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 6677 | . . . . . 6 ⊢ 𝐵 ∈ V |
4 | ovex 7182 | . . . . . 6 ⊢ (ℕ0 ↑m 1o) ∈ V | |
5 | 3, 4 | elmap 8428 | . . . . 5 ⊢ (𝐹 ∈ (𝐵 ↑m (ℕ0 ↑m 1o)) ↔ 𝐹:(ℕ0 ↑m 1o)⟶𝐵) |
6 | 1, 5 | sylibr 236 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m (ℕ0 ↑m 1o))) |
7 | df1o2 8109 | . . . . . . . . 9 ⊢ 1o = {∅} | |
8 | snfi 8587 | . . . . . . . . 9 ⊢ {∅} ∈ Fin | |
9 | 7, 8 | eqeltri 2908 | . . . . . . . 8 ⊢ 1o ∈ Fin |
10 | 9 | a1i 11 | . . . . . . 7 ⊢ (𝑓 ∈ (ℕ0 ↑m 1o) → 1o ∈ Fin) |
11 | elmapi 8421 | . . . . . . 7 ⊢ (𝑓 ∈ (ℕ0 ↑m 1o) → 𝑓:1o⟶ℕ0) | |
12 | 10, 11 | fisuppfi 8834 | . . . . . 6 ⊢ (𝑓 ∈ (ℕ0 ↑m 1o) → (◡𝑓 “ ℕ) ∈ Fin) |
13 | 12 | rabeqc 3674 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} = (ℕ0 ↑m 1o) |
14 | 13 | oveq2i 7160 | . . . 4 ⊢ (𝐵 ↑m {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin}) = (𝐵 ↑m (ℕ0 ↑m 1o)) |
15 | 6, 14 | eleqtrrdi 2923 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
16 | eqid 2820 | . . . 4 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
17 | eqid 2820 | . . . 4 ⊢ {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
18 | eqid 2820 | . . . 4 ⊢ (Base‘(1o mPwSer 𝑅)) = (Base‘(1o mPwSer 𝑅)) | |
19 | 1oex 8103 | . . . . 5 ⊢ 1o ∈ V | |
20 | 19 | a1i 11 | . . . 4 ⊢ (𝜑 → 1o ∈ V) |
21 | 16, 2, 17, 18, 20 | psrbas 20151 | . . 3 ⊢ (𝜑 → (Base‘(1o mPwSer 𝑅)) = (𝐵 ↑m {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
22 | 15, 21 | eleqtrrd 2915 | . 2 ⊢ (𝜑 → 𝐹 ∈ (Base‘(1o mPwSer 𝑅))) |
23 | fply1.5 | . 2 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
24 | eqid 2820 | . . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
25 | fply1.1 | . . 3 ⊢ 0 = (0g‘𝑅) | |
26 | eqid 2820 | . . . 4 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
27 | eqid 2820 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
28 | fply1.3 | . . . 4 ⊢ 𝑃 = (Base‘(Poly1‘𝑅)) | |
29 | 26, 27, 28 | ply1bas 20356 | . . 3 ⊢ 𝑃 = (Base‘(1o mPoly 𝑅)) |
30 | 24, 16, 18, 25, 29 | mplelbas 20203 | . 2 ⊢ (𝐹 ∈ 𝑃 ↔ (𝐹 ∈ (Base‘(1o mPwSer 𝑅)) ∧ 𝐹 finSupp 0 )) |
31 | 22, 23, 30 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝐹 ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 {crab 3141 Vcvv 3491 ∅c0 4284 {csn 4560 class class class wbr 5059 ◡ccnv 5547 “ cima 5551 ⟶wf 6344 ‘cfv 6348 (class class class)co 7149 1oc1o 8088 ↑m cmap 8399 Fincfn 8502 finSupp cfsupp 8826 ℕcn 11631 ℕ0cn0 11891 Basecbs 16476 0gc0g 16706 mPwSer cmps 20124 mPoly cmpl 20126 PwSer1cps1 20336 Poly1cpl1 20338 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12890 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-mulr 16572 df-sca 16574 df-vsca 16575 df-tset 16577 df-ple 16578 df-psr 20129 df-mpl 20131 df-opsr 20133 df-psr1 20341 df-ply1 20343 |
This theorem is referenced by: (None) |
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