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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 9212 | . . 3 ⊢ 𝐵 ∈ ℝ |
4 | numlt.4 | . . . 4 ⊢ 𝐶 ∈ ℕ | |
5 | 4 | nnrei 8953 | . . 3 ⊢ 𝐶 ∈ ℝ |
6 | numlt.1 | . . . . . 6 ⊢ 𝑇 ∈ ℕ | |
7 | 6 | nnnn0i 9209 | . . . . 5 ⊢ 𝑇 ∈ ℕ0 |
8 | numlt.2 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
9 | 7, 8 | nn0mulcli 9239 | . . . 4 ⊢ (𝑇 · 𝐴) ∈ ℕ0 |
10 | 9 | nn0rei 9212 | . . 3 ⊢ (𝑇 · 𝐴) ∈ ℝ |
11 | 3, 5, 10 | ltadd2i 8402 | . 2 ⊢ (𝐵 < 𝐶 ↔ ((𝑇 · 𝐴) + 𝐵) < ((𝑇 · 𝐴) + 𝐶)) |
12 | 1, 11 | mpbi 145 | 1 ⊢ ((𝑇 · 𝐴) + 𝐵) < ((𝑇 · 𝐴) + 𝐶) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 class class class wbr 4018 (class class class)co 5892 + caddc 7839 · cmul 7841 < clt 8017 ℕcn 8944 ℕ0cn0 9201 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7927 ax-resscn 7928 ax-1cn 7929 ax-1re 7930 ax-icn 7931 ax-addcl 7932 ax-addrcl 7933 ax-mulcl 7934 ax-addcom 7936 ax-mulcom 7937 ax-addass 7938 ax-mulass 7939 ax-distr 7940 ax-i2m1 7941 ax-1rid 7943 ax-0id 7944 ax-rnegex 7945 ax-cnre 7947 ax-pre-ltadd 7952 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5234 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-pnf 8019 df-mnf 8020 df-ltxr 8022 df-sub 8155 df-inn 8945 df-n0 9202 |
This theorem is referenced by: numltc 9434 declt 9436 |
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