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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dp2ltc | Structured version Visualization version GIF version |
Description: Comparing two decimal expansions (unequal higher places). (Contributed by Thierry Arnoux, 16-Dec-2021.) |
Ref | Expression |
---|---|
dp2lt.a | ⊢ 𝐴 ∈ ℕ0 |
dp2lt.b | ⊢ 𝐵 ∈ ℝ+ |
dp2ltc.c | ⊢ 𝐶 ∈ ℕ0 |
dp2ltc.d | ⊢ 𝐷 ∈ ℝ+ |
dp2ltc.s | ⊢ 𝐵 < ;10 |
dp2ltc.l | ⊢ 𝐴 < 𝐶 |
Ref | Expression |
---|---|
dp2ltc | ⊢ _𝐴𝐵 < _𝐶𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dp2ltc.s | . . . . . 6 ⊢ 𝐵 < ;10 | |
2 | rpssre 12046 | . . . . . . . 8 ⊢ ℝ+ ⊆ ℝ | |
3 | dp2lt.b | . . . . . . . 8 ⊢ 𝐵 ∈ ℝ+ | |
4 | 2, 3 | sselii 3749 | . . . . . . 7 ⊢ 𝐵 ∈ ℝ |
5 | 10re 11719 | . . . . . . . 8 ⊢ ;10 ∈ ℝ | |
6 | 10pos 11717 | . . . . . . . 8 ⊢ 0 < ;10 | |
7 | elrp 12037 | . . . . . . . 8 ⊢ (;10 ∈ ℝ+ ↔ (;10 ∈ ℝ ∧ 0 < ;10)) | |
8 | 5, 6, 7 | mpbir2an 690 | . . . . . . 7 ⊢ ;10 ∈ ℝ+ |
9 | divlt1lt 12102 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ ;10 ∈ ℝ+) → ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10)) | |
10 | 4, 8, 9 | mp2an 672 | . . . . . 6 ⊢ ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10) |
11 | 1, 10 | mpbir 221 | . . . . 5 ⊢ (𝐵 / ;10) < 1 |
12 | 5, 6 | gt0ne0ii 10766 | . . . . . . 7 ⊢ ;10 ≠ 0 |
13 | 4, 5, 12 | redivcli 10994 | . . . . . 6 ⊢ (𝐵 / ;10) ∈ ℝ |
14 | 1re 10241 | . . . . . 6 ⊢ 1 ∈ ℝ | |
15 | dp2lt.a | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
16 | 15 | nn0rei 11505 | . . . . . 6 ⊢ 𝐴 ∈ ℝ |
17 | ltadd2 10343 | . . . . . 6 ⊢ (((𝐵 / ;10) ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐵 / ;10) < 1 ↔ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1))) | |
18 | 13, 14, 16, 17 | mp3an 1572 | . . . . 5 ⊢ ((𝐵 / ;10) < 1 ↔ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1)) |
19 | 11, 18 | mpbi 220 | . . . 4 ⊢ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1) |
20 | dp2ltc.l | . . . . 5 ⊢ 𝐴 < 𝐶 | |
21 | 15 | nn0zi 11604 | . . . . . 6 ⊢ 𝐴 ∈ ℤ |
22 | dp2ltc.c | . . . . . . 7 ⊢ 𝐶 ∈ ℕ0 | |
23 | 22 | nn0zi 11604 | . . . . . 6 ⊢ 𝐶 ∈ ℤ |
24 | zltp1le 11629 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶)) | |
25 | 21, 23, 24 | mp2an 672 | . . . . 5 ⊢ (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶) |
26 | 20, 25 | mpbi 220 | . . . 4 ⊢ (𝐴 + 1) ≤ 𝐶 |
27 | 16, 13 | readdcli 10255 | . . . . 5 ⊢ (𝐴 + (𝐵 / ;10)) ∈ ℝ |
28 | 16, 14 | readdcli 10255 | . . . . 5 ⊢ (𝐴 + 1) ∈ ℝ |
29 | 22 | nn0rei 11505 | . . . . 5 ⊢ 𝐶 ∈ ℝ |
30 | 27, 28, 29 | ltletri 10367 | . . . 4 ⊢ (((𝐴 + (𝐵 / ;10)) < (𝐴 + 1) ∧ (𝐴 + 1) ≤ 𝐶) → (𝐴 + (𝐵 / ;10)) < 𝐶) |
31 | 19, 26, 30 | mp2an 672 | . . 3 ⊢ (𝐴 + (𝐵 / ;10)) < 𝐶 |
32 | dp2ltc.d | . . . . . 6 ⊢ 𝐷 ∈ ℝ+ | |
33 | 32, 8 | pm3.2i 447 | . . . . 5 ⊢ (𝐷 ∈ ℝ+ ∧ ;10 ∈ ℝ+) |
34 | rpdivcl 12059 | . . . . 5 ⊢ ((𝐷 ∈ ℝ+ ∧ ;10 ∈ ℝ+) → (𝐷 / ;10) ∈ ℝ+) | |
35 | 33, 34 | ax-mp 5 | . . . 4 ⊢ (𝐷 / ;10) ∈ ℝ+ |
36 | ltaddrp 12070 | . . . 4 ⊢ ((𝐶 ∈ ℝ ∧ (𝐷 / ;10) ∈ ℝ+) → 𝐶 < (𝐶 + (𝐷 / ;10))) | |
37 | 29, 35, 36 | mp2an 672 | . . 3 ⊢ 𝐶 < (𝐶 + (𝐷 / ;10)) |
38 | 2, 32 | sselii 3749 | . . . . . 6 ⊢ 𝐷 ∈ ℝ |
39 | 38, 5, 12 | redivcli 10994 | . . . . 5 ⊢ (𝐷 / ;10) ∈ ℝ |
40 | 29, 39 | readdcli 10255 | . . . 4 ⊢ (𝐶 + (𝐷 / ;10)) ∈ ℝ |
41 | 27, 29, 40 | lttri 10365 | . . 3 ⊢ (((𝐴 + (𝐵 / ;10)) < 𝐶 ∧ 𝐶 < (𝐶 + (𝐷 / ;10))) → (𝐴 + (𝐵 / ;10)) < (𝐶 + (𝐷 / ;10))) |
42 | 31, 37, 41 | mp2an 672 | . 2 ⊢ (𝐴 + (𝐵 / ;10)) < (𝐶 + (𝐷 / ;10)) |
43 | df-dp2 29918 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
44 | df-dp2 29918 | . 2 ⊢ _𝐶𝐷 = (𝐶 + (𝐷 / ;10)) | |
45 | 42, 43, 44 | 3brtr4i 4816 | 1 ⊢ _𝐴𝐵 < _𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 382 ∈ wcel 2145 class class class wbr 4786 (class class class)co 6793 ℝcr 10137 0cc0 10138 1c1 10139 + caddc 10141 < clt 10276 ≤ cle 10277 / cdiv 10886 ℕ0cn0 11494 ℤcz 11579 ;cdc 11695 ℝ+crp 12035 _cdp2 29917 |
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 7096 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 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 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-z 11580 df-dec 11696 df-rp 12036 df-dp2 29918 |
This theorem is referenced by: dpltc 29955 |
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