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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > reprf | Structured version Visualization version GIF version |
Description: Members of the representation of 𝑀 as the sum of 𝑆 nonnegative integers from set 𝐴 as functions. (Contributed by Thierry Arnoux, 5-Dec-2021.) |
Ref | Expression |
---|---|
reprval.a | ⊢ (𝜑 → 𝐴 ⊆ ℕ) |
reprval.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
reprval.s | ⊢ (𝜑 → 𝑆 ∈ ℕ0) |
reprf.c | ⊢ (𝜑 → 𝐶 ∈ (𝐴(repr‘𝑆)𝑀)) |
Ref | Expression |
---|---|
reprf | ⊢ (𝜑 → 𝐶:(0..^𝑆)⟶𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reprf.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴(repr‘𝑆)𝑀)) | |
2 | reprval.a | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℕ) | |
3 | reprval.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
4 | reprval.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
5 | 2, 3, 4 | reprval 34587 | . . 3 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
6 | 1, 5 | eleqtrd 2846 | . 2 ⊢ (𝜑 → 𝐶 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
7 | elrabi 3703 | . 2 ⊢ (𝐶 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀} → 𝐶 ∈ (𝐴 ↑m (0..^𝑆))) | |
8 | elmapi 8907 | . 2 ⊢ (𝐶 ∈ (𝐴 ↑m (0..^𝑆)) → 𝐶:(0..^𝑆)⟶𝐴) | |
9 | 6, 7, 8 | 3syl 18 | 1 ⊢ (𝜑 → 𝐶:(0..^𝑆)⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 {crab 3443 ⊆ wss 3976 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ↑m cmap 8884 0cc0 11184 ℕcn 12293 ℕ0cn0 12553 ℤcz 12639 ..^cfzo 13711 Σcsu 15734 reprcrepr 34585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-addcl 11244 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-map 8886 df-neg 11523 df-nn 12294 df-z 12640 df-seq 14053 df-sum 15735 df-repr 34586 |
This theorem is referenced by: reprle 34591 reprsuc 34592 hashreprin 34597 reprpmtf1o 34603 reprdifc 34604 breprexplema 34607 breprexplemc 34609 breprexpnat 34611 circlemeth 34617 circlevma 34619 circlemethhgt 34620 hgt750lemb 34633 hgt750lema 34634 hgt750leme 34635 tgoldbachgtde 34637 tgoldbachgt 34640 |
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