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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 5069 | . 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 20209 | . 2 ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } |
8 | 1, 7 | elrab2 3683 | 1 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 finSupp 0 )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 finSupp cfsupp 8833 Basecbs 16483 0gc0g 16713 mPwSer cmps 20131 mPoly cmpl 20133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-1cn 10595 ax-addcl 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-nn 11639 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-psr 20136 df-mpl 20138 |
This theorem is referenced by: mplsubrglem 20219 mplsubrg 20220 mvrcl 20229 mplmon 20244 mplcoe1 20246 mplbas2 20251 mplelsfi 20271 mhpinvcl 20339 fply1 30931 selvval2lem4 39156 |
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