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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 8992 | . 2 ⊢ 0 ∈ ℕ0 | |
4 | decrmanc.n | . 2 ⊢ 𝑁 ∈ ℕ0 | |
5 | decrmanc.m | . 2 ⊢ 𝑀 = ;𝐴𝐵 | |
6 | 4 | dec0h 9203 | . 2 ⊢ 𝑁 = ;0𝑁 |
7 | decrmanc.p | . 2 ⊢ 𝑃 ∈ ℕ0 | |
8 | decrmac.f | . 2 ⊢ 𝐹 ∈ ℕ0 | |
9 | decrmac.g | . 2 ⊢ 𝐺 ∈ ℕ0 | |
10 | 9 | nn0cni 8989 | . . . . 5 ⊢ 𝐺 ∈ ℂ |
11 | 10 | addid2i 7905 | . . . 4 ⊢ (0 + 𝐺) = 𝐺 |
12 | 11 | oveq2i 5785 | . . 3 ⊢ ((𝐴 · 𝑃) + (0 + 𝐺)) = ((𝐴 · 𝑃) + 𝐺) |
13 | decrmac.e | . . 3 ⊢ ((𝐴 · 𝑃) + 𝐺) = 𝐸 | |
14 | 12, 13 | eqtri 2160 | . 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 9233 | 1 ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 (class class class)co 5774 0cc0 7620 + caddc 7623 · cmul 7625 ℕ0cn0 8977 ;cdc 9182 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7935 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-5 8782 df-6 8783 df-7 8784 df-8 8785 df-9 8786 df-n0 8978 df-dec 9183 |
This theorem is referenced by: (None) |
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