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Mirrors > Home > MPE Home > Th. List > nummac | Structured version Visualization version GIF version |
Description: Perform a multiply-add of two decimal integers 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
numma.1 | ⊢ 𝑇 ∈ ℕ0 |
numma.2 | ⊢ 𝐴 ∈ ℕ0 |
numma.3 | ⊢ 𝐵 ∈ ℕ0 |
numma.4 | ⊢ 𝐶 ∈ ℕ0 |
numma.5 | ⊢ 𝐷 ∈ ℕ0 |
numma.6 | ⊢ 𝑀 = ((𝑇 · 𝐴) + 𝐵) |
numma.7 | ⊢ 𝑁 = ((𝑇 · 𝐶) + 𝐷) |
nummac.8 | ⊢ 𝑃 ∈ ℕ0 |
nummac.9 | ⊢ 𝐹 ∈ ℕ0 |
nummac.10 | ⊢ 𝐺 ∈ ℕ0 |
nummac.11 | ⊢ ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = 𝐸 |
nummac.12 | ⊢ ((𝐵 · 𝑃) + 𝐷) = ((𝑇 · 𝐺) + 𝐹) |
Ref | Expression |
---|---|
nummac | ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numma.1 | . . . . 5 ⊢ 𝑇 ∈ ℕ0 | |
2 | 1 | nn0cni 11897 | . . . 4 ⊢ 𝑇 ∈ ℂ |
3 | numma.2 | . . . . . . . . 9 ⊢ 𝐴 ∈ ℕ0 | |
4 | 3 | nn0cni 11897 | . . . . . . . 8 ⊢ 𝐴 ∈ ℂ |
5 | nummac.8 | . . . . . . . . 9 ⊢ 𝑃 ∈ ℕ0 | |
6 | 5 | nn0cni 11897 | . . . . . . . 8 ⊢ 𝑃 ∈ ℂ |
7 | 4, 6 | mulcli 10637 | . . . . . . 7 ⊢ (𝐴 · 𝑃) ∈ ℂ |
8 | numma.4 | . . . . . . . 8 ⊢ 𝐶 ∈ ℕ0 | |
9 | 8 | nn0cni 11897 | . . . . . . 7 ⊢ 𝐶 ∈ ℂ |
10 | nummac.10 | . . . . . . . 8 ⊢ 𝐺 ∈ ℕ0 | |
11 | 10 | nn0cni 11897 | . . . . . . 7 ⊢ 𝐺 ∈ ℂ |
12 | 7, 9, 11 | addassi 10640 | . . . . . 6 ⊢ (((𝐴 · 𝑃) + 𝐶) + 𝐺) = ((𝐴 · 𝑃) + (𝐶 + 𝐺)) |
13 | nummac.11 | . . . . . 6 ⊢ ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = 𝐸 | |
14 | 12, 13 | eqtri 2821 | . . . . 5 ⊢ (((𝐴 · 𝑃) + 𝐶) + 𝐺) = 𝐸 |
15 | 7, 9 | addcli 10636 | . . . . . 6 ⊢ ((𝐴 · 𝑃) + 𝐶) ∈ ℂ |
16 | 15, 11 | addcli 10636 | . . . . 5 ⊢ (((𝐴 · 𝑃) + 𝐶) + 𝐺) ∈ ℂ |
17 | 14, 16 | eqeltrri 2887 | . . . 4 ⊢ 𝐸 ∈ ℂ |
18 | 2, 17, 11 | subdii 11078 | . . 3 ⊢ (𝑇 · (𝐸 − 𝐺)) = ((𝑇 · 𝐸) − (𝑇 · 𝐺)) |
19 | 18 | oveq1i 7145 | . 2 ⊢ ((𝑇 · (𝐸 − 𝐺)) + ((𝑇 · 𝐺) + 𝐹)) = (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + ((𝑇 · 𝐺) + 𝐹)) |
20 | numma.3 | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
21 | numma.5 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
22 | numma.6 | . . 3 ⊢ 𝑀 = ((𝑇 · 𝐴) + 𝐵) | |
23 | numma.7 | . . 3 ⊢ 𝑁 = ((𝑇 · 𝐶) + 𝐷) | |
24 | 17, 11, 15 | subadd2i 10963 | . . . . 5 ⊢ ((𝐸 − 𝐺) = ((𝐴 · 𝑃) + 𝐶) ↔ (((𝐴 · 𝑃) + 𝐶) + 𝐺) = 𝐸) |
25 | 14, 24 | mpbir 234 | . . . 4 ⊢ (𝐸 − 𝐺) = ((𝐴 · 𝑃) + 𝐶) |
26 | 25 | eqcomi 2807 | . . 3 ⊢ ((𝐴 · 𝑃) + 𝐶) = (𝐸 − 𝐺) |
27 | nummac.12 | . . 3 ⊢ ((𝐵 · 𝑃) + 𝐷) = ((𝑇 · 𝐺) + 𝐹) | |
28 | 1, 3, 20, 8, 21, 22, 23, 5, 26, 27 | numma 12130 | . 2 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · (𝐸 − 𝐺)) + ((𝑇 · 𝐺) + 𝐹)) |
29 | 2, 17 | mulcli 10637 | . . . . 5 ⊢ (𝑇 · 𝐸) ∈ ℂ |
30 | 2, 11 | mulcli 10637 | . . . . 5 ⊢ (𝑇 · 𝐺) ∈ ℂ |
31 | npcan 10884 | . . . . 5 ⊢ (((𝑇 · 𝐸) ∈ ℂ ∧ (𝑇 · 𝐺) ∈ ℂ) → (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) = (𝑇 · 𝐸)) | |
32 | 29, 30, 31 | mp2an 691 | . . . 4 ⊢ (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) = (𝑇 · 𝐸) |
33 | 32 | oveq1i 7145 | . . 3 ⊢ ((((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) + 𝐹) = ((𝑇 · 𝐸) + 𝐹) |
34 | 29, 30 | subcli 10951 | . . . 4 ⊢ ((𝑇 · 𝐸) − (𝑇 · 𝐺)) ∈ ℂ |
35 | nummac.9 | . . . . 5 ⊢ 𝐹 ∈ ℕ0 | |
36 | 35 | nn0cni 11897 | . . . 4 ⊢ 𝐹 ∈ ℂ |
37 | 34, 30, 36 | addassi 10640 | . . 3 ⊢ ((((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) + 𝐹) = (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + ((𝑇 · 𝐺) + 𝐹)) |
38 | 33, 37 | eqtr3i 2823 | . 2 ⊢ ((𝑇 · 𝐸) + 𝐹) = (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + ((𝑇 · 𝐺) + 𝐹)) |
39 | 19, 28, 38 | 3eqtr4i 2831 | 1 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 + caddc 10529 · cmul 10531 − cmin 10859 ℕ0cn0 11885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 df-nn 11626 df-n0 11886 |
This theorem is referenced by: numma2c 12132 numaddc 12134 nummul1c 12135 decmac 12138 |
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