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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 11403 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐴) + 𝑇) |
6 | 1 | nnrei 10906 | . . . . . . 7 ⊢ 𝑇 ∈ ℝ |
7 | 6 | recni 9931 | . . . . . 6 ⊢ 𝑇 ∈ ℂ |
8 | 2 | nn0rei 11180 | . . . . . . 7 ⊢ 𝐴 ∈ ℝ |
9 | 8 | recni 9931 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
10 | ax-1cn 9873 | . . . . . 6 ⊢ 1 ∈ ℂ | |
11 | 7, 9, 10 | adddii 9929 | . . . . 5 ⊢ (𝑇 · (𝐴 + 1)) = ((𝑇 · 𝐴) + (𝑇 · 1)) |
12 | 7 | mulid1i 9921 | . . . . . 6 ⊢ (𝑇 · 1) = 𝑇 |
13 | 12 | oveq2i 6560 | . . . . 5 ⊢ ((𝑇 · 𝐴) + (𝑇 · 1)) = ((𝑇 · 𝐴) + 𝑇) |
14 | 11, 13 | eqtri 2632 | . . . 4 ⊢ (𝑇 · (𝐴 + 1)) = ((𝑇 · 𝐴) + 𝑇) |
15 | 5, 14 | breqtrri 4610 | . . 3 ⊢ ((𝑇 · 𝐴) + 𝐶) < (𝑇 · (𝐴 + 1)) |
16 | numltc.6 | . . . . 5 ⊢ 𝐴 < 𝐵 | |
17 | numlt.3 | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
18 | nn0ltp1le 11312 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵)) | |
19 | 2, 17, 18 | mp2an 704 | . . . . 5 ⊢ (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵) |
20 | 16, 19 | mpbi 219 | . . . 4 ⊢ (𝐴 + 1) ≤ 𝐵 |
21 | 1 | nngt0i 10931 | . . . . 5 ⊢ 0 < 𝑇 |
22 | peano2re 10088 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ) | |
23 | 8, 22 | ax-mp 5 | . . . . . 6 ⊢ (𝐴 + 1) ∈ ℝ |
24 | 17 | nn0rei 11180 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
25 | 23, 24, 6 | lemul2i 10826 | . . . . 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 219 | . . 3 ⊢ (𝑇 · (𝐴 + 1)) ≤ (𝑇 · 𝐵) |
28 | 6, 8 | remulcli 9933 | . . . . 5 ⊢ (𝑇 · 𝐴) ∈ ℝ |
29 | 3 | nn0rei 11180 | . . . . 5 ⊢ 𝐶 ∈ ℝ |
30 | 28, 29 | readdcli 9932 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐶) ∈ ℝ |
31 | 6, 23 | remulcli 9933 | . . . 4 ⊢ (𝑇 · (𝐴 + 1)) ∈ ℝ |
32 | 6, 24 | remulcli 9933 | . . . 4 ⊢ (𝑇 · 𝐵) ∈ ℝ |
33 | 30, 31, 32 | ltletri 10044 | . . 3 ⊢ ((((𝑇 · 𝐴) + 𝐶) < (𝑇 · (𝐴 + 1)) ∧ (𝑇 · (𝐴 + 1)) ≤ (𝑇 · 𝐵)) → ((𝑇 · 𝐴) + 𝐶) < (𝑇 · 𝐵)) |
34 | 15, 27, 33 | mp2an 704 | . 2 ⊢ ((𝑇 · 𝐴) + 𝐶) < (𝑇 · 𝐵) |
35 | numltc.4 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
36 | 32, 35 | nn0addge1i 11218 | . 2 ⊢ (𝑇 · 𝐵) ≤ ((𝑇 · 𝐵) + 𝐷) |
37 | 35 | nn0rei 11180 | . . . 4 ⊢ 𝐷 ∈ ℝ |
38 | 32, 37 | readdcli 9932 | . . 3 ⊢ ((𝑇 · 𝐵) + 𝐷) ∈ ℝ |
39 | 30, 32, 38 | ltletri 10044 | . 2 ⊢ ((((𝑇 · 𝐴) + 𝐶) < (𝑇 · 𝐵) ∧ (𝑇 · 𝐵) ≤ ((𝑇 · 𝐵) + 𝐷)) → ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐵) + 𝐷)) |
40 | 34, 36, 39 | mp2an 704 | 1 ⊢ ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐵) + 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 ℕcn 10897 ℕ0cn0 11169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 |
This theorem is referenced by: decltc 11408 decltcOLD 11409 numlti 11421 |
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