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Theorem mplbas 21198

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.)

**Hypotheses**
Ref	Expression
mplval.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mplval.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
mplval.b	⊢ 𝐵 = (Base‘𝑆)
mplval.z	⊢ 0 = (0_g‘𝑅)
mplbas.u	⊢ 𝑈 = (Base‘𝑃)

**Assertion**
Ref	Expression
mplbas	⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 }

Distinct variable groups: 𝐵,𝑓 𝑓,𝐼 𝑅,𝑓 0 ,𝑓 Allowed substitution hints: 𝑃(𝑓) 𝑆(𝑓) 𝑈(𝑓)

**Proof of Theorem mplbas**
Step	Hyp	Ref	Expression
1		mplbas.u	. 2 ⊢ 𝑈 = (Base‘𝑃)
2		ssrab2 4013	. . 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 = (0_g‘𝑅)
7		eqid 2738	. . . . 5 ⊢ {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 }
8	3, 4, 5, 6, 7	mplval 21197	. . . 4 ⊢ 𝑃 = (𝑆 ↾_s {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 })
9	8, 5	ressbas2 16949	. . 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 2769	1 ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 }

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 {crab 3068 ⊆ wss 3887 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 finSupp cfsupp 9128 Basecbs 16912 0_gc0g 17150 mPwSer cmps 21107 mPoly cmpl 21109

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-1cn 10929 ax-addcl 10931

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-nn 11974 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-psr 21112 df-mpl 21114

This theorem is referenced by: mplelbas 21199 mplval2 21202 mplbasss 21203 mplsubglem2 21207 ressmplbas2 21228 mplbaspropd 21408

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