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Theorem mplbas 21978

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 4021	. . 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 2737	. . . . 5 ⊢ {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 }
8	3, 4, 5, 6, 7	mplval 21977	. . . 4 ⊢ 𝑃 = (𝑆 ↾_s {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 })
9	8, 5	ressbas2 17199	. . 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 2763	1 ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 }

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 {crab 3390 ⊆ wss 3890 class class class wbr 5086 ‘cfv 6492 (class class class)co 7360 finSupp cfsupp 9267 Basecbs 17170 0_gc0g 17393 mPwSer cmps 21894 mPoly cmpl 21896

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-1cn 11087 ax-addcl 11089

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-nn 12166 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-psr 21899 df-mpl 21901

This theorem is referenced by: mplelbas 21979 mplval2 21984 mplbasss 21985 mplsubglem2 21989 ressmplbas2 22015 mplbaspropd 22210

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