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Theorem smufval 15676
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 11707 . . . 4 0 ∈ V
32elpw2 5098 . . 3 (𝐴 ∈ 𝒫 ℕ0𝐴 ⊆ ℕ0)
41, 3sylibr 226 . 2 (𝜑𝐴 ∈ 𝒫 ℕ0)
5 smuval.b . . 3 (𝜑𝐵 ⊆ ℕ0)
62elpw2 5098 . . 3 (𝐵 ∈ 𝒫 ℕ0𝐵 ⊆ ℕ0)
75, 6sylibr 226 . 2 (𝜑𝐵 ∈ 𝒫 ℕ0)
8 simp1l 1177 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → 𝑥 = 𝐴)
98eleq2d 2845 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → (𝑚𝑥𝑚𝐴))
10 simp1r 1178 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → 𝑦 = 𝐵)
1110eleq2d 2845 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → ((𝑛𝑚) ∈ 𝑦 ↔ (𝑛𝑚) ∈ 𝐵))
129, 11anbi12d 621 . . . . . . . . . . 11 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → ((𝑚𝑥 ∧ (𝑛𝑚) ∈ 𝑦) ↔ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)))
1312rabbidv 3397 . . . . . . . . . 10 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → {𝑛 ∈ ℕ0 ∣ (𝑚𝑥 ∧ (𝑛𝑚) ∈ 𝑦)} = {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})
1413oveq2d 6986 . . . . . . . . 9 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0𝑚 ∈ ℕ0) → (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝑥 ∧ (𝑛𝑚) ∈ 𝑦)}) = (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))
1514mpoeq3dva 7043 . . . . . . . 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 13184 . . . . . . 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, 17syl6eqr 2826 . . . . . 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 6495 . . . . 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 2845 . . . 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 3397 . . 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 15675 . . 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 5085 . . 3 {𝑘 ∈ ℕ0𝑘 ∈ (𝑃‘(𝑘 + 1))} ∈ V
2421, 22, 23ovmpoa 7115 . 2 ((𝐴 ∈ 𝒫 ℕ0𝐵 ∈ 𝒫 ℕ0) → (𝐴 smul 𝐵) = {𝑘 ∈ ℕ0𝑘 ∈ (𝑃‘(𝑘 + 1))})
254, 7, 24syl2anc 576 1 (𝜑 → (𝐴 smul 𝐵) = {𝑘 ∈ ℕ0𝑘 ∈ (𝑃‘(𝑘 + 1))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wcel 2048  {crab 3086  wss 3825  c0 4173  ifcif 4344  𝒫 cpw 4416  cmpt 5002  cfv 6182  (class class class)co 6970  cmpo 6972  0cc0 10327  1c1 10328   + caddc 10330  cmin 10662  0cn0 11700  seqcseq 13177   sadd csad 15619   smul csmu 15620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-1cn 10385  ax-addcl 10387
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-nn 11432  df-n0 11701  df-seq 13178  df-smu 15675
This theorem is referenced by:  smuval  15680  smupvallem  15682  smucl  15683
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