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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 31991 | . . 3 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
6 | 1, 5 | eleqtrd 2892 | . 2 ⊢ (𝜑 → 𝐶 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
7 | elrabi 3623 | . 2 ⊢ (𝐶 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀} → 𝐶 ∈ (𝐴 ↑m (0..^𝑆))) | |
8 | elmapi 8411 | . 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 1538 ∈ wcel 2111 {crab 3110 ⊆ wss 3881 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 0cc0 10526 ℕcn 11625 ℕ0cn0 11885 ℤcz 11969 ..^cfzo 13028 Σcsu 15034 reprcrepr 31989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-addcl 10586 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-map 8391 df-neg 10862 df-nn 11626 df-z 11970 df-seq 13365 df-sum 15035 df-repr 31990 |
This theorem is referenced by: reprle 31995 reprsuc 31996 hashreprin 32001 reprpmtf1o 32007 reprdifc 32008 breprexplema 32011 breprexplemc 32013 breprexpnat 32015 circlemeth 32021 circlevma 32023 circlemethhgt 32024 hgt750lemb 32037 hgt750lema 32038 hgt750leme 32039 tgoldbachgtde 32041 tgoldbachgt 32044 |
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