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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 12126 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐴) + 𝑇) |
6 | 1 | nnrei 11649 | . . . . . . 7 ⊢ 𝑇 ∈ ℝ |
7 | 6 | recni 10657 | . . . . . 6 ⊢ 𝑇 ∈ ℂ |
8 | 2 | nn0rei 11911 | . . . . . . 7 ⊢ 𝐴 ∈ ℝ |
9 | 8 | recni 10657 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
10 | ax-1cn 10597 | . . . . . 6 ⊢ 1 ∈ ℂ | |
11 | 7, 9, 10 | adddii 10655 | . . . . 5 ⊢ (𝑇 · (𝐴 + 1)) = ((𝑇 · 𝐴) + (𝑇 · 1)) |
12 | 7 | mulid1i 10647 | . . . . . 6 ⊢ (𝑇 · 1) = 𝑇 |
13 | 12 | oveq2i 7169 | . . . . 5 ⊢ ((𝑇 · 𝐴) + (𝑇 · 1)) = ((𝑇 · 𝐴) + 𝑇) |
14 | 11, 13 | eqtri 2846 | . . . 4 ⊢ (𝑇 · (𝐴 + 1)) = ((𝑇 · 𝐴) + 𝑇) |
15 | 5, 14 | breqtrri 5095 | . . 3 ⊢ ((𝑇 · 𝐴) + 𝐶) < (𝑇 · (𝐴 + 1)) |
16 | numltc.6 | . . . . 5 ⊢ 𝐴 < 𝐵 | |
17 | numlt.3 | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
18 | nn0ltp1le 12043 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵)) | |
19 | 2, 17, 18 | mp2an 690 | . . . . 5 ⊢ (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵) |
20 | 16, 19 | mpbi 232 | . . . 4 ⊢ (𝐴 + 1) ≤ 𝐵 |
21 | 1 | nngt0i 11679 | . . . . 5 ⊢ 0 < 𝑇 |
22 | peano2re 10815 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ) | |
23 | 8, 22 | ax-mp 5 | . . . . . 6 ⊢ (𝐴 + 1) ∈ ℝ |
24 | 17 | nn0rei 11911 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
25 | 23, 24, 6 | lemul2i 11565 | . . . . 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 232 | . . 3 ⊢ (𝑇 · (𝐴 + 1)) ≤ (𝑇 · 𝐵) |
28 | 6, 8 | remulcli 10659 | . . . . 5 ⊢ (𝑇 · 𝐴) ∈ ℝ |
29 | 3 | nn0rei 11911 | . . . . 5 ⊢ 𝐶 ∈ ℝ |
30 | 28, 29 | readdcli 10658 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐶) ∈ ℝ |
31 | 6, 23 | remulcli 10659 | . . . 4 ⊢ (𝑇 · (𝐴 + 1)) ∈ ℝ |
32 | 6, 24 | remulcli 10659 | . . . 4 ⊢ (𝑇 · 𝐵) ∈ ℝ |
33 | 30, 31, 32 | ltletri 10770 | . . 3 ⊢ ((((𝑇 · 𝐴) + 𝐶) < (𝑇 · (𝐴 + 1)) ∧ (𝑇 · (𝐴 + 1)) ≤ (𝑇 · 𝐵)) → ((𝑇 · 𝐴) + 𝐶) < (𝑇 · 𝐵)) |
34 | 15, 27, 33 | mp2an 690 | . 2 ⊢ ((𝑇 · 𝐴) + 𝐶) < (𝑇 · 𝐵) |
35 | numltc.4 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
36 | 32, 35 | nn0addge1i 11948 | . 2 ⊢ (𝑇 · 𝐵) ≤ ((𝑇 · 𝐵) + 𝐷) |
37 | 35 | nn0rei 11911 | . . . 4 ⊢ 𝐷 ∈ ℝ |
38 | 32, 37 | readdcli 10658 | . . 3 ⊢ ((𝑇 · 𝐵) + 𝐷) ∈ ℝ |
39 | 30, 32, 38 | ltletri 10770 | . 2 ⊢ ((((𝑇 · 𝐴) + 𝐶) < (𝑇 · 𝐵) ∧ (𝑇 · 𝐵) ≤ ((𝑇 · 𝐵) + 𝐷)) → ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐵) + 𝐷)) |
40 | 34, 36, 39 | mp2an 690 | 1 ⊢ ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐵) + 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 < clt 10677 ≤ cle 10678 ℕcn 11640 ℕ0cn0 11900 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 |
This theorem is referenced by: decltc 12130 numlti 12138 |
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