Theorem reprval 31529

Description: Value of the representations of 𝑀 as the sum of 𝑆 nonnegative integers in a given set 𝐴 (Contributed by Thierry Arnoux, 1-Dec-2021.)

Hypotheses

Ref	Expression
reprval.a	⊢ (𝜑 → 𝐴 ⊆ ℕ)
reprval.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
reprval.s	⊢ (𝜑 → 𝑆 ∈ ℕ0)

Assertion

Ref	Expression
reprval	⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})

Distinct variable groups:   𝐴,𝑐   𝑀,𝑐   𝑆,𝑎,𝑐   𝜑,𝑐

Allowed substitution hints:   𝜑(𝑎)   𝐴(𝑎)   𝑀(𝑎)

Proof of Theorem reprval

Dummy variables 𝑏 𝑚 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-repr 31528	. . 3 ⊢ repr = (𝑠 ∈ ℕ0 ↦ (𝑏 ∈ 𝒫 ℕ, 𝑚 ∈ ℤ ↦ {𝑐 ∈ (𝑏 ↑𝑚 (0..^𝑠)) ∣ Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = 𝑚}))
2		oveq2 6986	. . . . . 6 ⊢ (𝑠 = 𝑆 → (0..^𝑠) = (0..^𝑆))
3	2	oveq2d 6994	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑏 ↑𝑚 (0..^𝑠)) = (𝑏 ↑𝑚 (0..^𝑆)))
4	2	sumeq1d 14921	. . . . . 6 ⊢ (𝑠 = 𝑆 → Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎))
5	4	eqeq1d 2780	. . . . 5 ⊢ (𝑠 = 𝑆 → (Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = 𝑚 ↔ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑚))
6	3, 5	rabeqbidv 3408	. . . 4 ⊢ (𝑠 = 𝑆 → {𝑐 ∈ (𝑏 ↑𝑚 (0..^𝑠)) ∣ Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = 𝑚} = {𝑐 ∈ (𝑏 ↑𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑚})
7	6	mpoeq3dv 7053	. . 3 ⊢ (𝑠 = 𝑆 → (𝑏 ∈ 𝒫 ℕ, 𝑚 ∈ ℤ ↦ {𝑐 ∈ (𝑏 ↑𝑚 (0..^𝑠)) ∣ Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = 𝑚}) = (𝑏 ∈ 𝒫 ℕ, 𝑚 ∈ ℤ ↦ {𝑐 ∈ (𝑏 ↑𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑚}))
8		reprval.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
9		nnex 11448	. . . . . 6 ⊢ ℕ ∈ V
10	9	pwex 5135	. . . . 5 ⊢ 𝒫 ℕ ∈ V
11		zex 11805	. . . . 5 ⊢ ℤ ∈ V
12	10, 11	mpoex 7587	. . . 4 ⊢ (𝑏 ∈ 𝒫 ℕ, 𝑚 ∈ ℤ ↦ {𝑐 ∈ (𝑏 ↑𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑚}) ∈ V
13	12	a1i 11	. . 3 ⊢ (𝜑 → (𝑏 ∈ 𝒫 ℕ, 𝑚 ∈ ℤ ↦ {𝑐 ∈ (𝑏 ↑𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑚}) ∈ V)
14	1, 7, 8, 13	fvmptd3 6619	. 2 ⊢ (𝜑 → (repr‘𝑆) = (𝑏 ∈ 𝒫 ℕ, 𝑚 ∈ ℤ ↦ {𝑐 ∈ (𝑏 ↑𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑚}))
15		simprl 758	. . . 4 ⊢ ((𝜑 ∧ (𝑏 = 𝐴 ∧ 𝑚 = 𝑀)) → 𝑏 = 𝐴)
16	15	oveq1d 6993	. . 3 ⊢ ((𝜑 ∧ (𝑏 = 𝐴 ∧ 𝑚 = 𝑀)) → (𝑏 ↑𝑚 (0..^𝑆)) = (𝐴 ↑𝑚 (0..^𝑆)))
17		simprr 760	. . . 4 ⊢ ((𝜑 ∧ (𝑏 = 𝐴 ∧ 𝑚 = 𝑀)) → 𝑚 = 𝑀)
18	17	eqeq2d 2788	. . 3 ⊢ ((𝜑 ∧ (𝑏 = 𝐴 ∧ 𝑚 = 𝑀)) → (Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑚 ↔ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀))
19	16, 18	rabeqbidv 3408	. 2 ⊢ ((𝜑 ∧ (𝑏 = 𝐴 ∧ 𝑚 = 𝑀)) → {𝑐 ∈ (𝑏 ↑𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑚} = {𝑐 ∈ (𝐴 ↑𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
20	9	a1i 11	. . . 4 ⊢ (𝜑 → ℕ ∈ V)
21		reprval.a	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
22	20, 21	ssexd 5085	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
23	22, 21	elpwd 4432	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 ℕ)
24		reprval.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
25		ovex 7010	. . . 4 ⊢ (𝐴 ↑𝑚 (0..^𝑆)) ∈ V
26	25	rabex 5092	. . 3 ⊢ {𝑐 ∈ (𝐴 ↑𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀} ∈ V
27	26	a1i 11	. 2 ⊢ (𝜑 → {𝑐 ∈ (𝐴 ↑𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀} ∈ V)
28	14, 19, 23, 24, 27	ovmpod 7120	1 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050  {crab 3092  Vcvv 3415   ⊆ wss 3831  𝒫 cpw 4423  ‘cfv 6190  (class class class)co 6978   ∈ cmpo 6980   ↑_𝑚 cmap 8208  0cc0 10337  ℕcn 11441  ℕ₀cn0 11710  ℤcz 11796  ..^cfzo 12852  Σcsu 14906  reprcrepr 31527

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 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-cnex 10393  ax-resscn 10394  ax-1cn 10395  ax-addcl 10397

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 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-ov 6981  df-oprab 6982  df-mpo 6983  df-om 7399  df-1st 7503  df-2nd 7504  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-neg 10675  df-nn 11442  df-z 11797  df-seq 13188  df-sum 14907  df-repr 31528

This theorem is referenced by:  repr0  31530  reprf  31531  reprsum  31532  reprsuc  31534  reprfi  31535  reprss  31536  reprinrn  31537  reprlt  31538  reprgt  31540  reprinfz1  31541  reprpmtf1o  31545  reprdifc  31546  breprexplema  31549