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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 12225 | . . . . . . . 8 ⊢ ℕ ∈ V | |
2 | 1 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℕ ∈ V) |
3 | reprval.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℕ) | |
4 | 2, 3 | ssexd 5324 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V) |
5 | reprss.1 | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
6 | mapss 8889 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ⊆ 𝐴) → (𝐵 ↑m (0..^𝑆)) ⊆ (𝐴 ↑m (0..^𝑆))) | |
7 | 4, 5, 6 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (𝐵 ↑m (0..^𝑆)) ⊆ (𝐴 ↑m (0..^𝑆))) |
8 | 7 | sselda 3982 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵 ↑m (0..^𝑆))) → 𝑐 ∈ (𝐴 ↑m (0..^𝑆))) |
9 | 8 | adantrr 714 | . . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ (𝐵 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀)) → 𝑐 ∈ (𝐴 ↑m (0..^𝑆))) |
10 | 9 | rabss3d 4079 | . 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 3992 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ ℕ) |
12 | reprval.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
13 | reprval.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
14 | 11, 12, 13 | reprval 34086 | . 2 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐵 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
15 | 3, 12, 13 | reprval 34086 | . 2 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
16 | 10, 14, 15 | 3sstr4d 4029 | 1 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) ⊆ (𝐴(repr‘𝑆)𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 {crab 3431 Vcvv 3473 ⊆ wss 3948 ‘cfv 6543 (class class class)co 7412 ↑m cmap 8826 0cc0 11116 ℕcn 12219 ℕ0cn0 12479 ℤcz 12565 ..^cfzo 13634 Σcsu 15639 reprcrepr 34084 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-addcl 11176 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-map 8828 df-neg 11454 df-nn 12220 df-z 12566 df-seq 13974 df-sum 15640 df-repr 34085 |
This theorem is referenced by: hashreprin 34096 reprinfz1 34098 tgoldbachgtde 34136 |
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