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Mirrors > Home > MPE Home > Th. List > numltc | Structured version Visualization version GIF version |
Description: Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
numlt.1 | ⊢ 𝑇 ∈ ℕ |
numlt.2 | ⊢ 𝐴 ∈ ℕ0 |
numlt.3 | ⊢ 𝐵 ∈ ℕ0 |
numltc.3 | ⊢ 𝐶 ∈ ℕ0 |
numltc.4 | ⊢ 𝐷 ∈ ℕ0 |
numltc.5 | ⊢ 𝐶 < 𝑇 |
numltc.6 | ⊢ 𝐴 < 𝐵 |
Ref | Expression |
---|---|
numltc | ⊢ ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐵) + 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numlt.1 | . . . . 5 ⊢ 𝑇 ∈ ℕ | |
2 | numlt.2 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
3 | numltc.3 | . . . . 5 ⊢ 𝐶 ∈ ℕ0 | |
4 | numltc.5 | . . . . 5 ⊢ 𝐶 < 𝑇 | |
5 | 1, 2, 3, 1, 4 | numlt 12318 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐴) + 𝑇) |
6 | 1 | nnrei 11839 | . . . . . . 7 ⊢ 𝑇 ∈ ℝ |
7 | 6 | recni 10847 | . . . . . 6 ⊢ 𝑇 ∈ ℂ |
8 | 2 | nn0rei 12101 | . . . . . . 7 ⊢ 𝐴 ∈ ℝ |
9 | 8 | recni 10847 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
10 | ax-1cn 10787 | . . . . . 6 ⊢ 1 ∈ ℂ | |
11 | 7, 9, 10 | adddii 10845 | . . . . 5 ⊢ (𝑇 · (𝐴 + 1)) = ((𝑇 · 𝐴) + (𝑇 · 1)) |
12 | 7 | mulid1i 10837 | . . . . . 6 ⊢ (𝑇 · 1) = 𝑇 |
13 | 12 | oveq2i 7224 | . . . . 5 ⊢ ((𝑇 · 𝐴) + (𝑇 · 1)) = ((𝑇 · 𝐴) + 𝑇) |
14 | 11, 13 | eqtri 2765 | . . . 4 ⊢ (𝑇 · (𝐴 + 1)) = ((𝑇 · 𝐴) + 𝑇) |
15 | 5, 14 | breqtrri 5080 | . . 3 ⊢ ((𝑇 · 𝐴) + 𝐶) < (𝑇 · (𝐴 + 1)) |
16 | numltc.6 | . . . . 5 ⊢ 𝐴 < 𝐵 | |
17 | numlt.3 | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
18 | nn0ltp1le 12235 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵)) | |
19 | 2, 17, 18 | mp2an 692 | . . . . 5 ⊢ (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵) |
20 | 16, 19 | mpbi 233 | . . . 4 ⊢ (𝐴 + 1) ≤ 𝐵 |
21 | 1 | nngt0i 11869 | . . . . 5 ⊢ 0 < 𝑇 |
22 | peano2re 11005 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ) | |
23 | 8, 22 | ax-mp 5 | . . . . . 6 ⊢ (𝐴 + 1) ∈ ℝ |
24 | 17 | nn0rei 12101 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
25 | 23, 24, 6 | lemul2i 11755 | . . . . 5 ⊢ (0 < 𝑇 → ((𝐴 + 1) ≤ 𝐵 ↔ (𝑇 · (𝐴 + 1)) ≤ (𝑇 · 𝐵))) |
26 | 21, 25 | ax-mp 5 | . . . 4 ⊢ ((𝐴 + 1) ≤ 𝐵 ↔ (𝑇 · (𝐴 + 1)) ≤ (𝑇 · 𝐵)) |
27 | 20, 26 | mpbi 233 | . . 3 ⊢ (𝑇 · (𝐴 + 1)) ≤ (𝑇 · 𝐵) |
28 | 6, 8 | remulcli 10849 | . . . . 5 ⊢ (𝑇 · 𝐴) ∈ ℝ |
29 | 3 | nn0rei 12101 | . . . . 5 ⊢ 𝐶 ∈ ℝ |
30 | 28, 29 | readdcli 10848 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐶) ∈ ℝ |
31 | 6, 23 | remulcli 10849 | . . . 4 ⊢ (𝑇 · (𝐴 + 1)) ∈ ℝ |
32 | 6, 24 | remulcli 10849 | . . . 4 ⊢ (𝑇 · 𝐵) ∈ ℝ |
33 | 30, 31, 32 | ltletri 10960 | . . 3 ⊢ ((((𝑇 · 𝐴) + 𝐶) < (𝑇 · (𝐴 + 1)) ∧ (𝑇 · (𝐴 + 1)) ≤ (𝑇 · 𝐵)) → ((𝑇 · 𝐴) + 𝐶) < (𝑇 · 𝐵)) |
34 | 15, 27, 33 | mp2an 692 | . 2 ⊢ ((𝑇 · 𝐴) + 𝐶) < (𝑇 · 𝐵) |
35 | numltc.4 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
36 | 32, 35 | nn0addge1i 12138 | . 2 ⊢ (𝑇 · 𝐵) ≤ ((𝑇 · 𝐵) + 𝐷) |
37 | 35 | nn0rei 12101 | . . . 4 ⊢ 𝐷 ∈ ℝ |
38 | 32, 37 | readdcli 10848 | . . 3 ⊢ ((𝑇 · 𝐵) + 𝐷) ∈ ℝ |
39 | 30, 32, 38 | ltletri 10960 | . 2 ⊢ ((((𝑇 · 𝐴) + 𝐶) < (𝑇 · 𝐵) ∧ (𝑇 · 𝐵) ≤ ((𝑇 · 𝐵) + 𝐷)) → ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐵) + 𝐷)) |
40 | 34, 36, 39 | mp2an 692 | 1 ⊢ ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐵) + 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2110 class class class wbr 5053 (class class class)co 7213 ℝcr 10728 0cc0 10729 1c1 10730 + caddc 10732 · cmul 10734 < clt 10867 ≤ cle 10868 ℕcn 11830 ℕ0cn0 12090 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-z 12177 |
This theorem is referenced by: decltc 12322 numlti 12330 |
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