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Mirrors > Home > ILE Home > Th. List > numltc | 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 9406 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐴) + 𝑇) |
6 | 1 | nnrei 8926 | . . . . . . 7 ⊢ 𝑇 ∈ ℝ |
7 | 6 | recni 7968 | . . . . . 6 ⊢ 𝑇 ∈ ℂ |
8 | 2 | nn0rei 9185 | . . . . . . 7 ⊢ 𝐴 ∈ ℝ |
9 | 8 | recni 7968 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
10 | ax-1cn 7903 | . . . . . 6 ⊢ 1 ∈ ℂ | |
11 | 7, 9, 10 | adddii 7966 | . . . . 5 ⊢ (𝑇 · (𝐴 + 1)) = ((𝑇 · 𝐴) + (𝑇 · 1)) |
12 | 7 | mulid1i 7958 | . . . . . 6 ⊢ (𝑇 · 1) = 𝑇 |
13 | 12 | oveq2i 5885 | . . . . 5 ⊢ ((𝑇 · 𝐴) + (𝑇 · 1)) = ((𝑇 · 𝐴) + 𝑇) |
14 | 11, 13 | eqtri 2198 | . . . 4 ⊢ (𝑇 · (𝐴 + 1)) = ((𝑇 · 𝐴) + 𝑇) |
15 | 5, 14 | breqtrri 4030 | . . 3 ⊢ ((𝑇 · 𝐴) + 𝐶) < (𝑇 · (𝐴 + 1)) |
16 | numltc.6 | . . . . 5 ⊢ 𝐴 < 𝐵 | |
17 | numlt.3 | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
18 | nn0ltp1le 9313 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵)) | |
19 | 2, 17, 18 | mp2an 426 | . . . . 5 ⊢ (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵) |
20 | 16, 19 | mpbi 145 | . . . 4 ⊢ (𝐴 + 1) ≤ 𝐵 |
21 | 1 | nngt0i 8947 | . . . . 5 ⊢ 0 < 𝑇 |
22 | peano2re 8091 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ) | |
23 | 8, 22 | ax-mp 5 | . . . . . 6 ⊢ (𝐴 + 1) ∈ ℝ |
24 | 17 | nn0rei 9185 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
25 | 23, 24, 6 | lemul2i 8880 | . . . . 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 145 | . . 3 ⊢ (𝑇 · (𝐴 + 1)) ≤ (𝑇 · 𝐵) |
28 | 6, 8 | remulcli 7970 | . . . . 5 ⊢ (𝑇 · 𝐴) ∈ ℝ |
29 | 3 | nn0rei 9185 | . . . . 5 ⊢ 𝐶 ∈ ℝ |
30 | 28, 29 | readdcli 7969 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐶) ∈ ℝ |
31 | 6, 23 | remulcli 7970 | . . . 4 ⊢ (𝑇 · (𝐴 + 1)) ∈ ℝ |
32 | 6, 24 | remulcli 7970 | . . . 4 ⊢ (𝑇 · 𝐵) ∈ ℝ |
33 | 30, 31, 32 | ltletri 8062 | . . 3 ⊢ ((((𝑇 · 𝐴) + 𝐶) < (𝑇 · (𝐴 + 1)) ∧ (𝑇 · (𝐴 + 1)) ≤ (𝑇 · 𝐵)) → ((𝑇 · 𝐴) + 𝐶) < (𝑇 · 𝐵)) |
34 | 15, 27, 33 | mp2an 426 | . 2 ⊢ ((𝑇 · 𝐴) + 𝐶) < (𝑇 · 𝐵) |
35 | numltc.4 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
36 | 32, 35 | nn0addge1i 9222 | . 2 ⊢ (𝑇 · 𝐵) ≤ ((𝑇 · 𝐵) + 𝐷) |
37 | 35 | nn0rei 9185 | . . . 4 ⊢ 𝐷 ∈ ℝ |
38 | 32, 37 | readdcli 7969 | . . 3 ⊢ ((𝑇 · 𝐵) + 𝐷) ∈ ℝ |
39 | 30, 32, 38 | ltletri 8062 | . 2 ⊢ ((((𝑇 · 𝐴) + 𝐶) < (𝑇 · 𝐵) ∧ (𝑇 · 𝐵) ≤ ((𝑇 · 𝐵) + 𝐷)) → ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐵) + 𝐷)) |
40 | 34, 36, 39 | mp2an 426 | 1 ⊢ ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐵) + 𝐷) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2148 class class class wbr 4003 (class class class)co 5874 ℝcr 7809 0cc0 7810 1c1 7811 + caddc 7813 · cmul 7815 < clt 7990 ≤ cle 7991 ℕcn 8917 ℕ0cn0 9174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-inn 8918 df-n0 9175 df-z 9252 |
This theorem is referenced by: decltc 9410 numlti 9418 |
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