Theorem smufval 16560

Description: The multiplication of two bit sequences as repeated sequence addition. (Contributed by Mario Carneiro, 9-Sep-2016.)

Hypotheses

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))))

Assertion

Ref	Expression
smufval	⊢ (𝜑 → (𝐴 smul 𝐵) = {𝑘 ∈ ℕ0 ∣ 𝑘 ∈ (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.

Step	Hyp	Ref	Expression
1		smuval.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℕ0)
2		nn0ex 12528	. . . 4 ⊢ ℕ0 ∈ V
3	2	elpw2 5310	. . 3 ⊢ (𝐴 ∈ 𝒫 ℕ0 ↔ 𝐴 ⊆ ℕ0)
4	1, 3	sylibr 237	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 ℕ0)
5		smuval.b	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ℕ0)
6	2	elpw2 5310	. . 3 ⊢ (𝐵 ∈ 𝒫 ℕ0 ↔ 𝐵 ⊆ ℕ0)
7	5, 6	sylibr 237	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝒫 ℕ0)
8		simp1l 1216	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0 ∧ 𝑚 ∈ ℕ0) → 𝑥 = 𝐴)
9	8	eleq2d 2852	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0 ∧ 𝑚 ∈ ℕ0) → (𝑚 ∈ 𝑥 ↔ 𝑚 ∈ 𝐴))
10		simp1r 1217	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0 ∧ 𝑚 ∈ ℕ0) → 𝑦 = 𝐵)
11	10	eleq2d 2852	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0 ∧ 𝑚 ∈ ℕ0) → ((𝑛 − 𝑚) ∈ 𝑦 ↔ (𝑛 − 𝑚) ∈ 𝐵))
12	9, 11	anbi12d 644	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0 ∧ 𝑚 ∈ ℕ0) → ((𝑚 ∈ 𝑥 ∧ (𝑛 − 𝑚) ∈ 𝑦) ↔ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)))
13	12	rabbidv 3426	. . . . . . . . . 10 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0 ∧ 𝑚 ∈ ℕ0) → {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝑥 ∧ (𝑛 − 𝑚) ∈ 𝑦)} = {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})
14	13	oveq2d 7439	. . . . . . . . 9 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑝 ∈ 𝒫 ℕ0 ∧ 𝑚 ∈ ℕ0) → (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝑥 ∧ (𝑛 − 𝑚) ∈ 𝑦)}) = (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))
15	14	mpoeq3dva 7500	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝑥 ∧ (𝑛 − 𝑚) ∈ 𝑦)})) = (𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})))
16	15	seqeq2d 14064	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝑥 ∧ (𝑛 − 𝑚) ∈ 𝑦)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))))
17		smuval.p	. . . . . . 7 ⊢ 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
18	16, 17	eqtr4di 2819	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝑥 ∧ (𝑛 − 𝑚) ∈ 𝑦)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = 𝑃)
19	18	fveq1d 6890	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝑥 ∧ (𝑛 − 𝑚) ∈ 𝑦)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)))
20	19	eleq2d 2852	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑘 ∈ (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝑥 ∧ (𝑛 − 𝑚) ∈ 𝑦)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) ↔ 𝑘 ∈ (𝑃‘(𝑘 + 1))))
21	20	rabbidv 3426	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → {𝑘 ∈ ℕ0 ∣ 𝑘 ∈ (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝑥 ∧ (𝑛 − 𝑚) ∈ 𝑦)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1))} = {𝑘 ∈ ℕ0 ∣ 𝑘 ∈ (𝑃‘(𝑘 + 1))})
22		df-smu 16559	. . 3 ⊢ smul = (𝑥 ∈ 𝒫 ℕ0, 𝑦 ∈ 𝒫 ℕ0 ↦ {𝑘 ∈ ℕ0 ∣ 𝑘 ∈ (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝑥 ∧ (𝑛 − 𝑚) ∈ 𝑦)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1))})
23	2	rabex 5314	. . 3 ⊢ {𝑘 ∈ ℕ0 ∣ 𝑘 ∈ (𝑃‘(𝑘 + 1))} ∈ V
24	21, 22, 23	ovmpoa 7578	. 2 ⊢ ((𝐴 ∈ 𝒫 ℕ0 ∧ 𝐵 ∈ 𝒫 ℕ0) → (𝐴 smul 𝐵) = {𝑘 ∈ ℕ0 ∣ 𝑘 ∈ (𝑃‘(𝑘 + 1))})
25	4, 7, 24	syl2anc 596	1 ⊢ (𝜑 → (𝐴 smul 𝐵) = {𝑘 ∈ ℕ0 ∣ 𝑘 ∈ (𝑃‘(𝑘 + 1))})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  {crab 3419   ⊆ wss 3908  ∅c0 4289  ifcif 4492  𝒫 cpw 4567   ↦ cmpt 5197  ‘cfv 6543  (class class class)co 7423   ∈ cmpo 7425  0cc0 11118  1c1 11119   + caddc 11121   − cmin 11459  ℕ₀cn0 12522  seqcseq 14057   sadd csad 16503   smul csmu 16504

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-1cn 11176  ax-addcl 11178

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  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 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-nn 12252  df-n0 12523  df-seq 14058  df-smu 16559

This theorem is used by:  smuval  16564  smupvallem  16566  smucl  16567