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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 31883 | . . 3 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
6 | 1, 5 | eleqtrd 2917 | . 2 ⊢ (𝜑 → 𝐶 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
7 | elrabi 3677 | . 2 ⊢ (𝐶 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀} → 𝐶 ∈ (𝐴 ↑m (0..^𝑆))) | |
8 | elmapi 8430 | . 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 2114 {crab 3144 ⊆ wss 3938 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 0cc0 10539 ℕcn 11640 ℕ0cn0 11900 ℤcz 11984 ..^cfzo 13036 Σcsu 15044 reprcrepr 31881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-addcl 10599 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-map 8410 df-neg 10875 df-nn 11641 df-z 11985 df-seq 13373 df-sum 15045 df-repr 31882 |
This theorem is referenced by: reprle 31887 reprsuc 31888 hashreprin 31893 reprpmtf1o 31899 reprdifc 31900 breprexplema 31903 breprexplemc 31905 breprexpnat 31907 circlemeth 31913 circlevma 31915 circlemethhgt 31916 hgt750lemb 31929 hgt750lema 31930 hgt750leme 31931 tgoldbachgtde 31933 tgoldbachgt 31936 |
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