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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 12642 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐴) + 𝑇) |
6 | 1 | nnrei 12161 | . . . . . . 7 ⊢ 𝑇 ∈ ℝ |
7 | 6 | recni 11168 | . . . . . 6 ⊢ 𝑇 ∈ ℂ |
8 | 2 | nn0rei 12423 | . . . . . . 7 ⊢ 𝐴 ∈ ℝ |
9 | 8 | recni 11168 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
10 | ax-1cn 11108 | . . . . . 6 ⊢ 1 ∈ ℂ | |
11 | 7, 9, 10 | adddii 11166 | . . . . 5 ⊢ (𝑇 · (𝐴 + 1)) = ((𝑇 · 𝐴) + (𝑇 · 1)) |
12 | 7 | mulid1i 11158 | . . . . . 6 ⊢ (𝑇 · 1) = 𝑇 |
13 | 12 | oveq2i 7367 | . . . . 5 ⊢ ((𝑇 · 𝐴) + (𝑇 · 1)) = ((𝑇 · 𝐴) + 𝑇) |
14 | 11, 13 | eqtri 2764 | . . . 4 ⊢ (𝑇 · (𝐴 + 1)) = ((𝑇 · 𝐴) + 𝑇) |
15 | 5, 14 | breqtrri 5132 | . . 3 ⊢ ((𝑇 · 𝐴) + 𝐶) < (𝑇 · (𝐴 + 1)) |
16 | numltc.6 | . . . . 5 ⊢ 𝐴 < 𝐵 | |
17 | numlt.3 | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
18 | nn0ltp1le 12560 | . . . . . 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 229 | . . . 4 ⊢ (𝐴 + 1) ≤ 𝐵 |
21 | 1 | nngt0i 12191 | . . . . 5 ⊢ 0 < 𝑇 |
22 | peano2re 11327 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ) | |
23 | 8, 22 | ax-mp 5 | . . . . . 6 ⊢ (𝐴 + 1) ∈ ℝ |
24 | 17 | nn0rei 12423 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
25 | 23, 24, 6 | lemul2i 12077 | . . . . 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 229 | . . 3 ⊢ (𝑇 · (𝐴 + 1)) ≤ (𝑇 · 𝐵) |
28 | 6, 8 | remulcli 11170 | . . . . 5 ⊢ (𝑇 · 𝐴) ∈ ℝ |
29 | 3 | nn0rei 12423 | . . . . 5 ⊢ 𝐶 ∈ ℝ |
30 | 28, 29 | readdcli 11169 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐶) ∈ ℝ |
31 | 6, 23 | remulcli 11170 | . . . 4 ⊢ (𝑇 · (𝐴 + 1)) ∈ ℝ |
32 | 6, 24 | remulcli 11170 | . . . 4 ⊢ (𝑇 · 𝐵) ∈ ℝ |
33 | 30, 31, 32 | ltletri 11282 | . . 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 12460 | . 2 ⊢ (𝑇 · 𝐵) ≤ ((𝑇 · 𝐵) + 𝐷) |
37 | 35 | nn0rei 12423 | . . . 4 ⊢ 𝐷 ∈ ℝ |
38 | 32, 37 | readdcli 11169 | . . 3 ⊢ ((𝑇 · 𝐵) + 𝐷) ∈ ℝ |
39 | 30, 32, 38 | ltletri 11282 | . 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 205 ∈ wcel 2106 class class class wbr 5105 (class class class)co 7356 ℝcr 11049 0cc0 11050 1c1 11051 + caddc 11053 · cmul 11055 < clt 11188 ≤ cle 11189 ℕcn 12152 ℕ0cn0 12412 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-n0 12413 df-z 12499 |
This theorem is referenced by: decltc 12646 numlti 12654 |
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