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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 11644 | . . . . . . . 8 ⊢ ℕ ∈ V | |
2 | 1 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℕ ∈ V) |
3 | reprval.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℕ) | |
4 | 2, 3 | ssexd 5228 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V) |
5 | reprss.1 | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
6 | mapss 8453 | . . . . . 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 3967 | . . . 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 30276 | . 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 3977 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ ℕ) |
12 | reprval.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
13 | reprval.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
14 | 11, 12, 13 | reprval 31881 | . 2 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐵 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
15 | 3, 12, 13 | reprval 31881 | . 2 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
16 | 10, 14, 15 | 3sstr4d 4014 | 1 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) ⊆ (𝐴(repr‘𝑆)𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {crab 3142 Vcvv 3494 ⊆ wss 3936 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 0cc0 10537 ℕcn 11638 ℕ0cn0 11898 ℤcz 11982 ..^cfzo 13034 Σcsu 15042 reprcrepr 31879 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-addcl 10597 |
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 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-map 8408 df-neg 10873 df-nn 11639 df-z 11983 df-seq 13371 df-sum 15043 df-repr 31880 |
This theorem is referenced by: hashreprin 31891 reprinfz1 31893 tgoldbachgtde 31931 |
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