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Mirrors > Home > MPE Home > Th. List > coef3 | Structured version Visualization version GIF version |
Description: The domain and codomain of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
Ref | Expression |
---|---|
dgrval.1 | ⊢ 𝐴 = (coeff‘𝐹) |
Ref | Expression |
---|---|
coef3 | ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plyssc 25559 | . . 3 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) | |
2 | 1 | sseli 3940 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ)) |
3 | 0cn 11146 | . 2 ⊢ 0 ∈ ℂ | |
4 | dgrval.1 | . . 3 ⊢ 𝐴 = (coeff‘𝐹) | |
5 | 4 | coef2 25590 | . 2 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 0 ∈ ℂ) → 𝐴:ℕ0⟶ℂ) |
6 | 2, 3, 5 | sylancl 586 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⟶wf 6492 ‘cfv 6496 ℂcc 11048 0cc0 11050 ℕ0cn0 12412 Polycply 25543 coeffccoe 25545 |
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 7671 ax-inf2 9576 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-pre-sup 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-int 4908 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-se 5589 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-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7616 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-map 8766 df-pm 8767 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-sup 9377 df-inf 9378 df-oi 9445 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-n0 12413 df-z 12499 df-uz 12763 df-rp 12915 df-fz 13424 df-fzo 13567 df-fl 13696 df-seq 13906 df-exp 13967 df-hash 14230 df-cj 14983 df-re 14984 df-im 14985 df-sqrt 15119 df-abs 15120 df-clim 15369 df-rlim 15370 df-sum 15570 df-0p 25032 df-ply 25547 df-coe 25549 |
This theorem is referenced by: dgrub 25593 dgrub2 25594 dgrlb 25595 coeidlem 25596 coeid3 25599 plyco 25600 dgrle 25602 0dgrb 25605 coefv0 25607 coeaddlem 25608 coemullem 25609 coemulhi 25613 coemulc 25614 coe0 25615 coesub 25616 plycn 25620 dgreq0 25624 dgradd2 25627 dgrmul 25629 dgrcolem2 25633 plycjlem 25635 coecj 25637 plymul0or 25639 dvply2g 25643 plydivlem4 25654 plydiveu 25656 vieta1lem2 25669 vieta1 25670 elqaalem3 25679 aareccl 25684 ftalem1 26420 ftalem2 26421 ftalem4 26423 ftalem5 26424 signsplypnf 33102 dgrsub2 41439 mpaaeu 41454 |
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