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| Mirrors > Home > MPE Home > Th. List > mptnn0fsuppd | Structured version Visualization version GIF version | ||
| Description: A mapping from the nonnegative integers is finitely supported under certain conditions. (Contributed by AV, 2-Dec-2019.) (Revised by AV, 23-Dec-2019.) |
| Ref | Expression |
|---|---|
| mptnn0fsupp.0 | ⊢ (𝜑 → 0 ∈ 𝑉) |
| mptnn0fsupp.c | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐶 ∈ 𝐵) |
| mptnn0fsuppd.d | ⊢ (𝑘 = 𝑥 → 𝐶 = 𝐷) |
| mptnn0fsuppd.s | ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝐷 = 0 )) |
| Ref | Expression |
|---|---|
| mptnn0fsuppd | ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) finSupp 0 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mptnn0fsupp.0 | . 2 ⊢ (𝜑 → 0 ∈ 𝑉) | |
| 2 | mptnn0fsupp.c | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐶 ∈ 𝐵) | |
| 3 | mptnn0fsuppd.s | . . 3 ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝐷 = 0 )) | |
| 4 | vex 3484 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
| 5 | mptnn0fsuppd.d | . . . . . . . 8 ⊢ (𝑘 = 𝑥 → 𝐶 = 𝐷) | |
| 6 | 4, 5 | csbie 3934 | . . . . . . 7 ⊢ ⦋𝑥 / 𝑘⦌𝐶 = 𝐷 |
| 7 | id 22 | . . . . . . 7 ⊢ (𝐷 = 0 → 𝐷 = 0 ) | |
| 8 | 6, 7 | eqtrid 2789 | . . . . . 6 ⊢ (𝐷 = 0 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ) |
| 9 | 8 | imim2i 16 | . . . . 5 ⊢ ((𝑠 < 𝑥 → 𝐷 = 0 ) → (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 )) |
| 10 | 9 | ralimi 3083 | . . . 4 ⊢ (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝐷 = 0 ) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 )) |
| 11 | 10 | reximi 3084 | . . 3 ⊢ (∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝐷 = 0 ) → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 )) |
| 12 | 3, 11 | syl 17 | . 2 ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 )) |
| 13 | 1, 2, 12 | mptnn0fsupp 14038 | 1 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) finSupp 0 ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ⦋csb 3899 class class class wbr 5143 ↦ cmpt 5225 finSupp cfsupp 9401 < clt 11295 ℕ0cn0 12526 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 |
| This theorem is referenced by: evls1fpws 22373 decpmatfsupp 22775 decpmatmulsumfsupp 22779 pmatcollpw1lem1 22780 pm2mpmhmlem1 22824 cpmidpmatlem3 22878 evl1deg1 33601 evl1deg2 33602 evl1deg3 33603 |
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