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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 11440 | . . . . . . . 8 ⊢ ℕ ∈ V | |
2 | 1 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℕ ∈ V) |
3 | reprval.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℕ) | |
4 | 2, 3 | ssexd 5078 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V) |
5 | reprss.1 | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
6 | mapss 8245 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ⊆ 𝐴) → (𝐵 ↑𝑚 (0..^𝑆)) ⊆ (𝐴 ↑𝑚 (0..^𝑆))) | |
7 | 4, 5, 6 | syl2anc 576 | . . . . 5 ⊢ (𝜑 → (𝐵 ↑𝑚 (0..^𝑆)) ⊆ (𝐴 ↑𝑚 (0..^𝑆))) |
8 | 7 | sselda 3852 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵 ↑𝑚 (0..^𝑆))) → 𝑐 ∈ (𝐴 ↑𝑚 (0..^𝑆))) |
9 | 8 | adantrr 704 | . . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ (𝐵 ↑𝑚 (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀)) → 𝑐 ∈ (𝐴 ↑𝑚 (0..^𝑆))) |
10 | 9 | rabss3d 30046 | . 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 3862 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ ℕ) |
12 | reprval.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
13 | reprval.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
14 | 11, 12, 13 | reprval 31529 | . 2 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐵 ↑𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
15 | 3, 12, 13 | reprval 31529 | . 2 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
16 | 10, 14, 15 | 3sstr4d 3898 | 1 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) ⊆ (𝐴(repr‘𝑆)𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 {crab 3086 Vcvv 3409 ⊆ wss 3823 ‘cfv 6182 (class class class)co 6970 ↑𝑚 cmap 8200 0cc0 10329 ℕcn 11433 ℕ0cn0 11701 ℤcz 11787 ..^cfzo 12843 Σcsu 14897 reprcrepr 31527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-addcl 10389 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-map 8202 df-neg 10667 df-nn 11434 df-z 11788 df-seq 13179 df-sum 14898 df-repr 31528 |
This theorem is referenced by: hashreprin 31539 reprinfz1 31541 tgoldbachgtde 31579 |
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