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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 4586 | . 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 19250 | . 2 ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } |
8 | 1, 7 | elrab2 3333 | 1 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 finSupp 0 )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 finSupp cfsupp 8158 Basecbs 15695 0gc0g 15923 mPwSer cmps 19172 mPoly cmpl 19174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-psr 19177 df-mpl 19179 |
This theorem is referenced by: mplsubrglem 19260 mplsubrg 19261 mvrcl 19270 mplmon 19284 mplcoe1 19286 mplbas2 19291 mplelsfi 19312 |
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