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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 31780 | . . 3 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
6 | 1, 5 | eleqtrd 2912 | . 2 ⊢ (𝜑 → 𝐶 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
7 | elrabi 3672 | . 2 ⊢ (𝐶 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀} → 𝐶 ∈ (𝐴 ↑m (0..^𝑆))) | |
8 | elmapi 8417 | . 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 1528 ∈ wcel 2105 {crab 3139 ⊆ wss 3933 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 0cc0 10525 ℕcn 11626 ℕ0cn0 11885 ℤcz 11969 ..^cfzo 13021 Σcsu 15030 reprcrepr 31778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-addcl 10585 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-map 8397 df-neg 10861 df-nn 11627 df-z 11970 df-seq 13358 df-sum 15031 df-repr 31779 |
This theorem is referenced by: reprle 31784 reprsuc 31785 hashreprin 31790 reprpmtf1o 31796 reprdifc 31797 breprexplema 31800 breprexplemc 31802 breprexpnat 31804 circlemeth 31810 circlevma 31812 circlemethhgt 31813 hgt750lemb 31826 hgt750lema 31827 hgt750leme 31828 tgoldbachgtde 31830 tgoldbachgt 31833 |
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