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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 8674 | . . 3 ⊢ 𝐵 ∈ ℝ |
4 | numlt.4 | . . . 4 ⊢ 𝐶 ∈ ℕ | |
5 | 4 | nnrei 8421 | . . 3 ⊢ 𝐶 ∈ ℝ |
6 | numlt.1 | . . . . . 6 ⊢ 𝑇 ∈ ℕ | |
7 | 6 | nnnn0i 8671 | . . . . 5 ⊢ 𝑇 ∈ ℕ0 |
8 | numlt.2 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
9 | 7, 8 | nn0mulcli 8701 | . . . 4 ⊢ (𝑇 · 𝐴) ∈ ℕ0 |
10 | 9 | nn0rei 8674 | . . 3 ⊢ (𝑇 · 𝐴) ∈ ℝ |
11 | 3, 5, 10 | ltadd2i 7888 | . 2 ⊢ (𝐵 < 𝐶 ↔ ((𝑇 · 𝐴) + 𝐵) < ((𝑇 · 𝐴) + 𝐶)) |
12 | 1, 11 | mpbi 143 | 1 ⊢ ((𝑇 · 𝐴) + 𝐵) < ((𝑇 · 𝐴) + 𝐶) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1438 class class class wbr 3843 (class class class)co 5644 + caddc 7343 · cmul 7345 < clt 7512 ℕcn 8412 ℕ0cn0 8663 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3955 ax-pow 4007 ax-pr 4034 ax-un 4258 ax-setind 4351 ax-cnex 7426 ax-resscn 7427 ax-1cn 7428 ax-1re 7429 ax-icn 7430 ax-addcl 7431 ax-addrcl 7432 ax-mulcl 7433 ax-addcom 7435 ax-mulcom 7436 ax-addass 7437 ax-mulass 7438 ax-distr 7439 ax-i2m1 7440 ax-1rid 7442 ax-0id 7443 ax-rnegex 7444 ax-cnre 7446 ax-pre-ltadd 7451 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3429 df-sn 3450 df-pr 3451 df-op 3453 df-uni 3652 df-int 3687 df-br 3844 df-opab 3898 df-id 4118 df-xp 4442 df-rel 4443 df-cnv 4444 df-co 4445 df-dm 4446 df-iota 4975 df-fun 5012 df-fv 5018 df-riota 5600 df-ov 5647 df-oprab 5648 df-mpt2 5649 df-pnf 7514 df-mnf 7515 df-ltxr 7517 df-sub 7645 df-inn 8413 df-n0 8664 |
This theorem is referenced by: numltc 8892 declt 8894 |
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