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| 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 7441 | . . 3 ⊢ (𝑀 · 𝑃) = (((𝑇 · 𝐴) + 𝐵) · 𝑃) | 
| 3 | numma.7 | . . 3 ⊢ 𝑁 = ((𝑇 · 𝐶) + 𝐷) | |
| 4 | 2, 3 | oveq12i 7443 | . 2 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((((𝑇 · 𝐴) + 𝐵) · 𝑃) + ((𝑇 · 𝐶) + 𝐷)) | 
| 5 | numma.1 | . . . . . . 7 ⊢ 𝑇 ∈ ℕ0 | |
| 6 | 5 | nn0cni 12538 | . . . . . 6 ⊢ 𝑇 ∈ ℂ | 
| 7 | numma.2 | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
| 8 | 7 | nn0cni 12538 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ | 
| 9 | numma.8 | . . . . . . . 8 ⊢ 𝑃 ∈ ℕ0 | |
| 10 | 9 | nn0cni 12538 | . . . . . . 7 ⊢ 𝑃 ∈ ℂ | 
| 11 | 8, 10 | mulcli 11268 | . . . . . 6 ⊢ (𝐴 · 𝑃) ∈ ℂ | 
| 12 | numma.4 | . . . . . . 7 ⊢ 𝐶 ∈ ℕ0 | |
| 13 | 12 | nn0cni 12538 | . . . . . 6 ⊢ 𝐶 ∈ ℂ | 
| 14 | 6, 11, 13 | adddii 11273 | . . . . 5 ⊢ (𝑇 · ((𝐴 · 𝑃) + 𝐶)) = ((𝑇 · (𝐴 · 𝑃)) + (𝑇 · 𝐶)) | 
| 15 | 6, 8, 10 | mulassi 11272 | . . . . . 6 ⊢ ((𝑇 · 𝐴) · 𝑃) = (𝑇 · (𝐴 · 𝑃)) | 
| 16 | 15 | oveq1i 7441 | . . . . 5 ⊢ (((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) = ((𝑇 · (𝐴 · 𝑃)) + (𝑇 · 𝐶)) | 
| 17 | 14, 16 | eqtr4i 2768 | . . . 4 ⊢ (𝑇 · ((𝐴 · 𝑃) + 𝐶)) = (((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) | 
| 18 | 17 | oveq1i 7441 | . . 3 ⊢ ((𝑇 · ((𝐴 · 𝑃) + 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) = ((((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) | 
| 19 | 6, 8 | mulcli 11268 | . . . . . 6 ⊢ (𝑇 · 𝐴) ∈ ℂ | 
| 20 | numma.3 | . . . . . . 7 ⊢ 𝐵 ∈ ℕ0 | |
| 21 | 20 | nn0cni 12538 | . . . . . 6 ⊢ 𝐵 ∈ ℂ | 
| 22 | 19, 21, 10 | adddiri 11274 | . . . . 5 ⊢ (((𝑇 · 𝐴) + 𝐵) · 𝑃) = (((𝑇 · 𝐴) · 𝑃) + (𝐵 · 𝑃)) | 
| 23 | 22 | oveq1i 7441 | . . . 4 ⊢ ((((𝑇 · 𝐴) + 𝐵) · 𝑃) + ((𝑇 · 𝐶) + 𝐷)) = ((((𝑇 · 𝐴) · 𝑃) + (𝐵 · 𝑃)) + ((𝑇 · 𝐶) + 𝐷)) | 
| 24 | 19, 10 | mulcli 11268 | . . . . 5 ⊢ ((𝑇 · 𝐴) · 𝑃) ∈ ℂ | 
| 25 | 6, 13 | mulcli 11268 | . . . . 5 ⊢ (𝑇 · 𝐶) ∈ ℂ | 
| 26 | 21, 10 | mulcli 11268 | . . . . 5 ⊢ (𝐵 · 𝑃) ∈ ℂ | 
| 27 | numma.5 | . . . . . 6 ⊢ 𝐷 ∈ ℕ0 | |
| 28 | 27 | nn0cni 12538 | . . . . 5 ⊢ 𝐷 ∈ ℂ | 
| 29 | 24, 25, 26, 28 | add4i 11486 | . . . 4 ⊢ ((((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) = ((((𝑇 · 𝐴) · 𝑃) + (𝐵 · 𝑃)) + ((𝑇 · 𝐶) + 𝐷)) | 
| 30 | 23, 29 | eqtr4i 2768 | . . 3 ⊢ ((((𝑇 · 𝐴) + 𝐵) · 𝑃) + ((𝑇 · 𝐶) + 𝐷)) = ((((𝑇 · 𝐴) · 𝑃) + (𝑇 · 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) | 
| 31 | 18, 30 | eqtr4i 2768 | . 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 7442 | . . 3 ⊢ (𝑇 · ((𝐴 · 𝑃) + 𝐶)) = (𝑇 · 𝐸) | 
| 34 | numma.10 | . . 3 ⊢ ((𝐵 · 𝑃) + 𝐷) = 𝐹 | |
| 35 | 33, 34 | oveq12i 7443 | . 2 ⊢ ((𝑇 · ((𝐴 · 𝑃) + 𝐶)) + ((𝐵 · 𝑃) + 𝐷)) = ((𝑇 · 𝐸) + 𝐹) | 
| 36 | 4, 31, 35 | 3eqtr2i 2771 | 1 ⊢ ((𝑀 · 𝑃) + 𝑁) = ((𝑇 · 𝐸) + 𝐹) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7431 + caddc 11158 · cmul 11160 ℕ0cn0 12526 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-nn 12267 df-n0 12527 | 
| This theorem is referenced by: nummac 12778 numadd 12780 decma 12784 | 
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