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Mirrors > Home > MPE Home > Th. List > coef2 | Structured version Visualization version GIF version |
Description: The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
Ref | Expression |
---|---|
dgrval.1 | ⊢ 𝐴 = (coeff‘𝐹) |
Ref | Expression |
---|---|
coef2 | ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 0 ∈ 𝑆) → 𝐴:ℕ0⟶𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dgrval.1 | . . . 4 ⊢ 𝐴 = (coeff‘𝐹) | |
2 | 1 | coef 24384 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶(𝑆 ∪ {0})) |
3 | 2 | adantr 474 | . 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 0 ∈ 𝑆) → 𝐴:ℕ0⟶(𝑆 ∪ {0})) |
4 | simpr 479 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 0 ∈ 𝑆) → 0 ∈ 𝑆) | |
5 | 4 | snssd 4557 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 0 ∈ 𝑆) → {0} ⊆ 𝑆) |
6 | ssequn2 4012 | . . . 4 ⊢ ({0} ⊆ 𝑆 ↔ (𝑆 ∪ {0}) = 𝑆) | |
7 | 5, 6 | sylib 210 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 0 ∈ 𝑆) → (𝑆 ∪ {0}) = 𝑆) |
8 | 7 | feq3d 6264 | . 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 0 ∈ 𝑆) → (𝐴:ℕ0⟶(𝑆 ∪ {0}) ↔ 𝐴:ℕ0⟶𝑆)) |
9 | 3, 8 | mpbid 224 | 1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 0 ∈ 𝑆) → 𝐴:ℕ0⟶𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∪ cun 3795 ⊆ wss 3797 {csn 4396 ⟶wf 6118 ‘cfv 6122 0cc0 10251 ℕ0cn0 11617 Polycply 24338 coeffccoe 24340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-inf2 8814 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 ax-addf 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-se 5301 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-isom 6131 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-of 7156 df-om 7326 df-1st 7427 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-oadd 7829 df-er 8008 df-map 8123 df-pm 8124 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-sup 8616 df-inf 8617 df-oi 8683 df-card 9077 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-n0 11618 df-z 11704 df-uz 11968 df-rp 12112 df-fz 12619 df-fzo 12760 df-fl 12887 df-seq 13095 df-exp 13154 df-hash 13410 df-cj 14215 df-re 14216 df-im 14217 df-sqrt 14351 df-abs 14352 df-clim 14595 df-rlim 14596 df-sum 14793 df-0p 23835 df-ply 24342 df-coe 24344 |
This theorem is referenced by: coef3 24386 plyrecj 24433 dvply2g 24438 plydivlem4 24449 elqaalem1 24472 elqaalem3 24474 aareccl 24479 aannenlem1 24481 aannenlem2 24482 aalioulem1 24485 plymulx0 31170 signsply0 31174 mpaaeu 38562 cnsrplycl 38579 elaa2lem 41243 etransclem46 41290 etransclem47 41291 etransclem48 41292 |
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