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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 5542 | . . 3 |
3 | numma.7 | . . 3 | |
4 | 2, 3 | oveq12i 5544 | . 2 |
5 | numma.1 | . . . . . . 7 | |
6 | 5 | nn0cni 8300 | . . . . . 6 |
7 | numma.2 | . . . . . . . 8 | |
8 | 7 | nn0cni 8300 | . . . . . . 7 |
9 | numma.8 | . . . . . . . 8 | |
10 | 9 | nn0cni 8300 | . . . . . . 7 |
11 | 8, 10 | mulcli 7124 | . . . . . 6 |
12 | numma.4 | . . . . . . 7 | |
13 | 12 | nn0cni 8300 | . . . . . 6 |
14 | 6, 11, 13 | adddii 7129 | . . . . 5 |
15 | 6, 8, 10 | mulassi 7128 | . . . . . 6 |
16 | 15 | oveq1i 5542 | . . . . 5 |
17 | 14, 16 | eqtr4i 2104 | . . . 4 |
18 | 17 | oveq1i 5542 | . . 3 |
19 | 6, 8 | mulcli 7124 | . . . . . 6 |
20 | numma.3 | . . . . . . 7 | |
21 | 20 | nn0cni 8300 | . . . . . 6 |
22 | 19, 21, 10 | adddiri 7130 | . . . . 5 |
23 | 22 | oveq1i 5542 | . . . 4 |
24 | 19, 10 | mulcli 7124 | . . . . 5 |
25 | 6, 13 | mulcli 7124 | . . . . 5 |
26 | 21, 10 | mulcli 7124 | . . . . 5 |
27 | numma.5 | . . . . . 6 | |
28 | 27 | nn0cni 8300 | . . . . 5 |
29 | 24, 25, 26, 28 | add4i 7273 | . . . 4 |
30 | 23, 29 | eqtr4i 2104 | . . 3 |
31 | 18, 30 | eqtr4i 2104 | . 2 |
32 | numma.9 | . . . 4 | |
33 | 32 | oveq2i 5543 | . . 3 |
34 | numma.10 | . . 3 | |
35 | 33, 34 | oveq12i 5544 | . 2 |
36 | 4, 31, 35 | 3eqtr2i 2107 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1284 wcel 1433 (class class class)co 5532 caddc 6984 cmul 6986 cn0 8288 |
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-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-cnex 7067 ax-resscn 7068 ax-1re 7070 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-rnegex 7085 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 df-inn 8040 df-n0 8289 |
This theorem is referenced by: nummac 8521 numadd 8523 decma 8527 |
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