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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 9107 | . . . 4 |
3 | numma.2 | . . . . . . . . 9 | |
4 | 3 | nn0cni 9107 | . . . . . . . 8 |
5 | nummac.8 | . . . . . . . . 9 | |
6 | 5 | nn0cni 9107 | . . . . . . . 8 |
7 | 4, 6 | mulcli 7885 | . . . . . . 7 |
8 | numma.4 | . . . . . . . 8 | |
9 | 8 | nn0cni 9107 | . . . . . . 7 |
10 | nummac.10 | . . . . . . . 8 | |
11 | 10 | nn0cni 9107 | . . . . . . 7 |
12 | 7, 9, 11 | addassi 7888 | . . . . . 6 |
13 | nummac.11 | . . . . . 6 | |
14 | 12, 13 | eqtri 2178 | . . . . 5 |
15 | 7, 9 | addcli 7884 | . . . . . 6 |
16 | 15, 11 | addcli 7884 | . . . . 5 |
17 | 14, 16 | eqeltrri 2231 | . . . 4 |
18 | 2, 17, 11 | subdii 8286 | . . 3 |
19 | 18 | oveq1i 5836 | . 2 |
20 | numma.3 | . . 3 | |
21 | numma.5 | . . 3 | |
22 | numma.6 | . . 3 | |
23 | numma.7 | . . 3 | |
24 | 17, 11, 15 | subadd2i 8167 | . . . . 5 |
25 | 14, 24 | mpbir 145 | . . . 4 |
26 | 25 | eqcomi 2161 | . . 3 |
27 | nummac.12 | . . 3 | |
28 | 1, 3, 20, 8, 21, 22, 23, 5, 26, 27 | numma 9343 | . 2 |
29 | 2, 17 | mulcli 7885 | . . . . 5 |
30 | 2, 11 | mulcli 7885 | . . . . 5 |
31 | npcan 8088 | . . . . 5 | |
32 | 29, 30, 31 | mp2an 423 | . . . 4 |
33 | 32 | oveq1i 5836 | . . 3 |
34 | 29, 30 | subcli 8155 | . . . 4 |
35 | nummac.9 | . . . . 5 | |
36 | 35 | nn0cni 9107 | . . . 4 |
37 | 34, 30, 36 | addassi 7888 | . . 3 |
38 | 33, 37 | eqtr3i 2180 | . 2 |
39 | 19, 28, 38 | 3eqtr4i 2188 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1335 wcel 2128 (class class class)co 5826 cc 7732 caddc 7737 cmul 7739 cmin 8050 cn0 9095 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-setind 4498 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-addcom 7834 ax-mulcom 7835 ax-addass 7836 ax-mulass 7837 ax-distr 7838 ax-i2m1 7839 ax-0id 7842 ax-rnegex 7843 ax-cnre 7845 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-iota 5137 df-fun 5174 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-sub 8052 df-inn 8839 df-n0 9096 |
This theorem is referenced by: numma2c 9345 numaddc 9347 nummul1c 9348 decmac 9351 |
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