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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 34624 | . . 3 ⊢ repr = (𝑠 ∈ ℕ0 ↦ (𝑏 ∈ 𝒫 ℕ, 𝑚 ∈ ℤ ↦ {𝑐 ∈ (𝑏 ↑m (0..^𝑠)) ∣ Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = 𝑚})) | |
| 2 | oveq2 7439 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (0..^𝑠) = (0..^𝑆)) | |
| 3 | 2 | oveq2d 7447 | . . . . 5 ⊢ (𝑠 = 𝑆 → (𝑏 ↑m (0..^𝑠)) = (𝑏 ↑m (0..^𝑆))) | 
| 4 | 2 | sumeq1d 15736 | . . . . . 6 ⊢ (𝑠 = 𝑆 → Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎)) | 
| 5 | 4 | eqeq1d 2739 | . . . . 5 ⊢ (𝑠 = 𝑆 → (Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = 𝑚 ↔ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑚)) | 
| 6 | 3, 5 | rabeqbidv 3455 | . . . 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 7512 | . . 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 5380 | . . . . 5 ⊢ 𝒫 ℕ ∈ V | 
| 11 | zex 12622 | . . . . 5 ⊢ ℤ ∈ V | |
| 12 | 10, 11 | mpoex 8104 | . . . 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 7039 | . 2 ⊢ (𝜑 → (repr‘𝑆) = (𝑏 ∈ 𝒫 ℕ, 𝑚 ∈ ℤ ↦ {𝑐 ∈ (𝑏 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑚})) | 
| 15 | simprl 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑏 = 𝐴 ∧ 𝑚 = 𝑀)) → 𝑏 = 𝐴) | |
| 16 | 15 | oveq1d 7446 | . . 3 ⊢ ((𝜑 ∧ (𝑏 = 𝐴 ∧ 𝑚 = 𝑀)) → (𝑏 ↑m (0..^𝑆)) = (𝐴 ↑m (0..^𝑆))) | 
| 17 | simprr 773 | . . . 4 ⊢ ((𝜑 ∧ (𝑏 = 𝐴 ∧ 𝑚 = 𝑀)) → 𝑚 = 𝑀) | |
| 18 | 17 | eqeq2d 2748 | . . 3 ⊢ ((𝜑 ∧ (𝑏 = 𝐴 ∧ 𝑚 = 𝑀)) → (Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑚 ↔ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀)) | 
| 19 | 16, 18 | rabeqbidv 3455 | . 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 5324 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) | 
| 23 | 22, 21 | elpwd 4606 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 ℕ) | 
| 24 | reprval.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
| 25 | ovex 7464 | . . . 4 ⊢ (𝐴 ↑m (0..^𝑆)) ∈ V | |
| 26 | 25 | rabex 5339 | . . 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 7585 | 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 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ↑m cmap 8866 0cc0 11155 ℕcn 12266 ℕ0cn0 12526 ℤcz 12613 ..^cfzo 13694 Σcsu 15722 reprcrepr 34623 | 
| 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-rep 5279 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-addcl 11215 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 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-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-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-neg 11495 df-nn 12267 df-z 12614 df-seq 14043 df-sum 15723 df-repr 34624 | 
| This theorem is referenced by: repr0 34626 reprf 34627 reprsum 34628 reprsuc 34630 reprfi 34631 reprss 34632 reprinrn 34633 reprlt 34634 reprgt 34636 reprinfz1 34637 reprpmtf1o 34641 reprdifc 34642 breprexplema 34645 | 
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