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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 10636 | . . . . . . 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 10639 | . . . . . 6 ⊢ (((𝐴 · 𝑃) + 𝐶) + 𝐺) = ((𝐴 · 𝑃) + (𝐶 + 𝐺)) |
13 | nummac.11 | . . . . . 6 ⊢ ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = 𝐸 | |
14 | 12, 13 | eqtri 2841 | . . . . 5 ⊢ (((𝐴 · 𝑃) + 𝐶) + 𝐺) = 𝐸 |
15 | 7, 9 | addcli 10635 | . . . . . 6 ⊢ ((𝐴 · 𝑃) + 𝐶) ∈ ℂ |
16 | 15, 11 | addcli 10635 | . . . . 5 ⊢ (((𝐴 · 𝑃) + 𝐶) + 𝐺) ∈ ℂ |
17 | 14, 16 | eqeltrri 2907 | . . . 4 ⊢ 𝐸 ∈ ℂ |
18 | 2, 17, 11 | subdii 11077 | . . 3 ⊢ (𝑇 · (𝐸 − 𝐺)) = ((𝑇 · 𝐸) − (𝑇 · 𝐺)) |
19 | 18 | oveq1i 7155 | . 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 10962 | . . . . 5 ⊢ ((𝐸 − 𝐺) = ((𝐴 · 𝑃) + 𝐶) ↔ (((𝐴 · 𝑃) + 𝐶) + 𝐺) = 𝐸) |
25 | 14, 24 | mpbir 232 | . . . 4 ⊢ (𝐸 − 𝐺) = ((𝐴 · 𝑃) + 𝐶) |
26 | 25 | eqcomi 2827 | . . 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 10636 | . . . . 5 ⊢ (𝑇 · 𝐸) ∈ ℂ |
30 | 2, 11 | mulcli 10636 | . . . . 5 ⊢ (𝑇 · 𝐺) ∈ ℂ |
31 | npcan 10883 | . . . . 5 ⊢ (((𝑇 · 𝐸) ∈ ℂ ∧ (𝑇 · 𝐺) ∈ ℂ) → (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) = (𝑇 · 𝐸)) | |
32 | 29, 30, 31 | mp2an 688 | . . . 4 ⊢ (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) = (𝑇 · 𝐸) |
33 | 32 | oveq1i 7155 | . . 3 ⊢ ((((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) + 𝐹) = ((𝑇 · 𝐸) + 𝐹) |
34 | 29, 30 | subcli 10950 | . . . 4 ⊢ ((𝑇 · 𝐸) − (𝑇 · 𝐺)) ∈ ℂ |
35 | nummac.9 | . . . . 5 ⊢ 𝐹 ∈ ℕ0 | |
36 | 35 | nn0cni 11897 | . . . 4 ⊢ 𝐹 ∈ ℂ |
37 | 34, 30, 36 | addassi 10639 | . . 3 ⊢ ((((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) + 𝐹) = (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + ((𝑇 · 𝐺) + 𝐹)) |
38 | 33, 37 | eqtr3i 2843 | . 2 ⊢ ((𝑇 · 𝐸) + 𝐹) = (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + ((𝑇 · 𝐺) + 𝐹)) |
39 | 19, 28, 38 | 3eqtr4i 2851 | 1 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 (class class class)co 7145 ℂcc 10523 + caddc 10528 · cmul 10530 − cmin 10858 ℕ0cn0 11885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 df-sub 10860 df-nn 11627 df-n0 11886 |
This theorem is referenced by: numma2c 12132 numaddc 12134 nummul1c 12135 decmac 12138 |
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