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Theorem mplbas 21770

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 4078	. . 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 2730	. . . . 5 ⊢ {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 }
8	3, 4, 5, 6, 7	mplval 21769	. . . 4 ⊢ 𝑃 = (𝑆 ↾_s {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 })
9	8, 5	ressbas2 17188	. . 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 2761	1 ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 }

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 {crab 3430 ⊆ wss 3949 class class class wbr 5149 ‘cfv 6544 (class class class)co 7413 finSupp cfsupp 9365 Basecbs 17150 0_gc0g 17391 mPwSer cmps 21678 mPoly cmpl 21680

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-1cn 11172 ax-addcl 11174

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-nn 12219 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-psr 21683 df-mpl 21685

This theorem is referenced by: mplelbas 21771 mplval2 21776 mplbasss 21777 mplsubglem2 21781 ressmplbas2 21803 mplbaspropd 21981

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