Theorem smupf 16396

Description: The sequence of partial sums of the sequence multiplication. (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
smupf	⊢ (𝜑 → 𝑃:ℕ0⟶𝒫 ℕ0)

Distinct variable groups:   𝑚,𝑛,𝑝,𝐴   𝜑,𝑛   𝐵,𝑚,𝑛,𝑝

Allowed substitution hints:   𝜑(𝑚,𝑝)   𝑃(𝑚,𝑛,𝑝)

Proof of Theorem smupf

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nn0 12407	. . . . 5 ⊢ 0 ∈ ℕ0
2		iftrue 4482	. . . . . 6 ⊢ (𝑛 = 0 → if(𝑛 = 0, ∅, (𝑛 − 1)) = ∅)
3		eqid 2733	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))
4		0ex 5249	. . . . . 6 ⊢ ∅ ∈ V
5	2, 3, 4	fvmpt 6938	. . . . 5 ⊢ (0 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘0) = ∅)
6	1, 5	mp1i 13	. . . 4 ⊢ (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘0) = ∅)
7		0elpw 5298	. . . 4 ⊢ ∅ ∈ 𝒫 ℕ0
8	6, 7	eqeltrdi 2841	. . 3 ⊢ (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘0) ∈ 𝒫 ℕ0)
9		df-ov 7358	. . . . 5 ⊢ (𝑥(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))𝑦) = ((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))‘⟨𝑥, 𝑦⟩)
10		elpwi 4558	. . . . . . . . . . 11 ⊢ (𝑝 ∈ 𝒫 ℕ0 → 𝑝 ⊆ ℕ0)
11	10	adantr 480	. . . . . . . . . 10 ⊢ ((𝑝 ∈ 𝒫 ℕ0 ∧ 𝑚 ∈ ℕ0) → 𝑝 ⊆ ℕ0)
12		ssrab2 4029	. . . . . . . . . 10 ⊢ {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)} ⊆ ℕ0
13		sadcl 16380	. . . . . . . . . 10 ⊢ ((𝑝 ⊆ ℕ0 ∧ {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)} ⊆ ℕ0) → (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) ⊆ ℕ0)
14	11, 12, 13	sylancl 586	. . . . . . . . 9 ⊢ ((𝑝 ∈ 𝒫 ℕ0 ∧ 𝑚 ∈ ℕ0) → (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) ⊆ ℕ0)
15		nn0ex 12398	. . . . . . . . . 10 ⊢ ℕ0 ∈ V
16	15	elpw2 5276	. . . . . . . . 9 ⊢ ((𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) ∈ 𝒫 ℕ0 ↔ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) ⊆ ℕ0)
17	14, 16	sylibr 234	. . . . . . . 8 ⊢ ((𝑝 ∈ 𝒫 ℕ0 ∧ 𝑚 ∈ ℕ0) → (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) ∈ 𝒫 ℕ0)
18	17	rgen2 3173	. . . . . . 7 ⊢ ∀𝑝 ∈ 𝒫 ℕ0∀𝑚 ∈ ℕ0 (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) ∈ 𝒫 ℕ0
19		eqid 2733	. . . . . . . 8 ⊢ (𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})) = (𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))
20	19	fmpo 8009	. . . . . . 7 ⊢ (∀𝑝 ∈ 𝒫 ℕ0∀𝑚 ∈ ℕ0 (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) ∈ 𝒫 ℕ0 ↔ (𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})):(𝒫 ℕ0 × ℕ0)⟶𝒫 ℕ0)
21	18, 20	mpbi 230	. . . . . 6 ⊢ (𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})):(𝒫 ℕ0 × ℕ0)⟶𝒫 ℕ0
22	21, 7	f0cli 7040	. . . . 5 ⊢ ((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))‘⟨𝑥, 𝑦⟩) ∈ 𝒫 ℕ0
23	9, 22	eqeltri 2829	. . . 4 ⊢ (𝑥(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))𝑦) ∈ 𝒫 ℕ0
24	23	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝒫 ℕ0 ∧ 𝑦 ∈ V)) → (𝑥(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))𝑦) ∈ 𝒫 ℕ0)
25		nn0uz 12780	. . 3 ⊢ ℕ0 = (ℤ≥‘0)
26		0zd 12491	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
27		fvexd 6846	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(0 + 1))) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘𝑥) ∈ V)
28	8, 24, 25, 26, 27	seqf2 13935	. 2 ⊢ (𝜑 → seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))):ℕ0⟶𝒫 ℕ0)
29		smuval.p	. . 3 ⊢ 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
30	29	feq1i 6650	. 2 ⊢ (𝑃:ℕ0⟶𝒫 ℕ0 ↔ seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))):ℕ0⟶𝒫 ℕ0)
31	28, 30	sylibr 234	1 ⊢ (𝜑 → 𝑃:ℕ0⟶𝒫 ℕ0)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  {crab 3396  Vcvv 3437   ⊆ wss 3898  ∅c0 4282  ifcif 4476  𝒫 cpw 4551  ⟨cop 4583   ↦ cmpt 5176   × cxp 5619  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355   ∈ cmpo 7357  0cc0 11017  1c1 11018   + caddc 11020   − cmin 11355  ℕ₀cn0 12392  ℤ_≥cuz 12742  seqcseq 13915   sadd csad 16338

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1513  df-tru 1544  df-fal 1554  df-had 1595  df-cad 1608  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-er 8631  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-nn 12137  df-n0 12393  df-z 12480  df-uz 12743  df-fz 13415  df-seq 13916  df-sad 16369

This theorem is referenced by:  smupp1  16398  smuval2  16400  smupvallem  16401  smueqlem  16408