 47980 usgrexmpl2lem 47985 usgrexmpl2nb4 47994 usgrexmpl2nb5 47995 gpg5nbgrvtx13starlem2 48028 gpg5nbgr3star 48037 |
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