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Mirrors > Home > ILE Home > Th. List > decrmac | GIF version |
Description: Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by AV, 16-Sep-2021.) |
Ref | Expression |
---|---|
decrmanc.a | ⊢ 𝐴 ∈ ℕ0 |
decrmanc.b | ⊢ 𝐵 ∈ ℕ0 |
decrmanc.n | ⊢ 𝑁 ∈ ℕ0 |
decrmanc.m | ⊢ 𝑀 = ;𝐴𝐵 |
decrmanc.p | ⊢ 𝑃 ∈ ℕ0 |
decrmac.f | ⊢ 𝐹 ∈ ℕ0 |
decrmac.g | ⊢ 𝐺 ∈ ℕ0 |
decrmac.e | ⊢ ((𝐴 · 𝑃) + 𝐺) = 𝐸 |
decrmac.2 | ⊢ ((𝐵 · 𝑃) + 𝑁) = ;𝐺𝐹 |
Ref | Expression |
---|---|
decrmac | ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decrmanc.a | . 2 ⊢ 𝐴 ∈ ℕ0 | |
2 | decrmanc.b | . 2 ⊢ 𝐵 ∈ ℕ0 | |
3 | 0nn0 8985 | . 2 ⊢ 0 ∈ ℕ0 | |
4 | decrmanc.n | . 2 ⊢ 𝑁 ∈ ℕ0 | |
5 | decrmanc.m | . 2 ⊢ 𝑀 = ;𝐴𝐵 | |
6 | 4 | dec0h 9196 | . 2 ⊢ 𝑁 = ;0𝑁 |
7 | decrmanc.p | . 2 ⊢ 𝑃 ∈ ℕ0 | |
8 | decrmac.f | . 2 ⊢ 𝐹 ∈ ℕ0 | |
9 | decrmac.g | . 2 ⊢ 𝐺 ∈ ℕ0 | |
10 | 9 | nn0cni 8982 | . . . . 5 ⊢ 𝐺 ∈ ℂ |
11 | 10 | addid2i 7898 | . . . 4 ⊢ (0 + 𝐺) = 𝐺 |
12 | 11 | oveq2i 5778 | . . 3 ⊢ ((𝐴 · 𝑃) + (0 + 𝐺)) = ((𝐴 · 𝑃) + 𝐺) |
13 | decrmac.e | . . 3 ⊢ ((𝐴 · 𝑃) + 𝐺) = 𝐸 | |
14 | 12, 13 | eqtri 2158 | . 2 ⊢ ((𝐴 · 𝑃) + (0 + 𝐺)) = 𝐸 |
15 | decrmac.2 | . 2 ⊢ ((𝐵 · 𝑃) + 𝑁) = ;𝐺𝐹 | |
16 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15 | decmac 9226 | 1 ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 (class class class)co 5767 0cc0 7613 + caddc 7616 · cmul 7618 ℕ0cn0 8970 ;cdc 9175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-sub 7928 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-5 8775 df-6 8776 df-7 8777 df-8 8778 df-9 8779 df-n0 8971 df-dec 9176 |
This theorem is referenced by: (None) |
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