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Mirrors > Home > MPE Home > Th. List > 10re | Structured version Visualization version GIF version |
Description: The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 8-Oct-2022.) |
Ref | Expression |
---|---|
10re | ⊢ ;10 ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dec 12620 | . 2 ⊢ ;10 = (((9 + 1) · 1) + 0) | |
2 | 9re 12253 | . . . . 5 ⊢ 9 ∈ ℝ | |
3 | 1re 11156 | . . . . 5 ⊢ 1 ∈ ℝ | |
4 | 2, 3 | readdcli 11171 | . . . 4 ⊢ (9 + 1) ∈ ℝ |
5 | 4, 3 | remulcli 11172 | . . 3 ⊢ ((9 + 1) · 1) ∈ ℝ |
6 | 0re 11158 | . . 3 ⊢ 0 ∈ ℝ | |
7 | 5, 6 | readdcli 11171 | . 2 ⊢ (((9 + 1) · 1) + 0) ∈ ℝ |
8 | 1, 7 | eqeltri 2834 | 1 ⊢ ;10 ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 (class class class)co 7358 ℝcr 11051 0cc0 11052 1c1 11053 + caddc 11055 · cmul 11057 9c9 12216 ;cdc 12619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-i2m1 11120 ax-1ne0 11121 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-dec 12620 |
This theorem is referenced by: 8lt10 12751 7lt10 12752 6lt10 12753 5lt10 12754 4lt10 12755 3lt10 12756 2lt10 12757 1lt10 12758 0.999... 15767 bpoly4 15943 plendxnocndx 17266 slotsdifdsndx 17276 slotsdifunifndx 17283 slotsdifplendx2 17299 cnfldfunALTOLD 20813 thlleOLD 21106 bposlem4 26638 bposlem5 26639 dp2cl 31739 dp2lt10 31743 dp2lt 31744 dp2ltsuc 31745 dp2ltc 31746 dpfrac1 31751 dplti 31764 dpgti 31765 dpexpp1 31767 hgt750lem 33267 problem2 34257 lcmineqlem23 40511 aks4d1p1p7 40534 bgoldbtbndlem1 46004 tgblthelfgott 46014 tgoldbach 46016 prstclevalOLD 47096 |
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