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Mirrors > Home > MPE Home > Th. List > numma | Structured version Visualization version GIF version |
Description: Perform a multiply-add of two decimal integers 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (no 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) |
numma.8 | ⊢ 𝑃 ∈ ℕ0 |
numma.9 | ⊢ ((𝐴 · 𝑃) + 𝐶) = 𝐸 |
numma.10 | ⊢ ((𝐵 · 𝑃) + 𝐷) = 𝐹 |
Ref | Expression |
---|---|
numma | ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numma.6 | . . . 4 ⊢ 𝑀 = ((𝑇 · 𝐴) + 𝐵) | |
2 | 1 | oveq1i 7458 | . . 3 ⊢ (𝑀 · 𝑃) = (((𝑇 · 𝐴) + 𝐵) · 𝑃) |
3 | numma.7 | . . 3 ⊢ 𝑁 = ((𝑇 · 𝐶) + 𝐷) | |
4 | 2, 3 | oveq12i 7460 | . 2 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((((𝑇 · 𝐴) + 𝐵) · 𝑃) + ((𝑇 · 𝐶) + 𝐷)) |
5 | numma.1 | . . . . . . 7 ⊢ 𝑇 ∈ ℕ0 | |
6 | 5 | nn0cni 12565 | . . . . . 6 ⊢ 𝑇 ∈ ℂ |
7 | numma.2 | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
8 | 7 | nn0cni 12565 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
9 | numma.8 | . . . . . . . 8 ⊢ 𝑃 ∈ ℕ0 | |
10 | 9 | nn0cni 12565 | . . . . . . 7 ⊢ 𝑃 ∈ ℂ |
11 | 8, 10 | mulcli 11297 | . . . . . 6 ⊢ (𝐴 · 𝑃) ∈ ℂ |
12 | numma.4 | . . . . . . 7 ⊢ 𝐶 ∈ ℕ0 | |
13 | 12 | nn0cni 12565 | . . . . . 6 ⊢ 𝐶 ∈ ℂ |
14 | 6, 11, 13 | adddii 11302 | . . . . 5 ⊢ (𝑇 · ((𝐴 · 𝑃) + 𝐶)) = ((𝑇 · (𝐴 · 𝑃)) + (𝑇 · 𝐶)) |
15 | 6, 8, 10 | mulassi 11301 | . . . . . 6 ⊢ ((𝑇 · 𝐴) · 𝑃) = (𝑇 · (𝐴 · 𝑃)) |
16 | 15 | oveq1i 7458 | . . . . 5 ⊢ (((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) = ((𝑇 · (𝐴 · 𝑃)) + (𝑇 · 𝐶)) |
17 | 14, 16 | eqtr4i 2771 | . . . 4 ⊢ (𝑇 · ((𝐴 · 𝑃) + 𝐶)) = (((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) |
18 | 17 | oveq1i 7458 | . . 3 ⊢ ((𝑇 · ((𝐴 · 𝑃) + 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) = ((((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) |
19 | 6, 8 | mulcli 11297 | . . . . . 6 ⊢ (𝑇 · 𝐴) ∈ ℂ |
20 | numma.3 | . . . . . . 7 ⊢ 𝐵 ∈ ℕ0 | |
21 | 20 | nn0cni 12565 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
22 | 19, 21, 10 | adddiri 11303 | . . . . 5 ⊢ (((𝑇 · 𝐴) + 𝐵) · 𝑃) = (((𝑇 · 𝐴) · 𝑃) + (𝐵 · 𝑃)) |
23 | 22 | oveq1i 7458 | . . . 4 ⊢ ((((𝑇 · 𝐴) + 𝐵) · 𝑃) + ((𝑇 · 𝐶) + 𝐷)) = ((((𝑇 · 𝐴) · 𝑃) + (𝐵 · 𝑃)) + ((𝑇 · 𝐶) + 𝐷)) |
24 | 19, 10 | mulcli 11297 | . . . . 5 ⊢ ((𝑇 · 𝐴) · 𝑃) ∈ ℂ |
25 | 6, 13 | mulcli 11297 | . . . . 5 ⊢ (𝑇 · 𝐶) ∈ ℂ |
26 | 21, 10 | mulcli 11297 | . . . . 5 ⊢ (𝐵 · 𝑃) ∈ ℂ |
27 | numma.5 | . . . . . 6 ⊢ 𝐷 ∈ ℕ0 | |
28 | 27 | nn0cni 12565 | . . . . 5 ⊢ 𝐷 ∈ ℂ |
29 | 24, 25, 26, 28 | add4i 11514 | . . . 4 ⊢ ((((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) = ((((𝑇 · 𝐴) · 𝑃) + (𝐵 · 𝑃)) + ((𝑇 · 𝐶) + 𝐷)) |
30 | 23, 29 | eqtr4i 2771 | . . 3 ⊢ ((((𝑇 · 𝐴) + 𝐵) · 𝑃) + ((𝑇 · 𝐶) + 𝐷)) = ((((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) |
31 | 18, 30 | eqtr4i 2771 | . 2 ⊢ ((𝑇 · ((𝐴 · 𝑃) + 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) = ((((𝑇 · 𝐴) + 𝐵) · 𝑃) + ((𝑇 · 𝐶) + 𝐷)) |
32 | numma.9 | . . . 4 ⊢ ((𝐴 · 𝑃) + 𝐶) = 𝐸 | |
33 | 32 | oveq2i 7459 | . . 3 ⊢ (𝑇 · ((𝐴 · 𝑃) + 𝐶)) = (𝑇 · 𝐸) |
34 | numma.10 | . . 3 ⊢ ((𝐵 · 𝑃) + 𝐷) = 𝐹 | |
35 | 33, 34 | oveq12i 7460 | . 2 ⊢ ((𝑇 · ((𝐴 · 𝑃) + 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) = ((𝑇 · 𝐸) + 𝐹) |
36 | 4, 31, 35 | 3eqtr2i 2774 | 1 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2108 (class class class)co 7448 + caddc 11187 · cmul 11189 ℕ0cn0 12553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-ltxr 11329 df-nn 12294 df-n0 12554 |
This theorem is referenced by: nummac 12803 numadd 12805 decma 12809 |
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