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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 5847 | . . 3 |
3 | numma.7 | . . 3 | |
4 | 2, 3 | oveq12i 5849 | . 2 |
5 | numma.1 | . . . . . . 7 | |
6 | 5 | nn0cni 9118 | . . . . . 6 |
7 | numma.2 | . . . . . . . 8 | |
8 | 7 | nn0cni 9118 | . . . . . . 7 |
9 | numma.8 | . . . . . . . 8 | |
10 | 9 | nn0cni 9118 | . . . . . . 7 |
11 | 8, 10 | mulcli 7896 | . . . . . 6 |
12 | numma.4 | . . . . . . 7 | |
13 | 12 | nn0cni 9118 | . . . . . 6 |
14 | 6, 11, 13 | adddii 7901 | . . . . 5 |
15 | 6, 8, 10 | mulassi 7900 | . . . . . 6 |
16 | 15 | oveq1i 5847 | . . . . 5 |
17 | 14, 16 | eqtr4i 2188 | . . . 4 |
18 | 17 | oveq1i 5847 | . . 3 |
19 | 6, 8 | mulcli 7896 | . . . . . 6 |
20 | numma.3 | . . . . . . 7 | |
21 | 20 | nn0cni 9118 | . . . . . 6 |
22 | 19, 21, 10 | adddiri 7902 | . . . . 5 |
23 | 22 | oveq1i 5847 | . . . 4 |
24 | 19, 10 | mulcli 7896 | . . . . 5 |
25 | 6, 13 | mulcli 7896 | . . . . 5 |
26 | 21, 10 | mulcli 7896 | . . . . 5 |
27 | numma.5 | . . . . . 6 | |
28 | 27 | nn0cni 9118 | . . . . 5 |
29 | 24, 25, 26, 28 | add4i 8055 | . . . 4 |
30 | 23, 29 | eqtr4i 2188 | . . 3 |
31 | 18, 30 | eqtr4i 2188 | . 2 |
32 | numma.9 | . . . 4 | |
33 | 32 | oveq2i 5848 | . . 3 |
34 | numma.10 | . . 3 | |
35 | 33, 34 | oveq12i 5849 | . 2 |
36 | 4, 31, 35 | 3eqtr2i 2191 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1342 wcel 2135 (class class class)co 5837 caddc 7748 cmul 7750 cn0 9106 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-sep 4095 ax-cnex 7836 ax-resscn 7837 ax-1re 7839 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-addcom 7845 ax-mulcom 7846 ax-addass 7847 ax-mulass 7848 ax-distr 7849 ax-rnegex 7854 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2724 df-un 3116 df-in 3118 df-ss 3125 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-br 3978 df-iota 5148 df-fv 5191 df-ov 5840 df-inn 8850 df-n0 9107 |
This theorem is referenced by: nummac 9358 numadd 9360 decma 9364 |
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