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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 5042 | . 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 20908 | . 2 ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } |
8 | 1, 7 | elrab2 3594 | 1 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 finSupp 0 )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 finSupp cfsupp 8963 Basecbs 16666 0gc0g 16898 mPwSer cmps 20817 mPoly cmpl 20819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-1cn 10752 ax-addcl 10754 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-nn 11796 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-psr 20822 df-mpl 20824 |
This theorem is referenced by: mplelsfi 20911 mplsubrglem 20920 mplsubrg 20921 mvrcl 20931 mplmon 20946 mplcoe1 20948 mplbas2 20953 fply1 31335 selvval2lem4 39882 evlsbagval 39926 mhpind 39934 mhphf 39936 |
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