Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > decrmanc | GIF version |
Description: Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (no 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 |
decrmanc.e | ⊢ (𝐴 · 𝑃) = 𝐸 |
decrmanc.f | ⊢ ((𝐵 · 𝑃) + 𝑁) = 𝐹 |
Ref | Expression |
---|---|
decrmanc | ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decrmanc.a | . 2 ⊢ 𝐴 ∈ ℕ0 | |
2 | decrmanc.b | . 2 ⊢ 𝐵 ∈ ℕ0 | |
3 | 0nn0 8303 | . 2 ⊢ 0 ∈ ℕ0 | |
4 | decrmanc.n | . 2 ⊢ 𝑁 ∈ ℕ0 | |
5 | decrmanc.m | . 2 ⊢ 𝑀 = ;𝐴𝐵 | |
6 | 4 | dec0h 8498 | . 2 ⊢ 𝑁 = ;0𝑁 |
7 | decrmanc.p | . 2 ⊢ 𝑃 ∈ ℕ0 | |
8 | 1, 7 | nn0mulcli 8326 | . . . . 5 ⊢ (𝐴 · 𝑃) ∈ ℕ0 |
9 | 8 | nn0cni 8300 | . . . 4 ⊢ (𝐴 · 𝑃) ∈ ℂ |
10 | 9 | addid1i 7250 | . . 3 ⊢ ((𝐴 · 𝑃) + 0) = (𝐴 · 𝑃) |
11 | decrmanc.e | . . 3 ⊢ (𝐴 · 𝑃) = 𝐸 | |
12 | 10, 11 | eqtri 2101 | . 2 ⊢ ((𝐴 · 𝑃) + 0) = 𝐸 |
13 | decrmanc.f | . 2 ⊢ ((𝐵 · 𝑃) + 𝑁) = 𝐹 | |
14 | 1, 2, 3, 4, 5, 6, 7, 12, 13 | decma 8527 | 1 ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹 |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ∈ wcel 1433 (class class class)co 5532 0cc0 6981 + caddc 6984 · cmul 6986 ℕ0cn0 8288 ;cdc 8477 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-5 8101 df-6 8102 df-7 8103 df-8 8104 df-9 8105 df-n0 8289 df-dec 8478 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |