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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 11801 | . . . . . . . 8 ⊢ ℕ ∈ V | |
2 | 1 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℕ ∈ V) |
3 | reprval.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℕ) | |
4 | 2, 3 | ssexd 5202 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V) |
5 | reprss.1 | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
6 | mapss 8548 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ⊆ 𝐴) → (𝐵 ↑m (0..^𝑆)) ⊆ (𝐴 ↑m (0..^𝑆))) | |
7 | 4, 5, 6 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → (𝐵 ↑m (0..^𝑆)) ⊆ (𝐴 ↑m (0..^𝑆))) |
8 | 7 | sselda 3887 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵 ↑m (0..^𝑆))) → 𝑐 ∈ (𝐴 ↑m (0..^𝑆))) |
9 | 8 | adantrr 717 | . . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ (𝐵 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀)) → 𝑐 ∈ (𝐴 ↑m (0..^𝑆))) |
10 | 9 | rabss3d 30534 | . 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 3897 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ ℕ) |
12 | reprval.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
13 | reprval.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
14 | 11, 12, 13 | reprval 32256 | . 2 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐵 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
15 | 3, 12, 13 | reprval 32256 | . 2 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
16 | 10, 14, 15 | 3sstr4d 3934 | 1 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) ⊆ (𝐴(repr‘𝑆)𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 {crab 3055 Vcvv 3398 ⊆ wss 3853 ‘cfv 6358 (class class class)co 7191 ↑m cmap 8486 0cc0 10694 ℕcn 11795 ℕ0cn0 12055 ℤcz 12141 ..^cfzo 13203 Σcsu 15214 reprcrepr 32254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-addcl 10754 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-map 8488 df-neg 11030 df-nn 11796 df-z 12142 df-seq 13540 df-sum 15215 df-repr 32255 |
This theorem is referenced by: hashreprin 32266 reprinfz1 32268 tgoldbachgtde 32306 |
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