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Theorem smufval 16511
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 12530 . . . 4 0 ∈ V
32elpw2 5340 . . 3 (𝐴 ∈ 𝒫 ℕ0𝐴 ⊆ ℕ0)
41, 3sylibr 234 . 2 (𝜑𝐴 ∈ 𝒫 ℕ0)
5 smuval.b . . 3 (𝜑𝐵 ⊆ ℕ0)
62elpw2 5340 . . 3 (𝐵 ∈ 𝒫 ℕ0𝐵 ⊆ ℕ0)
75, 6sylibr 234 . 2 (𝜑𝐵 ∈ 𝒫 ℕ0)
8 simp1l 1196 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → 𝑥 = 𝐴)
98eleq2d 2825 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → (𝑚𝑥𝑚𝐴))
10 simp1r 1197 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → 𝑦 = 𝐵)
1110eleq2d 2825 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → ((𝑛𝑚) ∈ 𝑦 ↔ (𝑛𝑚) ∈ 𝐵))
129, 11anbi12d 632 . . . . . . . . . . 11 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → ((𝑚𝑥 ∧ (𝑛𝑚) ∈ 𝑦) ↔ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)))
1312rabbidv 3441 . . . . . . . . . 10 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → {𝑛 ∈ ℕ0 ∣ (𝑚𝑥 ∧ (𝑛𝑚) ∈ 𝑦)} = {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})
1413oveq2d 7447 . . . . . . . . 9 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝑥 ∧ (𝑛𝑚) ∈ 𝑦)}) = (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))
1514mpoeq3dva 7510 . . . . . . . 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 14046 . . . . . . 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 2793 . . . . . 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 6909 . . . . 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 2825 . . . 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 3441 . . 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 16510 . . 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 5345 . . 3 {𝑘 ∈ ℕ0𝑘 ∈ (𝑃‘(𝑘 + 1))} ∈ V
2421, 22, 23ovmpoa 7588 . 2 ((𝐴 ∈ 𝒫 ℕ0𝐵 ∈ 𝒫 ℕ0) → (𝐴 smul 𝐵) = {𝑘 ∈ ℕ0𝑘 ∈ (𝑃‘(𝑘 + 1))})
254, 7, 24syl2anc 584 1 (𝜑 → (𝐴 smul 𝐵) = {𝑘 ∈ ℕ0𝑘 ∈ (𝑃‘(𝑘 + 1))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  {crab 3433  wss 3963  c0 4339  ifcif 4531  𝒫 cpw 4605  cmpt 5231  cfv 6563  (class class class)co 7431  cmpo 7433  0cc0 11153  1c1 11154   + caddc 11156  cmin 11490  0cn0 12524  seqcseq 14039   sadd csad 16454   smul csmu 16455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265  df-n0 12525  df-seq 14040  df-smu 16510
This theorem is referenced by:  smuval  16515  smupvallem  16517  smucl  16518
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