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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 9147 | . . . 4 |
3 | numma.2 | . . . . . . . . 9 | |
4 | 3 | nn0cni 9147 | . . . . . . . 8 |
5 | nummac.8 | . . . . . . . . 9 | |
6 | 5 | nn0cni 9147 | . . . . . . . 8 |
7 | 4, 6 | mulcli 7925 | . . . . . . 7 |
8 | numma.4 | . . . . . . . 8 | |
9 | 8 | nn0cni 9147 | . . . . . . 7 |
10 | nummac.10 | . . . . . . . 8 | |
11 | 10 | nn0cni 9147 | . . . . . . 7 |
12 | 7, 9, 11 | addassi 7928 | . . . . . 6 |
13 | nummac.11 | . . . . . 6 | |
14 | 12, 13 | eqtri 2191 | . . . . 5 |
15 | 7, 9 | addcli 7924 | . . . . . 6 |
16 | 15, 11 | addcli 7924 | . . . . 5 |
17 | 14, 16 | eqeltrri 2244 | . . . 4 |
18 | 2, 17, 11 | subdii 8326 | . . 3 |
19 | 18 | oveq1i 5863 | . 2 |
20 | numma.3 | . . 3 | |
21 | numma.5 | . . 3 | |
22 | numma.6 | . . 3 | |
23 | numma.7 | . . 3 | |
24 | 17, 11, 15 | subadd2i 8207 | . . . . 5 |
25 | 14, 24 | mpbir 145 | . . . 4 |
26 | 25 | eqcomi 2174 | . . 3 |
27 | nummac.12 | . . 3 | |
28 | 1, 3, 20, 8, 21, 22, 23, 5, 26, 27 | numma 9386 | . 2 |
29 | 2, 17 | mulcli 7925 | . . . . 5 |
30 | 2, 11 | mulcli 7925 | . . . . 5 |
31 | npcan 8128 | . . . . 5 | |
32 | 29, 30, 31 | mp2an 424 | . . . 4 |
33 | 32 | oveq1i 5863 | . . 3 |
34 | 29, 30 | subcli 8195 | . . . 4 |
35 | nummac.9 | . . . . 5 | |
36 | 35 | nn0cni 9147 | . . . 4 |
37 | 34, 30, 36 | addassi 7928 | . . 3 |
38 | 33, 37 | eqtr3i 2193 | . 2 |
39 | 19, 28, 38 | 3eqtr4i 2201 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1348 wcel 2141 (class class class)co 5853 cc 7772 caddc 7777 cmul 7779 cmin 8090 cn0 9135 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-sub 8092 df-inn 8879 df-n0 9136 |
This theorem is referenced by: numma2c 9388 numaddc 9390 nummul1c 9391 decmac 9394 |
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