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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fply1 | Structured version Visualization version GIF version |
Description: Conditions for a function to be an univariate polynomial. (Contributed by Thierry Arnoux, 19-Aug-2023.) |
Ref | Expression |
---|---|
fply1.1 | ⊢ 0 = (0g‘𝑅) |
fply1.2 | ⊢ 𝐵 = (Base‘𝑅) |
fply1.3 | ⊢ 𝑃 = (Base‘(Poly1‘𝑅)) |
fply1.4 | ⊢ (𝜑 → 𝐹:(ℕ0 ↑𝑚 1o)⟶𝐵) |
fply1.5 | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
Ref | Expression |
---|---|
fply1 | ⊢ (𝜑 → 𝐹 ∈ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fply1.4 | . . . . 5 ⊢ (𝜑 → 𝐹:(ℕ0 ↑𝑚 1o)⟶𝐵) | |
2 | fply1.2 | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
3 | 2 | fvexi 6544 | . . . . . 6 ⊢ 𝐵 ∈ V |
4 | ovex 7039 | . . . . . 6 ⊢ (ℕ0 ↑𝑚 1o) ∈ V | |
5 | 3, 4 | elmap 8276 | . . . . 5 ⊢ (𝐹 ∈ (𝐵 ↑𝑚 (ℕ0 ↑𝑚 1o)) ↔ 𝐹:(ℕ0 ↑𝑚 1o)⟶𝐵) |
6 | 1, 5 | sylibr 235 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑𝑚 (ℕ0 ↑𝑚 1o))) |
7 | df1o2 7958 | . . . . . . . . 9 ⊢ 1o = {∅} | |
8 | snfi 8432 | . . . . . . . . 9 ⊢ {∅} ∈ Fin | |
9 | 7, 8 | eqeltri 2877 | . . . . . . . 8 ⊢ 1o ∈ Fin |
10 | 9 | a1i 11 | . . . . . . 7 ⊢ (𝑓 ∈ (ℕ0 ↑𝑚 1o) → 1o ∈ Fin) |
11 | elmapi 8269 | . . . . . . 7 ⊢ (𝑓 ∈ (ℕ0 ↑𝑚 1o) → 𝑓:1o⟶ℕ0) | |
12 | 10, 11 | fisuppfi 8677 | . . . . . 6 ⊢ (𝑓 ∈ (ℕ0 ↑𝑚 1o) → (◡𝑓 “ ℕ) ∈ Fin) |
13 | 12 | rabeqc 3611 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑𝑚 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} = (ℕ0 ↑𝑚 1o) |
14 | 13 | oveq2i 7018 | . . . 4 ⊢ (𝐵 ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin}) = (𝐵 ↑𝑚 (ℕ0 ↑𝑚 1o)) |
15 | 6, 14 | syl6eleqr 2892 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
16 | eqid 2793 | . . . 4 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
17 | eqid 2793 | . . . 4 ⊢ {𝑓 ∈ (ℕ0 ↑𝑚 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑𝑚 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
18 | eqid 2793 | . . . 4 ⊢ (Base‘(1o mPwSer 𝑅)) = (Base‘(1o mPwSer 𝑅)) | |
19 | 1oex 7952 | . . . . 5 ⊢ 1o ∈ V | |
20 | 19 | a1i 11 | . . . 4 ⊢ (𝜑 → 1o ∈ V) |
21 | 16, 2, 17, 18, 20 | psrbas 19834 | . . 3 ⊢ (𝜑 → (Base‘(1o mPwSer 𝑅)) = (𝐵 ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
22 | 15, 21 | eleqtrrd 2884 | . 2 ⊢ (𝜑 → 𝐹 ∈ (Base‘(1o mPwSer 𝑅))) |
23 | fply1.5 | . 2 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
24 | eqid 2793 | . . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
25 | fply1.1 | . . 3 ⊢ 0 = (0g‘𝑅) | |
26 | eqid 2793 | . . . 4 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
27 | eqid 2793 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
28 | fply1.3 | . . . 4 ⊢ 𝑃 = (Base‘(Poly1‘𝑅)) | |
29 | 26, 27, 28 | ply1bas 20034 | . . 3 ⊢ 𝑃 = (Base‘(1o mPoly 𝑅)) |
30 | 24, 16, 18, 25, 29 | mplelbas 19886 | . 2 ⊢ (𝐹 ∈ 𝑃 ↔ (𝐹 ∈ (Base‘(1o mPwSer 𝑅)) ∧ 𝐹 finSupp 0 )) |
31 | 22, 23, 30 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐹 ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1520 ∈ wcel 2079 {crab 3107 Vcvv 3432 ∅c0 4206 {csn 4466 class class class wbr 4956 ◡ccnv 5434 “ cima 5438 ⟶wf 6213 ‘cfv 6217 (class class class)co 7007 1oc1o 7937 ↑𝑚 cmap 8247 Fincfn 8347 finSupp cfsupp 8669 ℕcn 11475 ℕ0cn0 11734 Basecbs 16300 0gc0g 16530 mPwSer cmps 19807 mPoly cmpl 19809 PwSer1cps1 20014 Poly1cpl1 20016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-of 7258 df-om 7428 df-1st 7536 df-2nd 7537 df-supp 7673 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-oadd 7948 df-er 8130 df-map 8249 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-fsupp 8670 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-z 11819 df-dec 11937 df-uz 12083 df-fz 12732 df-struct 16302 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-ress 16308 df-plusg 16395 df-mulr 16396 df-sca 16398 df-vsca 16399 df-tset 16401 df-ple 16402 df-psr 19812 df-mpl 19814 df-opsr 19816 df-psr1 20019 df-ply1 20021 |
This theorem is referenced by: (None) |
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