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Mathbox for Thierry Arnoux |
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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 13014 | . . . . . 6 ⊢ ℝ+ ⊆ ℝ | |
2 | dp2lt.b | . . . . . 6 ⊢ 𝐵 ∈ ℝ+ | |
3 | 1, 2 | sselii 3977 | . . . . 5 ⊢ 𝐵 ∈ ℝ |
4 | 10re 12727 | . . . . 5 ⊢ ;10 ∈ ℝ | |
5 | 0re 11247 | . . . . . 6 ⊢ 0 ∈ ℝ | |
6 | 10pos 12725 | . . . . . 6 ⊢ 0 < ;10 | |
7 | 5, 6 | gtneii 11357 | . . . . 5 ⊢ ;10 ≠ 0 |
8 | redivcl 11964 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ ;10 ∈ ℝ ∧ ;10 ≠ 0) → (𝐵 / ;10) ∈ ℝ) | |
9 | 3, 4, 7, 8 | mp3an 1458 | . . . 4 ⊢ (𝐵 / ;10) ∈ ℝ |
10 | dp2lt.c | . . . . . 6 ⊢ 𝐶 ∈ ℝ+ | |
11 | 1, 10 | sselii 3977 | . . . . 5 ⊢ 𝐶 ∈ ℝ |
12 | redivcl 11964 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ ;10 ∈ ℝ ∧ ;10 ≠ 0) → (𝐶 / ;10) ∈ ℝ) | |
13 | 11, 4, 7, 12 | mp3an 1458 | . . . 4 ⊢ (𝐶 / ;10) ∈ ℝ |
14 | dp2lt.a | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
15 | 14 | nn0rei 12514 | . . . 4 ⊢ 𝐴 ∈ ℝ |
16 | 9, 13, 15 | 3pm3.2i 1337 | . . 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 12109 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ (;10 ∈ ℝ ∧ 0 < ;10)) → (𝐵 < 𝐶 ↔ (𝐵 / ;10) < (𝐶 / ;10))) | |
20 | 3, 11, 18, 19 | mp3an 1458 | . . . 4 ⊢ (𝐵 < 𝐶 ↔ (𝐵 / ;10) < (𝐶 / ;10)) |
21 | 17, 20 | mpbi 229 | . . 3 ⊢ (𝐵 / ;10) < (𝐶 / ;10) |
22 | axltadd 11318 | . . . 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 691 | . 2 ⊢ (𝐴 + (𝐵 / ;10)) < (𝐴 + (𝐶 / ;10)) |
25 | df-dp2 32608 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
26 | df-dp2 32608 | . 2 ⊢ _𝐴𝐶 = (𝐴 + (𝐶 / ;10)) | |
27 | 24, 25, 26 | 3brtr4i 5178 | 1 ⊢ _𝐴𝐵 < _𝐴𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∧ w3a 1085 ∈ wcel 2099 ≠ wne 2937 class class class wbr 5148 (class class class)co 7420 ℝcr 11138 0cc0 11139 1c1 11140 + caddc 11142 < clt 11279 / cdiv 11902 ℕ0cn0 12503 ;cdc 12708 ℝ+crp 13007 _cdp2 32607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-dec 12709 df-rp 13008 df-dp2 32608 |
This theorem is referenced by: dplt 32640 hgt750lem2 34284 |
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