Theorem reprval 34601

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 34600	. . 3 ⊢ repr = (𝑠 ∈ ℕ0 ↦ (𝑏 ∈ 𝒫 ℕ, 𝑚 ∈ ℤ ↦ {𝑐 ∈ (𝑏 ↑m (0..^𝑠)) ∣ Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = 𝑚}))
2		oveq2 7395	. . . . . 6 ⊢ (𝑠 = 𝑆 → (0..^𝑠) = (0..^𝑆))
3	2	oveq2d 7403	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑏 ↑m (0..^𝑠)) = (𝑏 ↑m (0..^𝑆)))
4	2	sumeq1d 15666	. . . . . 6 ⊢ (𝑠 = 𝑆 → Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎))
5	4	eqeq1d 2731	. . . . 5 ⊢ (𝑠 = 𝑆 → (Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = 𝑚 ↔ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑚))
6	3, 5	rabeqbidv 3424	. . . 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 7468	. . 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 12192	. . . . . 6 ⊢ ℕ ∈ V
10	9	pwex 5335	. . . . 5 ⊢ 𝒫 ℕ ∈ V
11		zex 12538	. . . . 5 ⊢ ℤ ∈ V
12	10, 11	mpoex 8058	. . . 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 6991	. 2 ⊢ (𝜑 → (repr‘𝑆) = (𝑏 ∈ 𝒫 ℕ, 𝑚 ∈ ℤ ↦ {𝑐 ∈ (𝑏 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑚}))
15		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑏 = 𝐴 ∧ 𝑚 = 𝑀)) → 𝑏 = 𝐴)
16	15	oveq1d 7402	. . 3 ⊢ ((𝜑 ∧ (𝑏 = 𝐴 ∧ 𝑚 = 𝑀)) → (𝑏 ↑m (0..^𝑆)) = (𝐴 ↑m (0..^𝑆)))
17		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑏 = 𝐴 ∧ 𝑚 = 𝑀)) → 𝑚 = 𝑀)
18	17	eqeq2d 2740	. . 3 ⊢ ((𝜑 ∧ (𝑏 = 𝐴 ∧ 𝑚 = 𝑀)) → (Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑚 ↔ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀))
19	16, 18	rabeqbidv 3424	. 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 5279	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
23	22, 21	elpwd 4569	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 ℕ)
24		reprval.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
25		ovex 7420	. . . 4 ⊢ (𝐴 ↑m (0..^𝑆)) ∈ V
26	25	rabex 5294	. . 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 7541	1 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3405  Vcvv 3447   ⊆ wss 3914  𝒫 cpw 4563  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389   ↑_m cmap 8799  0cc0 11068  ℕcn 12186  ℕ₀cn0 12442  ℤcz 12529  ..^cfzo 13615  Σcsu 15652  reprcrepr 34599

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-addcl 11128

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-neg 11408  df-nn 12187  df-z 12530  df-seq 13967  df-sum 15653  df-repr 34600

This theorem is referenced by:  repr0  34602  reprf  34603  reprsum  34604  reprsuc  34606  reprfi  34607  reprss  34608  reprinrn  34609  reprlt  34610  reprgt  34612  reprinfz1  34613  reprpmtf1o  34617  reprdifc  34618  breprexplema  34621