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Mirrors > Home > MPE Home > Th. List > Mathboxes > reprss | Structured version Visualization version GIF version |
Description: Representations with terms in a subset. (Contributed by Thierry Arnoux, 11-Dec-2021.) |
Ref | Expression |
---|---|
reprval.a | ⊢ (𝜑 → 𝐴 ⊆ ℕ) |
reprval.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
reprval.s | ⊢ (𝜑 → 𝑆 ∈ ℕ0) |
reprss.1 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
Ref | Expression |
---|---|
reprss | ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) ⊆ (𝐴(repr‘𝑆)𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnex 11638 | . . . . . . . 8 ⊢ ℕ ∈ V | |
2 | 1 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℕ ∈ V) |
3 | reprval.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℕ) | |
4 | 2, 3 | ssexd 5220 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V) |
5 | reprss.1 | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
6 | mapss 8447 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ⊆ 𝐴) → (𝐵 ↑m (0..^𝑆)) ⊆ (𝐴 ↑m (0..^𝑆))) | |
7 | 4, 5, 6 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝐵 ↑m (0..^𝑆)) ⊆ (𝐴 ↑m (0..^𝑆))) |
8 | 7 | sselda 3966 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵 ↑m (0..^𝑆))) → 𝑐 ∈ (𝐴 ↑m (0..^𝑆))) |
9 | 8 | adantrr 715 | . . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ (𝐵 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀)) → 𝑐 ∈ (𝐴 ↑m (0..^𝑆))) |
10 | 9 | rabss3d 30270 | . 2 ⊢ (𝜑 → {𝑐 ∈ (𝐵 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀} ⊆ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
11 | 5, 3 | sstrd 3976 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ ℕ) |
12 | reprval.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
13 | reprval.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
14 | 11, 12, 13 | reprval 31876 | . 2 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐵 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
15 | 3, 12, 13 | reprval 31876 | . 2 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
16 | 10, 14, 15 | 3sstr4d 4013 | 1 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) ⊆ (𝐴(repr‘𝑆)𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3494 ⊆ wss 3935 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 0cc0 10531 ℕcn 11632 ℕ0cn0 11891 ℤcz 11975 ..^cfzo 13027 Σcsu 15036 reprcrepr 31874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-addcl 10591 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-map 8402 df-neg 10867 df-nn 11633 df-z 11976 df-seq 13364 df-sum 15037 df-repr 31875 |
This theorem is referenced by: hashreprin 31886 reprinfz1 31888 tgoldbachgtde 31926 |
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