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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dp2lt10 | Structured version Visualization version GIF version |
Description: Decimal fraction builds real numbers less than 10. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
Ref | Expression |
---|---|
dp2lt10.a | ⊢ 𝐴 ∈ ℕ0 |
dp2lt10.b | ⊢ 𝐵 ∈ ℝ+ |
dp2lt10.1 | ⊢ 𝐴 < ;10 |
dp2lt10.2 | ⊢ 𝐵 < ;10 |
Ref | Expression |
---|---|
dp2lt10 | ⊢ _𝐴𝐵 < ;10 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dp2 29915 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
2 | dp2lt10.1 | . . . . . 6 ⊢ 𝐴 < ;10 | |
3 | 9p1e10 11697 | . . . . . 6 ⊢ (9 + 1) = ;10 | |
4 | 2, 3 | breqtrri 4813 | . . . . 5 ⊢ 𝐴 < (9 + 1) |
5 | dp2lt10.a | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
6 | 5 | nn0zi 11603 | . . . . . 6 ⊢ 𝐴 ∈ ℤ |
7 | 9nn0 11517 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
8 | 7 | nn0zi 11603 | . . . . . 6 ⊢ 9 ∈ ℤ |
9 | zleltp1 11629 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 9 ∈ ℤ) → (𝐴 ≤ 9 ↔ 𝐴 < (9 + 1))) | |
10 | 6, 8, 9 | mp2an 664 | . . . . 5 ⊢ (𝐴 ≤ 9 ↔ 𝐴 < (9 + 1)) |
11 | 4, 10 | mpbir 221 | . . . 4 ⊢ 𝐴 ≤ 9 |
12 | dp2lt10.2 | . . . . 5 ⊢ 𝐵 < ;10 | |
13 | rpssre 12045 | . . . . . . 7 ⊢ ℝ+ ⊆ ℝ | |
14 | dp2lt10.b | . . . . . . 7 ⊢ 𝐵 ∈ ℝ+ | |
15 | 13, 14 | sselii 3749 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
16 | 10re 11718 | . . . . . . 7 ⊢ ;10 ∈ ℝ | |
17 | 10pos 11716 | . . . . . . 7 ⊢ 0 < ;10 | |
18 | 16, 17 | elrpii 12037 | . . . . . 6 ⊢ ;10 ∈ ℝ+ |
19 | divlt1lt 12101 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ ;10 ∈ ℝ+) → ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10)) | |
20 | 15, 18, 19 | mp2an 664 | . . . . 5 ⊢ ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10) |
21 | 12, 20 | mpbir 221 | . . . 4 ⊢ (𝐵 / ;10) < 1 |
22 | 5 | nn0rei 11504 | . . . . . 6 ⊢ 𝐴 ∈ ℝ |
23 | 0re 10241 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
24 | 23, 17 | gtneii 10350 | . . . . . . 7 ⊢ ;10 ≠ 0 |
25 | 15, 16, 24 | redivcli 10993 | . . . . . 6 ⊢ (𝐵 / ;10) ∈ ℝ |
26 | 22, 25 | pm3.2i 447 | . . . . 5 ⊢ (𝐴 ∈ ℝ ∧ (𝐵 / ;10) ∈ ℝ) |
27 | 9re 11308 | . . . . . 6 ⊢ 9 ∈ ℝ | |
28 | 1re 10240 | . . . . . 6 ⊢ 1 ∈ ℝ | |
29 | 27, 28 | pm3.2i 447 | . . . . 5 ⊢ (9 ∈ ℝ ∧ 1 ∈ ℝ) |
30 | leltadd 10713 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (𝐵 / ;10) ∈ ℝ) ∧ (9 ∈ ℝ ∧ 1 ∈ ℝ)) → ((𝐴 ≤ 9 ∧ (𝐵 / ;10) < 1) → (𝐴 + (𝐵 / ;10)) < (9 + 1))) | |
31 | 26, 29, 30 | mp2an 664 | . . . 4 ⊢ ((𝐴 ≤ 9 ∧ (𝐵 / ;10) < 1) → (𝐴 + (𝐵 / ;10)) < (9 + 1)) |
32 | 11, 21, 31 | mp2an 664 | . . 3 ⊢ (𝐴 + (𝐵 / ;10)) < (9 + 1) |
33 | 32, 3 | breqtri 4811 | . 2 ⊢ (𝐴 + (𝐵 / ;10)) < ;10 |
34 | 1, 33 | eqbrtri 4807 | 1 ⊢ _𝐴𝐵 < ;10 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∈ wcel 2145 class class class wbr 4786 (class class class)co 6792 ℝcr 10136 0cc0 10137 1c1 10138 + caddc 10140 < clt 10275 ≤ cle 10276 / cdiv 10885 9c9 11278 ℕ0cn0 11493 ℤcz 11578 ;cdc 11694 ℝ+crp 12034 _cdp2 29914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-div 10886 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-7 11285 df-8 11286 df-9 11287 df-n0 11494 df-z 11579 df-dec 11695 df-rp 12035 df-dp2 29915 |
This theorem is referenced by: hgt750lem 31066 hgt750lem2 31067 |
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