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 12390 | . . . . . 6 ⊢ ℝ+ ⊆ ℝ | |
2 | dp2lt.b | . . . . . 6 ⊢ 𝐵 ∈ ℝ+ | |
3 | 1, 2 | sselii 3963 | . . . . 5 ⊢ 𝐵 ∈ ℝ |
4 | 10re 12111 | . . . . 5 ⊢ ;10 ∈ ℝ | |
5 | 0re 10637 | . . . . . 6 ⊢ 0 ∈ ℝ | |
6 | 10pos 12109 | . . . . . 6 ⊢ 0 < ;10 | |
7 | 5, 6 | gtneii 10746 | . . . . 5 ⊢ ;10 ≠ 0 |
8 | redivcl 11353 | . . . . 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 3963 | . . . . 5 ⊢ 𝐶 ∈ ℝ |
12 | redivcl 11353 | . . . . 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 11902 | . . . 4 ⊢ 𝐴 ∈ ℝ |
16 | 9, 13, 15 | 3pm3.2i 1335 | . . 3 ⊢ ((𝐵 / ;10) ∈ ℝ ∧ (𝐶 / ;10) ∈ ℝ ∧ 𝐴 ∈ ℝ) |
17 | dp2lt.l | . . . 4 ⊢ 𝐵 < 𝐶 | |
18 | 4, 6 | pm3.2i 473 | . . . . 5 ⊢ (;10 ∈ ℝ ∧ 0 < ;10) |
19 | ltdiv1 11498 | . . . . 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 232 | . . 3 ⊢ (𝐵 / ;10) < (𝐶 / ;10) |
22 | axltadd 10708 | . . . 4 ⊢ (((𝐵 / ;10) ∈ ℝ ∧ (𝐶 / ;10) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐵 / ;10) < (𝐶 / ;10) → (𝐴 + (𝐵 / ;10)) < (𝐴 + (𝐶 / ;10)))) | |
23 | 22 | imp 409 | . . 3 ⊢ ((((𝐵 / ;10) ∈ ℝ ∧ (𝐶 / ;10) ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (𝐵 / ;10) < (𝐶 / ;10)) → (𝐴 + (𝐵 / ;10)) < (𝐴 + (𝐶 / ;10))) |
24 | 16, 21, 23 | mp2an 690 | . 2 ⊢ (𝐴 + (𝐵 / ;10)) < (𝐴 + (𝐶 / ;10)) |
25 | df-dp2 30543 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
26 | df-dp2 30543 | . 2 ⊢ _𝐴𝐶 = (𝐴 + (𝐶 / ;10)) | |
27 | 24, 25, 26 | 3brtr4i 5088 | 1 ⊢ _𝐴𝐵 < _𝐴𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5058 (class class class)co 7150 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 < clt 10669 / cdiv 11291 ℕ0cn0 11891 ;cdc 12092 ℝ+crp 12383 _cdp2 30542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-dec 12093 df-rp 12384 df-dp2 30543 |
This theorem is referenced by: dplt 30575 hgt750lem2 31918 |
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