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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 11181 | . . . 4 ⊢ 𝑇 ∈ ℂ |
3 | numma.2 | . . . . . . . . 9 ⊢ 𝐴 ∈ ℕ0 | |
4 | 3 | nn0cni 11181 | . . . . . . . 8 ⊢ 𝐴 ∈ ℂ |
5 | nummac.8 | . . . . . . . . 9 ⊢ 𝑃 ∈ ℕ0 | |
6 | 5 | nn0cni 11181 | . . . . . . . 8 ⊢ 𝑃 ∈ ℂ |
7 | 4, 6 | mulcli 9924 | . . . . . . 7 ⊢ (𝐴 · 𝑃) ∈ ℂ |
8 | numma.4 | . . . . . . . 8 ⊢ 𝐶 ∈ ℕ0 | |
9 | 8 | nn0cni 11181 | . . . . . . 7 ⊢ 𝐶 ∈ ℂ |
10 | nummac.10 | . . . . . . . 8 ⊢ 𝐺 ∈ ℕ0 | |
11 | 10 | nn0cni 11181 | . . . . . . 7 ⊢ 𝐺 ∈ ℂ |
12 | 7, 9, 11 | addassi 9927 | . . . . . 6 ⊢ (((𝐴 · 𝑃) + 𝐶) + 𝐺) = ((𝐴 · 𝑃) + (𝐶 + 𝐺)) |
13 | nummac.11 | . . . . . 6 ⊢ ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = 𝐸 | |
14 | 12, 13 | eqtri 2632 | . . . . 5 ⊢ (((𝐴 · 𝑃) + 𝐶) + 𝐺) = 𝐸 |
15 | 7, 9 | addcli 9923 | . . . . . 6 ⊢ ((𝐴 · 𝑃) + 𝐶) ∈ ℂ |
16 | 15, 11 | addcli 9923 | . . . . 5 ⊢ (((𝐴 · 𝑃) + 𝐶) + 𝐺) ∈ ℂ |
17 | 14, 16 | eqeltrri 2685 | . . . 4 ⊢ 𝐸 ∈ ℂ |
18 | 2, 17, 11 | subdii 10358 | . . 3 ⊢ (𝑇 · (𝐸 − 𝐺)) = ((𝑇 · 𝐸) − (𝑇 · 𝐺)) |
19 | 18 | oveq1i 6559 | . 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 10248 | . . . . 5 ⊢ ((𝐸 − 𝐺) = ((𝐴 · 𝑃) + 𝐶) ↔ (((𝐴 · 𝑃) + 𝐶) + 𝐺) = 𝐸) |
25 | 14, 24 | mpbir 220 | . . . 4 ⊢ (𝐸 − 𝐺) = ((𝐴 · 𝑃) + 𝐶) |
26 | 25 | eqcomi 2619 | . . 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 11433 | . 2 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · (𝐸 − 𝐺)) + ((𝑇 · 𝐺) + 𝐹)) |
29 | 2, 17 | mulcli 9924 | . . . . 5 ⊢ (𝑇 · 𝐸) ∈ ℂ |
30 | 2, 11 | mulcli 9924 | . . . . 5 ⊢ (𝑇 · 𝐺) ∈ ℂ |
31 | npcan 10169 | . . . . 5 ⊢ (((𝑇 · 𝐸) ∈ ℂ ∧ (𝑇 · 𝐺) ∈ ℂ) → (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) = (𝑇 · 𝐸)) | |
32 | 29, 30, 31 | mp2an 704 | . . . 4 ⊢ (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) = (𝑇 · 𝐸) |
33 | 32 | oveq1i 6559 | . . 3 ⊢ ((((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) + 𝐹) = ((𝑇 · 𝐸) + 𝐹) |
34 | 29, 30 | subcli 10236 | . . . 4 ⊢ ((𝑇 · 𝐸) − (𝑇 · 𝐺)) ∈ ℂ |
35 | nummac.9 | . . . . 5 ⊢ 𝐹 ∈ ℕ0 | |
36 | 35 | nn0cni 11181 | . . . 4 ⊢ 𝐹 ∈ ℂ |
37 | 34, 30, 36 | addassi 9927 | . . 3 ⊢ ((((𝑇 · 𝐸) − (𝑇 · 𝐺)) + (𝑇 · 𝐺)) + 𝐹) = (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + ((𝑇 · 𝐺) + 𝐹)) |
38 | 33, 37 | eqtr3i 2634 | . 2 ⊢ ((𝑇 · 𝐸) + 𝐹) = (((𝑇 · 𝐸) − (𝑇 · 𝐺)) + ((𝑇 · 𝐺) + 𝐹)) |
39 | 19, 28, 38 | 3eqtr4i 2642 | 1 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 (class class class)co 6549 ℂcc 9813 + caddc 9818 · cmul 9820 − cmin 10145 ℕ0cn0 11169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-nn 10898 df-n0 11170 |
This theorem is referenced by: numma2c 11435 numaddc 11437 nummul1c 11438 decmac 11442 decmacOLD 11443 |
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