![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mplelbas | Structured version Visualization version GIF version |
Description: Property of being a polynomial. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.) |
Ref | Expression |
---|---|
mplval.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplval.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
mplval.b | ⊢ 𝐵 = (Base‘𝑆) |
mplval.z | ⊢ 0 = (0g‘𝑅) |
mplbas.u | ⊢ 𝑈 = (Base‘𝑃) |
Ref | Expression |
---|---|
mplelbas | ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 finSupp 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5150 | . 2 ⊢ (𝑓 = 𝑋 → (𝑓 finSupp 0 ↔ 𝑋 finSupp 0 )) | |
2 | mplval.p | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
3 | mplval.s | . . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
4 | mplval.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
5 | mplval.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
6 | mplbas.u | . . 3 ⊢ 𝑈 = (Base‘𝑃) | |
7 | 2, 3, 4, 5, 6 | mplbas 22027 | . 2 ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } |
8 | 1, 7 | elrab2 3697 | 1 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 finSupp 0 )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 finSupp cfsupp 9398 Basecbs 17244 0gc0g 17485 mPwSer cmps 21941 mPoly cmpl 21943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-1cn 11210 ax-addcl 11212 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-nn 12264 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-psr 21946 df-mpl 21948 |
This theorem is referenced by: mvrcl 22029 mplelsfi 22032 mplsubrglem 22041 mplsubrg 22042 mplmon 22070 mplcoe1 22072 mplbas2 22077 psdmplcl 22183 mhmcompl 22399 fply1 33563 evlsbagval 42552 mhpind 42580 |
Copyright terms: Public domain | W3C validator |