![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dp2lt | Structured version Visualization version GIF version |
Description: Comparing two decimal fractions (equal unit places). (Contributed by Thierry Arnoux, 16-Dec-2021.) |
Ref | Expression |
---|---|
dp2lt.a | ⊢ 𝐴 ∈ ℕ0 |
dp2lt.b | ⊢ 𝐵 ∈ ℝ+ |
dp2lt.c | ⊢ 𝐶 ∈ ℝ+ |
dp2lt.l | ⊢ 𝐵 < 𝐶 |
Ref | Expression |
---|---|
dp2lt | ⊢ _𝐴𝐵 < _𝐴𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpssre 12982 | . . . . . 6 ⊢ ℝ+ ⊆ ℝ | |
2 | dp2lt.b | . . . . . 6 ⊢ 𝐵 ∈ ℝ+ | |
3 | 1, 2 | sselii 3972 | . . . . 5 ⊢ 𝐵 ∈ ℝ |
4 | 10re 12695 | . . . . 5 ⊢ ;10 ∈ ℝ | |
5 | 0re 11215 | . . . . . 6 ⊢ 0 ∈ ℝ | |
6 | 10pos 12693 | . . . . . 6 ⊢ 0 < ;10 | |
7 | 5, 6 | gtneii 11325 | . . . . 5 ⊢ ;10 ≠ 0 |
8 | redivcl 11932 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ ;10 ∈ ℝ ∧ ;10 ≠ 0) → (𝐵 / ;10) ∈ ℝ) | |
9 | 3, 4, 7, 8 | mp3an 1457 | . . . 4 ⊢ (𝐵 / ;10) ∈ ℝ |
10 | dp2lt.c | . . . . . 6 ⊢ 𝐶 ∈ ℝ+ | |
11 | 1, 10 | sselii 3972 | . . . . 5 ⊢ 𝐶 ∈ ℝ |
12 | redivcl 11932 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ ;10 ∈ ℝ ∧ ;10 ≠ 0) → (𝐶 / ;10) ∈ ℝ) | |
13 | 11, 4, 7, 12 | mp3an 1457 | . . . 4 ⊢ (𝐶 / ;10) ∈ ℝ |
14 | dp2lt.a | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
15 | 14 | nn0rei 12482 | . . . 4 ⊢ 𝐴 ∈ ℝ |
16 | 9, 13, 15 | 3pm3.2i 1336 | . . 3 ⊢ ((𝐵 / ;10) ∈ ℝ ∧ (𝐶 / ;10) ∈ ℝ ∧ 𝐴 ∈ ℝ) |
17 | dp2lt.l | . . . 4 ⊢ 𝐵 < 𝐶 | |
18 | 4, 6 | pm3.2i 470 | . . . . 5 ⊢ (;10 ∈ ℝ ∧ 0 < ;10) |
19 | ltdiv1 12077 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ (;10 ∈ ℝ ∧ 0 < ;10)) → (𝐵 < 𝐶 ↔ (𝐵 / ;10) < (𝐶 / ;10))) | |
20 | 3, 11, 18, 19 | mp3an 1457 | . . . 4 ⊢ (𝐵 < 𝐶 ↔ (𝐵 / ;10) < (𝐶 / ;10)) |
21 | 17, 20 | mpbi 229 | . . 3 ⊢ (𝐵 / ;10) < (𝐶 / ;10) |
22 | axltadd 11286 | . . . 4 ⊢ (((𝐵 / ;10) ∈ ℝ ∧ (𝐶 / ;10) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐵 / ;10) < (𝐶 / ;10) → (𝐴 + (𝐵 / ;10)) < (𝐴 + (𝐶 / ;10)))) | |
23 | 22 | imp 406 | . . 3 ⊢ ((((𝐵 / ;10) ∈ ℝ ∧ (𝐶 / ;10) ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (𝐵 / ;10) < (𝐶 / ;10)) → (𝐴 + (𝐵 / ;10)) < (𝐴 + (𝐶 / ;10))) |
24 | 16, 21, 23 | mp2an 689 | . 2 ⊢ (𝐴 + (𝐵 / ;10)) < (𝐴 + (𝐶 / ;10)) |
25 | df-dp2 32531 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
26 | df-dp2 32531 | . 2 ⊢ _𝐴𝐶 = (𝐴 + (𝐶 / ;10)) | |
27 | 24, 25, 26 | 3brtr4i 5169 | 1 ⊢ _𝐴𝐵 < _𝐴𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∧ w3a 1084 ∈ wcel 2098 ≠ wne 2932 class class class wbr 5139 (class class class)co 7402 ℝcr 11106 0cc0 11107 1c1 11108 + caddc 11110 < clt 11247 / cdiv 11870 ℕ0cn0 12471 ;cdc 12676 ℝ+crp 12975 _cdp2 32530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-dec 12677 df-rp 12976 df-dp2 32531 |
This theorem is referenced by: dplt 32563 hgt750lem2 34183 |
Copyright terms: Public domain | W3C validator |