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Mirrors > Home > MPE Home > Th. List > mptnn0fsupp | Structured version Visualization version GIF version |
Description: A mapping from the nonnegative integers is finitely supported under certain conditions. (Contributed by AV, 5-Oct-2019.) (Revised by AV, 23-Dec-2019.) |
Ref | Expression |
---|---|
mptnn0fsupp.0 | ⊢ (𝜑 → 0 ∈ 𝑉) |
mptnn0fsupp.c | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐶 ∈ 𝐵) |
mptnn0fsupp.s | ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 )) |
Ref | Expression |
---|---|
mptnn0fsupp | ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) finSupp 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptnn0fsupp.c | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐶 ∈ 𝐵) | |
2 | 1 | ralrimiva 3143 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵) |
3 | eqid 2736 | . . . . . 6 ⊢ (𝑘 ∈ ℕ0 ↦ 𝐶) = (𝑘 ∈ ℕ0 ↦ 𝐶) | |
4 | 3 | fnmpt 6641 | . . . . 5 ⊢ (∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵 → (𝑘 ∈ ℕ0 ↦ 𝐶) Fn ℕ0) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) Fn ℕ0) |
6 | nn0ex 12419 | . . . . 5 ⊢ ℕ0 ∈ V | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
8 | mptnn0fsupp.0 | . . . . 5 ⊢ (𝜑 → 0 ∈ 𝑉) | |
9 | 8 | elexd 3465 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
10 | suppvalfn 8100 | . . . 4 ⊢ (((𝑘 ∈ ℕ0 ↦ 𝐶) Fn ℕ0 ∧ ℕ0 ∈ V ∧ 0 ∈ V) → ((𝑘 ∈ ℕ0 ↦ 𝐶) supp 0 ) = {𝑥 ∈ ℕ0 ∣ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 }) | |
11 | 5, 7, 9, 10 | syl3anc 1371 | . . 3 ⊢ (𝜑 → ((𝑘 ∈ ℕ0 ↦ 𝐶) supp 0 ) = {𝑥 ∈ ℕ0 ∣ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 }) |
12 | mptnn0fsupp.s | . . . . 5 ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 )) | |
13 | nne 2947 | . . . . . . . . 9 ⊢ (¬ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 ↔ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = 0 ) | |
14 | simpr 485 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0) | |
15 | 2 | ad2antrr 724 | . . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵) |
16 | rspcsbela 4395 | . . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵) → ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵) | |
17 | 14, 15, 16 | syl2anc 584 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵) |
18 | 3 | fvmpts 6951 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ0 ∧ ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵) → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = ⦋𝑥 / 𝑘⦌𝐶) |
19 | 14, 17, 18 | syl2anc 584 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = ⦋𝑥 / 𝑘⦌𝐶) |
20 | 19 | eqeq1d 2738 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = 0 ↔ ⦋𝑥 / 𝑘⦌𝐶 = 0 )) |
21 | 13, 20 | bitrid 282 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (¬ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 ↔ ⦋𝑥 / 𝑘⦌𝐶 = 0 )) |
22 | 21 | imbi2d 340 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑠 < 𝑥 → ¬ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 ) ↔ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ))) |
23 | 22 | ralbidva 3172 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 ) ↔ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ))) |
24 | 23 | rexbidva 3173 | . . . . 5 ⊢ (𝜑 → (∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 ) ↔ ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ))) |
25 | 12, 24 | mpbird 256 | . . . 4 ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 )) |
26 | rabssnn0fi 13891 | . . . 4 ⊢ ({𝑥 ∈ ℕ0 ∣ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 } ∈ Fin ↔ ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 )) | |
27 | 25, 26 | sylibr 233 | . . 3 ⊢ (𝜑 → {𝑥 ∈ ℕ0 ∣ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 } ∈ Fin) |
28 | 11, 27 | eqeltrd 2838 | . 2 ⊢ (𝜑 → ((𝑘 ∈ ℕ0 ↦ 𝐶) supp 0 ) ∈ Fin) |
29 | funmpt 6539 | . . 3 ⊢ Fun (𝑘 ∈ ℕ0 ↦ 𝐶) | |
30 | 6 | mptex 7173 | . . 3 ⊢ (𝑘 ∈ ℕ0 ↦ 𝐶) ∈ V |
31 | funisfsupp 9310 | . . 3 ⊢ ((Fun (𝑘 ∈ ℕ0 ↦ 𝐶) ∧ (𝑘 ∈ ℕ0 ↦ 𝐶) ∈ V ∧ 0 ∈ V) → ((𝑘 ∈ ℕ0 ↦ 𝐶) finSupp 0 ↔ ((𝑘 ∈ ℕ0 ↦ 𝐶) supp 0 ) ∈ Fin)) | |
32 | 29, 30, 9, 31 | mp3an12i 1465 | . 2 ⊢ (𝜑 → ((𝑘 ∈ ℕ0 ↦ 𝐶) finSupp 0 ↔ ((𝑘 ∈ ℕ0 ↦ 𝐶) supp 0 ) ∈ Fin)) |
33 | 28, 32 | mpbird 256 | 1 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 {crab 3407 Vcvv 3445 ⦋csb 3855 class class class wbr 5105 ↦ cmpt 5188 Fun wfun 6490 Fn wfn 6491 ‘cfv 6496 (class class class)co 7357 supp csupp 8092 Fincfn 8883 finSupp cfsupp 9305 < clt 11189 ℕ0cn0 12413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 |
This theorem is referenced by: mptnn0fsuppd 13903 mptcoe1fsupp 21586 mptcoe1matfsupp 22151 pm2mp 22174 chfacffsupp 22205 chfacfscmulfsupp 22208 chfacfpmmulfsupp 22212 cayhamlem4 22237 ply1mulgsumlem3 46459 ply1mulgsumlem4 46460 |
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