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Mirrors > Home > MPE Home > Th. List > 8re | Structured version Visualization version GIF version |
Description: The number 8 is real. (Contributed by NM, 27-May-1999.) |
Ref | Expression |
---|---|
8re | ⊢ 8 ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-8 11694 | . 2 ⊢ 8 = (7 + 1) | |
2 | 7re 11718 | . . 3 ⊢ 7 ∈ ℝ | |
3 | 1re 10629 | . . 3 ⊢ 1 ∈ ℝ | |
4 | 2, 3 | readdcli 10644 | . 2 ⊢ (7 + 1) ∈ ℝ |
5 | 1, 4 | eqeltri 2906 | 1 ⊢ 8 ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 (class class class)co 7145 ℝcr 10524 1c1 10526 + caddc 10528 7c7 11685 8c8 11686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-i2m1 10593 ax-1ne0 10594 ax-rrecex 10597 ax-cnre 10598 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 |
This theorem is referenced by: 9re 11724 9pos 11738 6lt8 11818 5lt8 11819 4lt8 11820 3lt8 11821 2lt8 11822 1lt8 11823 8lt9 11824 7lt9 11825 8th4div3 11845 8lt10 12218 7lt10 12219 ef01bndlem 15525 cos2bnd 15529 sralem 19878 chtub 25715 bposlem8 25794 bposlem9 25795 lgsdir2lem1 25828 lgsdir2lem4 25831 lgsdir2lem5 25832 2lgsoddprmlem1 25911 2lgsoddprmlem2 25912 chebbnd1lem2 25973 chebbnd1lem3 25974 chebbnd1 25975 pntlemf 26108 cchhllem 26600 hgt750lem 31821 hgt750lem2 31822 hgt750leme 31828 fmtnoprmfac2lem1 43605 mod42tp1mod8 43644 nnsum3primesle9 43836 nnsum4primesoddALTV 43839 nnsum4primesevenALTV 43843 bgoldbtbndlem1 43847 tgoldbach 43859 |
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