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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 1587 and df-cad 1600 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 16415 | . . 3 ⊢ (𝜑 → (𝐴 smul 𝐵) = {𝑘 ∈ ℕ0 ∣ 𝑘 ∈ (𝑃‘(𝑘 + 1))}) |
5 | 4 | eleq2d 2811 | . 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 7424 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑁 + 1))) | |
9 | 7, 8 | eleq12d 2819 | . . . 4 ⊢ (𝑘 = 𝑁 → (𝑘 ∈ (𝑃‘(𝑘 + 1)) ↔ 𝑁 ∈ (𝑃‘(𝑁 + 1)))) |
10 | 9 | elrab3 3676 | . . 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 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3424 ⊆ wss 3940 ∅c0 4314 ifcif 4520 𝒫 cpw 4594 ↦ cmpt 5221 ‘cfv 6533 (class class class)co 7401 ∈ cmpo 7403 0cc0 11106 1c1 11107 + caddc 11109 − cmin 11441 ℕ0cn0 12469 seqcseq 13963 sadd csad 16358 smul csmu 16359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-1cn 11164 ax-addcl 11166 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-nn 12210 df-n0 12470 df-seq 13964 df-smu 16414 |
This theorem is referenced by: smuval2 16420 smupvallem 16421 smu01lem 16423 |
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