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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > reprval | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
reprval.a | ⊢ (𝜑 → 𝐴 ⊆ ℕ) |
reprval.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
reprval.s | ⊢ (𝜑 → 𝑆 ∈ ℕ0) |
Ref | Expression |
---|---|
reprval | ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
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}) |
Colors of variables: wff setvar class |
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 |
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