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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 7160 | . . 3 ⊢ (𝑀 · 𝑃) = (((𝑇 · 𝐴) + 𝐵) · 𝑃) |
3 | numma.7 | . . 3 ⊢ 𝑁 = ((𝑇 · 𝐶) + 𝐷) | |
4 | 2, 3 | oveq12i 7162 | . 2 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((((𝑇 · 𝐴) + 𝐵) · 𝑃) + ((𝑇 · 𝐶) + 𝐷)) |
5 | numma.1 | . . . . . . 7 ⊢ 𝑇 ∈ ℕ0 | |
6 | 5 | nn0cni 11903 | . . . . . 6 ⊢ 𝑇 ∈ ℂ |
7 | numma.2 | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
8 | 7 | nn0cni 11903 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
9 | numma.8 | . . . . . . . 8 ⊢ 𝑃 ∈ ℕ0 | |
10 | 9 | nn0cni 11903 | . . . . . . 7 ⊢ 𝑃 ∈ ℂ |
11 | 8, 10 | mulcli 10642 | . . . . . 6 ⊢ (𝐴 · 𝑃) ∈ ℂ |
12 | numma.4 | . . . . . . 7 ⊢ 𝐶 ∈ ℕ0 | |
13 | 12 | nn0cni 11903 | . . . . . 6 ⊢ 𝐶 ∈ ℂ |
14 | 6, 11, 13 | adddii 10647 | . . . . 5 ⊢ (𝑇 · ((𝐴 · 𝑃) + 𝐶)) = ((𝑇 · (𝐴 · 𝑃)) + (𝑇 · 𝐶)) |
15 | 6, 8, 10 | mulassi 10646 | . . . . . 6 ⊢ ((𝑇 · 𝐴) · 𝑃) = (𝑇 · (𝐴 · 𝑃)) |
16 | 15 | oveq1i 7160 | . . . . 5 ⊢ (((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) = ((𝑇 · (𝐴 · 𝑃)) + (𝑇 · 𝐶)) |
17 | 14, 16 | eqtr4i 2847 | . . . 4 ⊢ (𝑇 · ((𝐴 · 𝑃) + 𝐶)) = (((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) |
18 | 17 | oveq1i 7160 | . . 3 ⊢ ((𝑇 · ((𝐴 · 𝑃) + 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) = ((((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) |
19 | 6, 8 | mulcli 10642 | . . . . . 6 ⊢ (𝑇 · 𝐴) ∈ ℂ |
20 | numma.3 | . . . . . . 7 ⊢ 𝐵 ∈ ℕ0 | |
21 | 20 | nn0cni 11903 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
22 | 19, 21, 10 | adddiri 10648 | . . . . 5 ⊢ (((𝑇 · 𝐴) + 𝐵) · 𝑃) = (((𝑇 · 𝐴) · 𝑃) + (𝐵 · 𝑃)) |
23 | 22 | oveq1i 7160 | . . . 4 ⊢ ((((𝑇 · 𝐴) + 𝐵) · 𝑃) + ((𝑇 · 𝐶) + 𝐷)) = ((((𝑇 · 𝐴) · 𝑃) + (𝐵 · 𝑃)) + ((𝑇 · 𝐶) + 𝐷)) |
24 | 19, 10 | mulcli 10642 | . . . . 5 ⊢ ((𝑇 · 𝐴) · 𝑃) ∈ ℂ |
25 | 6, 13 | mulcli 10642 | . . . . 5 ⊢ (𝑇 · 𝐶) ∈ ℂ |
26 | 21, 10 | mulcli 10642 | . . . . 5 ⊢ (𝐵 · 𝑃) ∈ ℂ |
27 | numma.5 | . . . . . 6 ⊢ 𝐷 ∈ ℕ0 | |
28 | 27 | nn0cni 11903 | . . . . 5 ⊢ 𝐷 ∈ ℂ |
29 | 24, 25, 26, 28 | add4i 10858 | . . . 4 ⊢ ((((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) = ((((𝑇 · 𝐴) · 𝑃) + (𝐵 · 𝑃)) + ((𝑇 · 𝐶) + 𝐷)) |
30 | 23, 29 | eqtr4i 2847 | . . 3 ⊢ ((((𝑇 · 𝐴) + 𝐵) · 𝑃) + ((𝑇 · 𝐶) + 𝐷)) = ((((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) |
31 | 18, 30 | eqtr4i 2847 | . 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 7161 | . . 3 ⊢ (𝑇 · ((𝐴 · 𝑃) + 𝐶)) = (𝑇 · 𝐸) |
34 | numma.10 | . . 3 ⊢ ((𝐵 · 𝑃) + 𝐷) = 𝐹 | |
35 | 33, 34 | oveq12i 7162 | . 2 ⊢ ((𝑇 · ((𝐴 · 𝑃) + 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) = ((𝑇 · 𝐸) + 𝐹) |
36 | 4, 31, 35 | 3eqtr2i 2850 | 1 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 (class class class)co 7150 + caddc 10534 · cmul 10536 ℕ0cn0 11891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 df-nn 11633 df-n0 11892 |
This theorem is referenced by: nummac 12137 numadd 12139 decma 12143 |
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