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 12397 | . . . . . . . 8 ⊢ ℝ+ ⊆ ℝ | |
3 | dp2lt.b | . . . . . . . 8 ⊢ 𝐵 ∈ ℝ+ | |
4 | 2, 3 | sselii 3964 | . . . . . . 7 ⊢ 𝐵 ∈ ℝ |
5 | 10re 12118 | . . . . . . . 8 ⊢ ;10 ∈ ℝ | |
6 | 10pos 12116 | . . . . . . . 8 ⊢ 0 < ;10 | |
7 | elrp 12392 | . . . . . . . 8 ⊢ (;10 ∈ ℝ+ ↔ (;10 ∈ ℝ ∧ 0 < ;10)) | |
8 | 5, 6, 7 | mpbir2an 709 | . . . . . . 7 ⊢ ;10 ∈ ℝ+ |
9 | divlt1lt 12459 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ ;10 ∈ ℝ+) → ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10)) | |
10 | 4, 8, 9 | mp2an 690 | . . . . . 6 ⊢ ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10) |
11 | 1, 10 | mpbir 233 | . . . . 5 ⊢ (𝐵 / ;10) < 1 |
12 | 5, 6 | gt0ne0ii 11176 | . . . . . . 7 ⊢ ;10 ≠ 0 |
13 | 4, 5, 12 | redivcli 11407 | . . . . . 6 ⊢ (𝐵 / ;10) ∈ ℝ |
14 | 1re 10641 | . . . . . 6 ⊢ 1 ∈ ℝ | |
15 | dp2lt.a | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
16 | 15 | nn0rei 11909 | . . . . . 6 ⊢ 𝐴 ∈ ℝ |
17 | ltadd2 10744 | . . . . . 6 ⊢ (((𝐵 / ;10) ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐵 / ;10) < 1 ↔ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1))) | |
18 | 13, 14, 16, 17 | mp3an 1457 | . . . . 5 ⊢ ((𝐵 / ;10) < 1 ↔ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1)) |
19 | 11, 18 | mpbi 232 | . . . 4 ⊢ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1) |
20 | dp2ltc.l | . . . . 5 ⊢ 𝐴 < 𝐶 | |
21 | 15 | nn0zi 12008 | . . . . . 6 ⊢ 𝐴 ∈ ℤ |
22 | dp2ltc.c | . . . . . . 7 ⊢ 𝐶 ∈ ℕ0 | |
23 | 22 | nn0zi 12008 | . . . . . 6 ⊢ 𝐶 ∈ ℤ |
24 | zltp1le 12033 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶)) | |
25 | 21, 23, 24 | mp2an 690 | . . . . 5 ⊢ (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶) |
26 | 20, 25 | mpbi 232 | . . . 4 ⊢ (𝐴 + 1) ≤ 𝐶 |
27 | 16, 13 | readdcli 10656 | . . . . 5 ⊢ (𝐴 + (𝐵 / ;10)) ∈ ℝ |
28 | 16, 14 | readdcli 10656 | . . . . 5 ⊢ (𝐴 + 1) ∈ ℝ |
29 | 22 | nn0rei 11909 | . . . . 5 ⊢ 𝐶 ∈ ℝ |
30 | 27, 28, 29 | ltletri 10768 | . . . 4 ⊢ (((𝐴 + (𝐵 / ;10)) < (𝐴 + 1) ∧ (𝐴 + 1) ≤ 𝐶) → (𝐴 + (𝐵 / ;10)) < 𝐶) |
31 | 19, 26, 30 | mp2an 690 | . . 3 ⊢ (𝐴 + (𝐵 / ;10)) < 𝐶 |
32 | dp2ltc.d | . . . . . 6 ⊢ 𝐷 ∈ ℝ+ | |
33 | 32, 8 | pm3.2i 473 | . . . . 5 ⊢ (𝐷 ∈ ℝ+ ∧ ;10 ∈ ℝ+) |
34 | rpdivcl 12415 | . . . . 5 ⊢ ((𝐷 ∈ ℝ+ ∧ ;10 ∈ ℝ+) → (𝐷 / ;10) ∈ ℝ+) | |
35 | 33, 34 | ax-mp 5 | . . . 4 ⊢ (𝐷 / ;10) ∈ ℝ+ |
36 | ltaddrp 12427 | . . . 4 ⊢ ((𝐶 ∈ ℝ ∧ (𝐷 / ;10) ∈ ℝ+) → 𝐶 < (𝐶 + (𝐷 / ;10))) | |
37 | 29, 35, 36 | mp2an 690 | . . 3 ⊢ 𝐶 < (𝐶 + (𝐷 / ;10)) |
38 | 2, 32 | sselii 3964 | . . . . . 6 ⊢ 𝐷 ∈ ℝ |
39 | 38, 5, 12 | redivcli 11407 | . . . . 5 ⊢ (𝐷 / ;10) ∈ ℝ |
40 | 29, 39 | readdcli 10656 | . . . 4 ⊢ (𝐶 + (𝐷 / ;10)) ∈ ℝ |
41 | 27, 29, 40 | lttri 10766 | . . 3 ⊢ (((𝐴 + (𝐵 / ;10)) < 𝐶 ∧ 𝐶 < (𝐶 + (𝐷 / ;10))) → (𝐴 + (𝐵 / ;10)) < (𝐶 + (𝐷 / ;10))) |
42 | 31, 37, 41 | mp2an 690 | . 2 ⊢ (𝐴 + (𝐵 / ;10)) < (𝐶 + (𝐷 / ;10)) |
43 | df-dp2 30548 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
44 | df-dp2 30548 | . 2 ⊢ _𝐶𝐷 = (𝐶 + (𝐷 / ;10)) | |
45 | 42, 43, 44 | 3brtr4i 5096 | 1 ⊢ _𝐴𝐵 < _𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 < clt 10675 ≤ cle 10676 / cdiv 11297 ℕ0cn0 11898 ℤcz 11982 ;cdc 12099 ℝ+crp 12390 _cdp2 30547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-rp 12391 df-dp2 30548 |
This theorem is referenced by: dpltc 30583 |
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