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Mirrors > Home > MPE Home > Th. List > numma2c | 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) |
numma2c.8 | ⊢ 𝑃 ∈ ℕ0 |
numma2c.9 | ⊢ 𝐹 ∈ ℕ0 |
numma2c.10 | ⊢ 𝐺 ∈ ℕ0 |
numma2c.11 | ⊢ ((𝑃 · 𝐴) + (𝐶 + 𝐺)) = 𝐸 |
numma2c.12 | ⊢ ((𝑃 · 𝐵) + 𝐷) = ((𝑇 · 𝐺) + 𝐹) |
Ref | Expression |
---|---|
numma2c | ⊢ ((𝑃 · 𝑀) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numma2c.8 | . . . . 5 ⊢ 𝑃 ∈ ℕ0 | |
2 | 1 | nn0cni 11912 | . . . 4 ⊢ 𝑃 ∈ ℂ |
3 | numma.6 | . . . . . 6 ⊢ 𝑀 = ((𝑇 · 𝐴) + 𝐵) | |
4 | numma.1 | . . . . . . 7 ⊢ 𝑇 ∈ ℕ0 | |
5 | numma.2 | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
6 | numma.3 | . . . . . . 7 ⊢ 𝐵 ∈ ℕ0 | |
7 | 4, 5, 6 | numcl 12114 | . . . . . 6 ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0 |
8 | 3, 7 | eqeltri 2911 | . . . . 5 ⊢ 𝑀 ∈ ℕ0 |
9 | 8 | nn0cni 11912 | . . . 4 ⊢ 𝑀 ∈ ℂ |
10 | 2, 9 | mulcomi 10651 | . . 3 ⊢ (𝑃 · 𝑀) = (𝑀 · 𝑃) |
11 | 10 | oveq1i 7168 | . 2 ⊢ ((𝑃 · 𝑀) + 𝑁) = ((𝑀 · 𝑃) + 𝑁) |
12 | numma.4 | . . 3 ⊢ 𝐶 ∈ ℕ0 | |
13 | numma.5 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
14 | numma.7 | . . 3 ⊢ 𝑁 = ((𝑇 · 𝐶) + 𝐷) | |
15 | numma2c.9 | . . 3 ⊢ 𝐹 ∈ ℕ0 | |
16 | numma2c.10 | . . 3 ⊢ 𝐺 ∈ ℕ0 | |
17 | 5 | nn0cni 11912 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
18 | 17, 2 | mulcomi 10651 | . . . . 5 ⊢ (𝐴 · 𝑃) = (𝑃 · 𝐴) |
19 | 18 | oveq1i 7168 | . . . 4 ⊢ ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = ((𝑃 · 𝐴) + (𝐶 + 𝐺)) |
20 | numma2c.11 | . . . 4 ⊢ ((𝑃 · 𝐴) + (𝐶 + 𝐺)) = 𝐸 | |
21 | 19, 20 | eqtri 2846 | . . 3 ⊢ ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = 𝐸 |
22 | 6 | nn0cni 11912 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
23 | 22, 2 | mulcomi 10651 | . . . . 5 ⊢ (𝐵 · 𝑃) = (𝑃 · 𝐵) |
24 | 23 | oveq1i 7168 | . . . 4 ⊢ ((𝐵 · 𝑃) + 𝐷) = ((𝑃 · 𝐵) + 𝐷) |
25 | numma2c.12 | . . . 4 ⊢ ((𝑃 · 𝐵) + 𝐷) = ((𝑇 · 𝐺) + 𝐹) | |
26 | 24, 25 | eqtri 2846 | . . 3 ⊢ ((𝐵 · 𝑃) + 𝐷) = ((𝑇 · 𝐺) + 𝐹) |
27 | 4, 5, 6, 12, 13, 3, 14, 1, 15, 16, 21, 26 | nummac 12146 | . 2 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
28 | 11, 27 | eqtri 2846 | 1 ⊢ ((𝑃 · 𝑀) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7158 + caddc 10542 · cmul 10544 ℕ0cn0 11900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 df-nn 11641 df-n0 11901 |
This theorem is referenced by: decma2c 12154 |
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