Theorem smupf 16515

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 12541	. . . . 5 ⊢ 0 ∈ ℕ0
2		iftrue 4531	. . . . . 6 ⊢ (𝑛 = 0 → if(𝑛 = 0, ∅, (𝑛 − 1)) = ∅)
3		eqid 2737	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))
4		0ex 5307	. . . . . 6 ⊢ ∅ ∈ V
5	2, 3, 4	fvmpt 7016	. . . . 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 5356	. . . 4 ⊢ ∅ ∈ 𝒫 ℕ0
8	6, 7	eqeltrdi 2849	. . 3 ⊢ (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘0) ∈ 𝒫 ℕ0)
9		df-ov 7434	. . . . 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 4607	. . . . . . . . . . 11 ⊢ (𝑝 ∈ 𝒫 ℕ0 → 𝑝 ⊆ ℕ0)
11	10	adantr 480	. . . . . . . . . 10 ⊢ ((𝑝 ∈ 𝒫 ℕ0 ∧ 𝑚 ∈ ℕ0) → 𝑝 ⊆ ℕ0)
12		ssrab2 4080	. . . . . . . . . 10 ⊢ {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)} ⊆ ℕ0
13		sadcl 16499	. . . . . . . . . 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 12532	. . . . . . . . . 10 ⊢ ℕ0 ∈ V
16	15	elpw2 5334	. . . . . . . . 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 3199	. . . . . . 7 ⊢ ∀𝑝 ∈ 𝒫 ℕ0∀𝑚 ∈ ℕ0 (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) ∈ 𝒫 ℕ0
19		eqid 2737	. . . . . . . 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 8093	. . . . . . 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 7118	. . . . 5 ⊢ ((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))‘⟨𝑥, 𝑦⟩) ∈ 𝒫 ℕ0
23	9, 22	eqeltri 2837	. . . 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 12920	. . 3 ⊢ ℕ0 = (ℤ≥‘0)
26		0zd 12625	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
27		fvexd 6921	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(0 + 1))) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘𝑥) ∈ V)
28	8, 24, 25, 26, 27	seqf2 14062	. 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 6727	. 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 1540   ∈ wcel 2108  ∀wral 3061  {crab 3436  Vcvv 3480   ⊆ wss 3951  ∅c0 4333  ifcif 4525  𝒫 cpw 4600  ⟨cop 4632   ↦ cmpt 5225   × cxp 5683  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  0cc0 11155  1c1 11156   + caddc 11158   − cmin 11492  ℕ₀cn0 12526  ℤ_≥cuz 12878  seqcseq 14042   sadd csad 16457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-xor 1512  df-tru 1543  df-fal 1553  df-had 1594  df-cad 1607  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-seq 14043  df-sad 16488

This theorem is referenced by:  smupp1  16517  smuval2  16519  smupvallem  16520  smueqlem  16527