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Theorem smufval 16422
Description: The multiplication of two bit sequences as repeated sequence addition. (Contributed by Mario Carneiro, 9-Sep-2016.)
Hypotheses
Ref Expression
smuval.a (𝜑𝐴 ⊆ ℕ0)
smuval.b (𝜑𝐵 ⊆ ℕ0)
smuval.p 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
Assertion
Ref Expression
smufval (𝜑 → (𝐴 smul 𝐵) = {𝑘 ∈ ℕ0𝑘 ∈ (𝑃‘(𝑘 + 1))})
Distinct variable groups:   𝑘,𝑚,𝑛,𝑝,𝐴   𝜑,𝑘,𝑛   𝐵,𝑘,𝑚,𝑛,𝑝   𝑃,𝑘
Allowed substitution hints:   𝜑(𝑚,𝑝)   𝑃(𝑚,𝑛,𝑝)

Proof of Theorem smufval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smuval.a . . 3 (𝜑𝐴 ⊆ ℕ0)
2 nn0ex 12479 . . . 4 0 ∈ V
32elpw2 5338 . . 3 (𝐴 ∈ 𝒫 ℕ0𝐴 ⊆ ℕ0)
41, 3sylibr 233 . 2 (𝜑𝐴 ∈ 𝒫 ℕ0)
5 smuval.b . . 3 (𝜑𝐵 ⊆ ℕ0)
62elpw2 5338 . . 3 (𝐵 ∈ 𝒫 ℕ0𝐵 ⊆ ℕ0)
75, 6sylibr 233 . 2 (𝜑𝐵 ∈ 𝒫 ℕ0)
8 simp1l 1194 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → 𝑥 = 𝐴)
98eleq2d 2813 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → (𝑚𝑥𝑚𝐴))
10 simp1r 1195 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → 𝑦 = 𝐵)
1110eleq2d 2813 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → ((𝑛𝑚) ∈ 𝑦 ↔ (𝑛𝑚) ∈ 𝐵))
129, 11anbi12d 630 . . . . . . . . . . 11 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → ((𝑚𝑥 ∧ (𝑛𝑚) ∈ 𝑦) ↔ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)))
1312rabbidv 3434 . . . . . . . . . 10 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → {𝑛 ∈ ℕ0 ∣ (𝑚𝑥 ∧ (𝑛𝑚) ∈ 𝑦)} = {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})
1413oveq2d 7420 . . . . . . . . 9 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝑥 ∧ (𝑛𝑚) ∈ 𝑦)}) = (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))
1514mpoeq3dva 7481 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝑥 ∧ (𝑛𝑚) ∈ 𝑦)})) = (𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})))
1615seqeq2d 13976 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝑥 ∧ (𝑛𝑚) ∈ 𝑦)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))))
17 smuval.p . . . . . . 7 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
1816, 17eqtr4di 2784 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝑥 ∧ (𝑛𝑚) ∈ 𝑦)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = 𝑃)
1918fveq1d 6886 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝑥 ∧ (𝑛𝑚) ∈ 𝑦)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)))
2019eleq2d 2813 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑘 ∈ (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝑥 ∧ (𝑛𝑚) ∈ 𝑦)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) ↔ 𝑘 ∈ (𝑃‘(𝑘 + 1))))
2120rabbidv 3434 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → {𝑘 ∈ ℕ0𝑘 ∈ (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝑥 ∧ (𝑛𝑚) ∈ 𝑦)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1))} = {𝑘 ∈ ℕ0𝑘 ∈ (𝑃‘(𝑘 + 1))})
22 df-smu 16421 . . 3 smul = (𝑥 ∈ 𝒫 ℕ0, 𝑦 ∈ 𝒫 ℕ0 ↦ {𝑘 ∈ ℕ0𝑘 ∈ (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝑥 ∧ (𝑛𝑚) ∈ 𝑦)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1))})
232rabex 5325 . . 3 {𝑘 ∈ ℕ0𝑘 ∈ (𝑃‘(𝑘 + 1))} ∈ V
2421, 22, 23ovmpoa 7558 . 2 ((𝐴 ∈ 𝒫 ℕ0𝐵 ∈ 𝒫 ℕ0) → (𝐴 smul 𝐵) = {𝑘 ∈ ℕ0𝑘 ∈ (𝑃‘(𝑘 + 1))})
254, 7, 24syl2anc 583 1 (𝜑 → (𝐴 smul 𝐵) = {𝑘 ∈ ℕ0𝑘 ∈ (𝑃‘(𝑘 + 1))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  {crab 3426  wss 3943  c0 4317  ifcif 4523  𝒫 cpw 4597  cmpt 5224  cfv 6536  (class class class)co 7404  cmpo 7406  0cc0 11109  1c1 11110   + caddc 11112  cmin 11445  0cn0 12473  seqcseq 13969   sadd csad 16365   smul csmu 16366
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-1cn 11167  ax-addcl 11169
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-nn 12214  df-n0 12474  df-seq 13970  df-smu 16421
This theorem is referenced by:  smuval  16426  smupvallem  16428  smucl  16429
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