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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 12929 | . . . . . . . 8 ⊢ ℝ+ ⊆ ℝ | |
3 | dp2lt.b | . . . . . . . 8 ⊢ 𝐵 ∈ ℝ+ | |
4 | 2, 3 | sselii 3946 | . . . . . . 7 ⊢ 𝐵 ∈ ℝ |
5 | 10re 12644 | . . . . . . . 8 ⊢ ;10 ∈ ℝ | |
6 | 10pos 12642 | . . . . . . . 8 ⊢ 0 < ;10 | |
7 | elrp 12924 | . . . . . . . 8 ⊢ (;10 ∈ ℝ+ ↔ (;10 ∈ ℝ ∧ 0 < ;10)) | |
8 | 5, 6, 7 | mpbir2an 710 | . . . . . . 7 ⊢ ;10 ∈ ℝ+ |
9 | divlt1lt 12991 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ ;10 ∈ ℝ+) → ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10)) | |
10 | 4, 8, 9 | mp2an 691 | . . . . . 6 ⊢ ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10) |
11 | 1, 10 | mpbir 230 | . . . . 5 ⊢ (𝐵 / ;10) < 1 |
12 | 5, 6 | gt0ne0ii 11698 | . . . . . . 7 ⊢ ;10 ≠ 0 |
13 | 4, 5, 12 | redivcli 11929 | . . . . . 6 ⊢ (𝐵 / ;10) ∈ ℝ |
14 | 1re 11162 | . . . . . 6 ⊢ 1 ∈ ℝ | |
15 | dp2lt.a | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
16 | 15 | nn0rei 12431 | . . . . . 6 ⊢ 𝐴 ∈ ℝ |
17 | ltadd2 11266 | . . . . . 6 ⊢ (((𝐵 / ;10) ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐵 / ;10) < 1 ↔ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1))) | |
18 | 13, 14, 16, 17 | mp3an 1462 | . . . . 5 ⊢ ((𝐵 / ;10) < 1 ↔ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1)) |
19 | 11, 18 | mpbi 229 | . . . 4 ⊢ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1) |
20 | dp2ltc.l | . . . . 5 ⊢ 𝐴 < 𝐶 | |
21 | 15 | nn0zi 12535 | . . . . . 6 ⊢ 𝐴 ∈ ℤ |
22 | dp2ltc.c | . . . . . . 7 ⊢ 𝐶 ∈ ℕ0 | |
23 | 22 | nn0zi 12535 | . . . . . 6 ⊢ 𝐶 ∈ ℤ |
24 | zltp1le 12560 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶)) | |
25 | 21, 23, 24 | mp2an 691 | . . . . 5 ⊢ (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶) |
26 | 20, 25 | mpbi 229 | . . . 4 ⊢ (𝐴 + 1) ≤ 𝐶 |
27 | 16, 13 | readdcli 11177 | . . . . 5 ⊢ (𝐴 + (𝐵 / ;10)) ∈ ℝ |
28 | 16, 14 | readdcli 11177 | . . . . 5 ⊢ (𝐴 + 1) ∈ ℝ |
29 | 22 | nn0rei 12431 | . . . . 5 ⊢ 𝐶 ∈ ℝ |
30 | 27, 28, 29 | ltletri 11290 | . . . 4 ⊢ (((𝐴 + (𝐵 / ;10)) < (𝐴 + 1) ∧ (𝐴 + 1) ≤ 𝐶) → (𝐴 + (𝐵 / ;10)) < 𝐶) |
31 | 19, 26, 30 | mp2an 691 | . . 3 ⊢ (𝐴 + (𝐵 / ;10)) < 𝐶 |
32 | dp2ltc.d | . . . . . 6 ⊢ 𝐷 ∈ ℝ+ | |
33 | 32, 8 | pm3.2i 472 | . . . . 5 ⊢ (𝐷 ∈ ℝ+ ∧ ;10 ∈ ℝ+) |
34 | rpdivcl 12947 | . . . . 5 ⊢ ((𝐷 ∈ ℝ+ ∧ ;10 ∈ ℝ+) → (𝐷 / ;10) ∈ ℝ+) | |
35 | 33, 34 | ax-mp 5 | . . . 4 ⊢ (𝐷 / ;10) ∈ ℝ+ |
36 | ltaddrp 12959 | . . . 4 ⊢ ((𝐶 ∈ ℝ ∧ (𝐷 / ;10) ∈ ℝ+) → 𝐶 < (𝐶 + (𝐷 / ;10))) | |
37 | 29, 35, 36 | mp2an 691 | . . 3 ⊢ 𝐶 < (𝐶 + (𝐷 / ;10)) |
38 | 2, 32 | sselii 3946 | . . . . . 6 ⊢ 𝐷 ∈ ℝ |
39 | 38, 5, 12 | redivcli 11929 | . . . . 5 ⊢ (𝐷 / ;10) ∈ ℝ |
40 | 29, 39 | readdcli 11177 | . . . 4 ⊢ (𝐶 + (𝐷 / ;10)) ∈ ℝ |
41 | 27, 29, 40 | lttri 11288 | . . 3 ⊢ (((𝐴 + (𝐵 / ;10)) < 𝐶 ∧ 𝐶 < (𝐶 + (𝐷 / ;10))) → (𝐴 + (𝐵 / ;10)) < (𝐶 + (𝐷 / ;10))) |
42 | 31, 37, 41 | mp2an 691 | . 2 ⊢ (𝐴 + (𝐵 / ;10)) < (𝐶 + (𝐷 / ;10)) |
43 | df-dp2 31770 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
44 | df-dp2 31770 | . 2 ⊢ _𝐶𝐷 = (𝐶 + (𝐷 / ;10)) | |
45 | 42, 43, 44 | 3brtr4i 5140 | 1 ⊢ _𝐴𝐵 < _𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∈ wcel 2107 class class class wbr 5110 (class class class)co 7362 ℝcr 11057 0cc0 11058 1c1 11059 + caddc 11061 < clt 11196 ≤ cle 11197 / cdiv 11819 ℕ0cn0 12420 ℤcz 12506 ;cdc 12625 ℝ+crp 12922 _cdp2 31769 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-rp 12923 df-dp2 31770 |
This theorem is referenced by: dpltc 31805 |
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