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Mirrors > Home > ILE Home > Th. List > numlt | GIF version |
Description: Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
numlt.1 | ⊢ 𝑇 ∈ ℕ |
numlt.2 | ⊢ 𝐴 ∈ ℕ0 |
numlt.3 | ⊢ 𝐵 ∈ ℕ0 |
numlt.4 | ⊢ 𝐶 ∈ ℕ |
numlt.5 | ⊢ 𝐵 < 𝐶 |
Ref | Expression |
---|---|
numlt | ⊢ ((𝑇 · 𝐴) + 𝐵) < ((𝑇 · 𝐴) + 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numlt.5 | . 2 ⊢ 𝐵 < 𝐶 | |
2 | numlt.3 | . . . 4 ⊢ 𝐵 ∈ ℕ0 | |
3 | 2 | nn0rei 8988 | . . 3 ⊢ 𝐵 ∈ ℝ |
4 | numlt.4 | . . . 4 ⊢ 𝐶 ∈ ℕ | |
5 | 4 | nnrei 8729 | . . 3 ⊢ 𝐶 ∈ ℝ |
6 | numlt.1 | . . . . . 6 ⊢ 𝑇 ∈ ℕ | |
7 | 6 | nnnn0i 8985 | . . . . 5 ⊢ 𝑇 ∈ ℕ0 |
8 | numlt.2 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
9 | 7, 8 | nn0mulcli 9015 | . . . 4 ⊢ (𝑇 · 𝐴) ∈ ℕ0 |
10 | 9 | nn0rei 8988 | . . 3 ⊢ (𝑇 · 𝐴) ∈ ℝ |
11 | 3, 5, 10 | ltadd2i 8182 | . 2 ⊢ (𝐵 < 𝐶 ↔ ((𝑇 · 𝐴) + 𝐵) < ((𝑇 · 𝐴) + 𝐶)) |
12 | 1, 11 | mpbi 144 | 1 ⊢ ((𝑇 · 𝐴) + 𝐵) < ((𝑇 · 𝐴) + 𝐶) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 class class class wbr 3929 (class class class)co 5774 + caddc 7623 · cmul 7625 < clt 7800 ℕcn 8720 ℕ0cn0 8977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-sub 7935 df-inn 8721 df-n0 8978 |
This theorem is referenced by: numltc 9207 declt 9209 |
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