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Mirrors > Home > MPE Home > Th. List > mplbas | Structured version Visualization version GIF version |
Description: Base set of the set of multivariate polynomials. (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 |
---|---|
mplbas | ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mplbas.u | . 2 ⊢ 𝑈 = (Base‘𝑃) | |
2 | ssrab2 4056 | . . 3 ⊢ {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } ⊆ 𝐵 | |
3 | mplval.p | . . . . 5 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
4 | mplval.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
5 | mplval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑆) | |
6 | mplval.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
7 | eqid 2821 | . . . . 5 ⊢ {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } | |
8 | 3, 4, 5, 6, 7 | mplval 20202 | . . . 4 ⊢ 𝑃 = (𝑆 ↾s {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 }) |
9 | 8, 5 | ressbas2 16549 | . . 3 ⊢ ({𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } ⊆ 𝐵 → {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } = (Base‘𝑃)) |
10 | 2, 9 | ax-mp 5 | . 2 ⊢ {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } = (Base‘𝑃) |
11 | 1, 10 | eqtr4i 2847 | 1 ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 {crab 3142 ⊆ wss 3936 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 finSupp cfsupp 8827 Basecbs 16477 0gc0g 16707 mPwSer cmps 20125 mPoly cmpl 20127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-1cn 10589 ax-addcl 10591 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-nn 11633 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-psr 20130 df-mpl 20132 |
This theorem is referenced by: mplelbas 20204 mplval2 20205 mplbasss 20206 mplsubglem2 20210 ressmplbas2 20230 mhpinvcl 20333 mplbaspropd 20399 |
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