![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 34457 | . . 3 ⊢ repr = (𝑠 ∈ ℕ0 ↦ (𝑏 ∈ 𝒫 ℕ, 𝑚 ∈ ℤ ↦ {𝑐 ∈ (𝑏 ↑m (0..^𝑠)) ∣ Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = 𝑚})) | |
2 | oveq2 7434 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (0..^𝑠) = (0..^𝑆)) | |
3 | 2 | oveq2d 7442 | . . . . 5 ⊢ (𝑠 = 𝑆 → (𝑏 ↑m (0..^𝑠)) = (𝑏 ↑m (0..^𝑆))) |
4 | 2 | sumeq1d 15707 | . . . . . 6 ⊢ (𝑠 = 𝑆 → Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎)) |
5 | 4 | eqeq1d 2728 | . . . . 5 ⊢ (𝑠 = 𝑆 → (Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = 𝑚 ↔ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑚)) |
6 | 3, 5 | rabeqbidv 3437 | . . . 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 7506 | . . 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 12272 | . . . . . 6 ⊢ ℕ ∈ V | |
10 | 9 | pwex 5386 | . . . . 5 ⊢ 𝒫 ℕ ∈ V |
11 | zex 12621 | . . . . 5 ⊢ ℤ ∈ V | |
12 | 10, 11 | mpoex 8095 | . . . 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 7034 | . 2 ⊢ (𝜑 → (repr‘𝑆) = (𝑏 ∈ 𝒫 ℕ, 𝑚 ∈ ℤ ↦ {𝑐 ∈ (𝑏 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑚})) |
15 | simprl 769 | . . . 4 ⊢ ((𝜑 ∧ (𝑏 = 𝐴 ∧ 𝑚 = 𝑀)) → 𝑏 = 𝐴) | |
16 | 15 | oveq1d 7441 | . . 3 ⊢ ((𝜑 ∧ (𝑏 = 𝐴 ∧ 𝑚 = 𝑀)) → (𝑏 ↑m (0..^𝑆)) = (𝐴 ↑m (0..^𝑆))) |
17 | simprr 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑏 = 𝐴 ∧ 𝑚 = 𝑀)) → 𝑚 = 𝑀) | |
18 | 17 | eqeq2d 2737 | . . 3 ⊢ ((𝜑 ∧ (𝑏 = 𝐴 ∧ 𝑚 = 𝑀)) → (Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑚 ↔ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀)) |
19 | 16, 18 | rabeqbidv 3437 | . 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 5331 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
23 | 22, 21 | elpwd 4613 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 ℕ) |
24 | reprval.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
25 | ovex 7459 | . . . 4 ⊢ (𝐴 ↑m (0..^𝑆)) ∈ V | |
26 | 25 | rabex 5341 | . . 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 7580 | 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 394 = wceq 1534 ∈ wcel 2099 {crab 3419 Vcvv 3462 ⊆ wss 3947 𝒫 cpw 4607 ‘cfv 6556 (class class class)co 7426 ∈ cmpo 7428 ↑m cmap 8857 0cc0 11160 ℕcn 12266 ℕ0cn0 12526 ℤcz 12612 ..^cfzo 13683 Σcsu 15692 reprcrepr 34456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-addcl 11220 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-neg 11499 df-nn 12267 df-z 12613 df-seq 14024 df-sum 15693 df-repr 34457 |
This theorem is referenced by: repr0 34459 reprf 34460 reprsum 34461 reprsuc 34463 reprfi 34464 reprss 34465 reprinrn 34466 reprlt 34467 reprgt 34469 reprinfz1 34470 reprpmtf1o 34474 reprdifc 34475 breprexplema 34478 |
Copyright terms: Public domain | W3C validator |