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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 3451 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
5 | mptnn0fsuppd.d | . . . . . . . 8 ⊢ (𝑘 = 𝑥 → 𝐶 = 𝐷) | |
6 | 4, 5 | csbie 3895 | . . . . . . 7 ⊢ ⦋𝑥 / 𝑘⦌𝐶 = 𝐷 |
7 | id 22 | . . . . . . 7 ⊢ (𝐷 = 0 → 𝐷 = 0 ) | |
8 | 6, 7 | eqtrid 2785 | . . . . . 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 13911 | 1 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ∃wrex 3070 ⦋csb 3859 class class class wbr 5109 ↦ cmpt 5192 finSupp cfsupp 9311 < clt 11197 ℕ0cn0 12421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 |
This theorem is referenced by: decpmatfsupp 22141 decpmatmulsumfsupp 22145 pmatcollpw1lem1 22146 pm2mpmhmlem1 22190 cpmidpmatlem3 22244 evls1fpws 32327 |
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