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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 5107 | . 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 21394 | . 2 ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } |
8 | 1, 7 | elrab2 3647 | 1 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 finSupp 0 )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5104 ‘cfv 6494 (class class class)co 7354 finSupp cfsupp 9302 Basecbs 17080 0gc0g 17318 mPwSer cmps 21302 mPoly cmpl 21304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-1cn 11106 ax-addcl 11108 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-nn 12151 df-sets 17033 df-slot 17051 df-ndx 17063 df-base 17081 df-ress 17110 df-psr 21307 df-mpl 21309 |
This theorem is referenced by: mplelsfi 21397 mplsubrglem 21406 mplsubrg 21407 mvrcl 21417 mplmon 21432 mplcoe1 21434 mplbas2 21439 fply1 32156 mhmcompl 40714 evlsbagval 40726 mhpind 40745 mhphf 40747 |
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