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Theorem mplmon 21991

Description: A monomial is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)

**Hypotheses**
Ref	Expression
mplmon.s	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mplmon.b	⊢ 𝐵 = (Base‘𝑃)
mplmon.z	⊢ 0 = (0_g‘𝑅)
mplmon.o	⊢ 1 = (1_r‘𝑅)
mplmon.d	⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
mplmon.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
mplmon.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mplmon.x	⊢ (𝜑 → 𝑋 ∈ 𝐷)

**Assertion**
Ref	Expression
mplmon	⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵)

Distinct variable groups: 𝑦,𝐷 𝑓,𝐼 𝜑,𝑦 𝑦,𝑓,𝑋 𝑦, 0 𝑦, 1 𝑦,𝑅 Allowed substitution hints: 𝜑(𝑓) 𝐵(𝑦,𝑓) 𝐷(𝑓) 𝑃(𝑦,𝑓) 𝑅(𝑓) 1 (𝑓) 𝐼(𝑦) 𝑊(𝑦,𝑓) 0 (𝑓)

**Proof of Theorem mplmon**
Step	Hyp	Ref	Expression
1		mplmon.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
2		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		mplmon.o	. . . . . . . . 9 ⊢ 1 = (1_r‘𝑅)
4	2, 3	ringidcl 20204	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
5		mplmon.z	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑅)
6	2, 5	ring0cl 20206	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
7	4, 6	ifcld 4514	. . . . . . 7 ⊢ (𝑅 ∈ Ring → if(𝑦 = 𝑋, 1 , 0 ) ∈ (Base‘𝑅))
8	1, 7	syl 17	. . . . . 6 ⊢ (𝜑 → if(𝑦 = 𝑋, 1 , 0 ) ∈ (Base‘𝑅))
9	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → if(𝑦 = 𝑋, 1 , 0 ) ∈ (Base‘𝑅))
10	9	fmpttd 7059	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
11		fvex 6845	. . . . 5 ⊢ (Base‘𝑅) ∈ V
12		mplmon.d	. . . . . 6 ⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
13		ovex 7391	. . . . . 6 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
14	12, 13	rabex2 5276	. . . . 5 ⊢ 𝐷 ∈ V
15	11, 14	elmap 8810	. . . 4 ⊢ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ ((Base‘𝑅) ↑_m 𝐷) ↔ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
16	10, 15	sylibr 234	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ ((Base‘𝑅) ↑_m 𝐷))
17		eqid 2737	. . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
18		eqid 2737	. . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
19		mplmon.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
20	17, 2, 12, 18, 19	psrbas 21890	. . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑_m 𝐷))
21	16, 20	eleqtrrd 2840	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅)))
22	14	mptex 7169	. . . . 5 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ V
23		funmpt 6528	. . . . 5 ⊢ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))
24	5	fvexi 6846	. . . . 5 ⊢ 0 ∈ V
25	22, 23, 24	3pm3.2i 1341	. . . 4 ⊢ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ V ∧ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∧ 0 ∈ V)
26	25	a1i 11	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ V ∧ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∧ 0 ∈ V))
27		snfi 8981	. . . 4 ⊢ {𝑋} ∈ Fin
28	27	a1i 11	. . 3 ⊢ (𝜑 → {𝑋} ∈ Fin)
29		eldifsni 4734	. . . . . . 7 ⊢ (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦 ≠ 𝑋)
30	29	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦 ≠ 𝑋)
31	30	neneqd 2938	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋)
32	31	iffalsed 4478	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 )
33	14	a1i 11	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
34	32, 33	suppss2 8141	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋})
35		suppssfifsupp 9284	. . 3 ⊢ ((((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ V ∧ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∧ 0 ∈ V) ∧ ({𝑋} ∈ Fin ∧ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋})) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) finSupp 0 )
36	26, 28, 34, 35	syl12anc 837	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) finSupp 0 )
37		mplmon.s	. . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
38		mplmon.b	. . 3 ⊢ 𝐵 = (Base‘𝑃)
39	37, 17, 18, 5, 38	mplelbas 21947	. 2 ⊢ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵 ↔ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) finSupp 0 ))
40	21, 36, 39	sylanbrc 584	1 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 {crab 3390 Vcvv 3430 ∖ cdif 3887 ⊆ wss 3890 ifcif 4467 {csn 4568 class class class wbr 5086 ↦ cmpt 5167 ^◡ccnv 5621 “ cima 5625 Fun wfun 6484 ⟶wf 6486 ‘cfv 6490 (class class class)co 7358 supp csupp 8101 ↑_m cmap 8764 Fincfn 8884 finSupp cfsupp 9265 ℕcn 12146 ℕ₀cn0 12402 Basecbs 17137 0_gc0g 17360 1_rcur 20120 Ringcrg 20172 mPwSer cmps 21861 mPoly cmpl 21863

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-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

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-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 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-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-er 8634 df-map 8766 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-uz 12753 df-fz 13425 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17138 df-ress 17159 df-plusg 17191 df-mulr 17192 df-sca 17194 df-vsca 17195 df-tset 17197 df-0g 17362 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18870 df-mgp 20080 df-ur 20121 df-ring 20174 df-psr 21866 df-mpl 21868

This theorem is referenced by: mplmonmul 21992 mplcoe1 21993 mplbas2 21998 mplmon2 22017 mplmon2cl 22024 mplmon2mul 22025 selvvvval 43017

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