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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 1594 and df-cad 1608 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 15826 | . . 3 ⊢ (𝜑 → (𝐴 smul 𝐵) = {𝑘 ∈ ℕ0 ∣ 𝑘 ∈ (𝑃‘(𝑘 + 1))}) |
5 | 4 | eleq2d 2898 | . 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 7179 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑁 + 1))) | |
9 | 7, 8 | eleq12d 2907 | . . . 4 ⊢ (𝑘 = 𝑁 → (𝑘 ∈ (𝑃‘(𝑘 + 1)) ↔ 𝑁 ∈ (𝑃‘(𝑁 + 1)))) |
10 | 9 | elrab3 3681 | . . 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 281 | 1 ⊢ (𝜑 → (𝑁 ∈ (𝐴 smul 𝐵) ↔ 𝑁 ∈ (𝑃‘(𝑁 + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3142 ⊆ wss 3936 ∅c0 4291 ifcif 4467 𝒫 cpw 4539 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 0cc0 10537 1c1 10538 + caddc 10540 − cmin 10870 ℕ0cn0 11898 seqcseq 13370 sadd csad 15769 smul csmu 15770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-1cn 10595 ax-addcl 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-nn 11639 df-n0 11899 df-seq 13371 df-smu 15825 |
This theorem is referenced by: smuval2 15831 smupvallem 15832 smu01lem 15834 |
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