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Mirrors > Home > ILE Home > Th. List > nummac | Unicode version |
Description: Perform a multiply-add of two decimal integers and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
numma.1 | |
numma.2 | |
numma.3 | |
numma.4 | |
numma.5 | |
numma.6 | |
numma.7 | |
nummac.8 | |
nummac.9 | |
nummac.10 | |
nummac.11 | |
nummac.12 |
Ref | Expression |
---|---|
nummac |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numma.1 | . . . . 5 | |
2 | 1 | nn0cni 8989 | . . . 4 |
3 | numma.2 | . . . . . . . . 9 | |
4 | 3 | nn0cni 8989 | . . . . . . . 8 |
5 | nummac.8 | . . . . . . . . 9 | |
6 | 5 | nn0cni 8989 | . . . . . . . 8 |
7 | 4, 6 | mulcli 7771 | . . . . . . 7 |
8 | numma.4 | . . . . . . . 8 | |
9 | 8 | nn0cni 8989 | . . . . . . 7 |
10 | nummac.10 | . . . . . . . 8 | |
11 | 10 | nn0cni 8989 | . . . . . . 7 |
12 | 7, 9, 11 | addassi 7774 | . . . . . 6 |
13 | nummac.11 | . . . . . 6 | |
14 | 12, 13 | eqtri 2160 | . . . . 5 |
15 | 7, 9 | addcli 7770 | . . . . . 6 |
16 | 15, 11 | addcli 7770 | . . . . 5 |
17 | 14, 16 | eqeltrri 2213 | . . . 4 |
18 | 2, 17, 11 | subdii 8169 | . . 3 |
19 | 18 | oveq1i 5784 | . 2 |
20 | numma.3 | . . 3 | |
21 | numma.5 | . . 3 | |
22 | numma.6 | . . 3 | |
23 | numma.7 | . . 3 | |
24 | 17, 11, 15 | subadd2i 8050 | . . . . 5 |
25 | 14, 24 | mpbir 145 | . . . 4 |
26 | 25 | eqcomi 2143 | . . 3 |
27 | nummac.12 | . . 3 | |
28 | 1, 3, 20, 8, 21, 22, 23, 5, 26, 27 | numma 9225 | . 2 |
29 | 2, 17 | mulcli 7771 | . . . . 5 |
30 | 2, 11 | mulcli 7771 | . . . . 5 |
31 | npcan 7971 | . . . . 5 | |
32 | 29, 30, 31 | mp2an 422 | . . . 4 |
33 | 32 | oveq1i 5784 | . . 3 |
34 | 29, 30 | subcli 8038 | . . . 4 |
35 | nummac.9 | . . . . 5 | |
36 | 35 | nn0cni 8989 | . . . 4 |
37 | 34, 30, 36 | addassi 7774 | . . 3 |
38 | 33, 37 | eqtr3i 2162 | . 2 |
39 | 19, 28, 38 | 3eqtr4i 2170 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1331 wcel 1480 (class class class)co 5774 cc 7618 caddc 7623 cmul 7625 cmin 7933 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-in1 603 ax-in2 604 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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 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-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7935 df-inn 8721 df-n0 8978 |
This theorem is referenced by: numma2c 9227 numaddc 9229 nummul1c 9230 decmac 9233 |
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