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Mirrors > Home > ILE Home > Th. List > numma | Unicode version |
Description: Perform a multiply-add of two decimal integers and against a fixed multiplicand (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
numma.1 | |
numma.2 | |
numma.3 | |
numma.4 | |
numma.5 | |
numma.6 | |
numma.7 | |
numma.8 | |
numma.9 | |
numma.10 |
Ref | Expression |
---|---|
numma |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numma.6 | . . . 4 | |
2 | 1 | oveq1i 5784 | . . 3 |
3 | numma.7 | . . 3 | |
4 | 2, 3 | oveq12i 5786 | . 2 |
5 | numma.1 | . . . . . . 7 | |
6 | 5 | nn0cni 8989 | . . . . . 6 |
7 | numma.2 | . . . . . . . 8 | |
8 | 7 | nn0cni 8989 | . . . . . . 7 |
9 | numma.8 | . . . . . . . 8 | |
10 | 9 | nn0cni 8989 | . . . . . . 7 |
11 | 8, 10 | mulcli 7771 | . . . . . 6 |
12 | numma.4 | . . . . . . 7 | |
13 | 12 | nn0cni 8989 | . . . . . 6 |
14 | 6, 11, 13 | adddii 7776 | . . . . 5 |
15 | 6, 8, 10 | mulassi 7775 | . . . . . 6 |
16 | 15 | oveq1i 5784 | . . . . 5 |
17 | 14, 16 | eqtr4i 2163 | . . . 4 |
18 | 17 | oveq1i 5784 | . . 3 |
19 | 6, 8 | mulcli 7771 | . . . . . 6 |
20 | numma.3 | . . . . . . 7 | |
21 | 20 | nn0cni 8989 | . . . . . 6 |
22 | 19, 21, 10 | adddiri 7777 | . . . . 5 |
23 | 22 | oveq1i 5784 | . . . 4 |
24 | 19, 10 | mulcli 7771 | . . . . 5 |
25 | 6, 13 | mulcli 7771 | . . . . 5 |
26 | 21, 10 | mulcli 7771 | . . . . 5 |
27 | numma.5 | . . . . . 6 | |
28 | 27 | nn0cni 8989 | . . . . 5 |
29 | 24, 25, 26, 28 | add4i 7927 | . . . 4 |
30 | 23, 29 | eqtr4i 2163 | . . 3 |
31 | 18, 30 | eqtr4i 2163 | . 2 |
32 | numma.9 | . . . 4 | |
33 | 32 | oveq2i 5785 | . . 3 |
34 | numma.10 | . . 3 | |
35 | 33, 34 | oveq12i 5786 | . 2 |
36 | 4, 31, 35 | 3eqtr2i 2166 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1331 wcel 1480 (class class class)co 5774 caddc 7623 cmul 7625 cn0 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-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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 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-rnegex 7729 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 df-inn 8721 df-n0 8978 |
This theorem is referenced by: nummac 9226 numadd 9228 decma 9232 |
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