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| Mirrors > Home > MPE Home > Th. List > smuval | Structured version Visualization version GIF version | ||
| Description: Define the addition of two bit sequences, using df-had 1593 and df-cad 1606 bit operations. (Contributed by Mario Carneiro, 9-Sep-2016.) | 
| Ref | Expression | 
|---|---|
| smuval.a | ⊢ (𝜑 → 𝐴 ⊆ ℕ0) | 
| smuval.b | ⊢ (𝜑 → 𝐵 ⊆ ℕ0) | 
| smuval.p | ⊢ 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) | 
| smuval.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) | 
| Ref | Expression | 
|---|---|
| smuval | ⊢ (𝜑 → (𝑁 ∈ (𝐴 smul 𝐵) ↔ 𝑁 ∈ (𝑃‘(𝑁 + 1)))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | smuval.a | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℕ0) | |
| 2 | smuval.b | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℕ0) | |
| 3 | smuval.p | . . . 4 ⊢ 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) | |
| 4 | 1, 2, 3 | smufval 16515 | . . 3 ⊢ (𝜑 → (𝐴 smul 𝐵) = {𝑘 ∈ ℕ0 ∣ 𝑘 ∈ (𝑃‘(𝑘 + 1))}) | 
| 5 | 4 | eleq2d 2826 | . 2 ⊢ (𝜑 → (𝑁 ∈ (𝐴 smul 𝐵) ↔ 𝑁 ∈ {𝑘 ∈ ℕ0 ∣ 𝑘 ∈ (𝑃‘(𝑘 + 1))})) | 
| 6 | smuval.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
| 7 | id 22 | . . . . 5 ⊢ (𝑘 = 𝑁 → 𝑘 = 𝑁) | |
| 8 | fvoveq1 7455 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑁 + 1))) | |
| 9 | 7, 8 | eleq12d 2834 | . . . 4 ⊢ (𝑘 = 𝑁 → (𝑘 ∈ (𝑃‘(𝑘 + 1)) ↔ 𝑁 ∈ (𝑃‘(𝑁 + 1)))) | 
| 10 | 9 | elrab3 3692 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ {𝑘 ∈ ℕ0 ∣ 𝑘 ∈ (𝑃‘(𝑘 + 1))} ↔ 𝑁 ∈ (𝑃‘(𝑁 + 1)))) | 
| 11 | 6, 10 | syl 17 | . 2 ⊢ (𝜑 → (𝑁 ∈ {𝑘 ∈ ℕ0 ∣ 𝑘 ∈ (𝑃‘(𝑘 + 1))} ↔ 𝑁 ∈ (𝑃‘(𝑁 + 1)))) | 
| 12 | 5, 11 | bitrd 279 | 1 ⊢ (𝜑 → (𝑁 ∈ (𝐴 smul 𝐵) ↔ 𝑁 ∈ (𝑃‘(𝑁 + 1)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {crab 3435 ⊆ wss 3950 ∅c0 4332 ifcif 4524 𝒫 cpw 4599 ↦ cmpt 5224 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 0cc0 11156 1c1 11157 + caddc 11159 − cmin 11493 ℕ0cn0 12528 seqcseq 14043 sadd csad 16458 smul csmu 16459 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-1cn 11214 ax-addcl 11216 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-nn 12268 df-n0 12529 df-seq 14044 df-smu 16514 | 
| This theorem is referenced by: smuval2 16520 smupvallem 16521 smu01lem 16523 | 
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