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Theorem mplmon 22014

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 2736	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		mplmon.o	. . . . . . . . 9 ⊢ 1 = (1_r‘𝑅)
4	2, 3	ringidcl 20240	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
5		mplmon.z	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑅)
6	2, 5	ring0cl 20242	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
7	4, 6	ifcld 4504	. . . . . . 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 481	. . . . 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 6843	. . . . 5 ⊢ (Base‘𝑅) ∈ V
12		mplmon.d	. . . . . 6 ⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
13		ovex 7392	. . . . . 6 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
14	12, 13	rabex2 5272	. . . . 5 ⊢ 𝐷 ∈ V
15	11, 14	elmap 8812	. . . 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 235	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ ((Base‘𝑅) ↑_m 𝐷))
17		eqid 2736	. . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
18		eqid 2736	. . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
19		mplmon.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
20	17, 2, 12, 18, 19	psrbas 21912	. . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑_m 𝐷))
21	16, 20	eleqtrrd 2839	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅)))
22	14	mptex 7170	. . . . 5 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ V
23		funmpt 6526	. . . . 5 ⊢ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))
24	5	fvexi 6844	. . . . 5 ⊢ 0 ∈ V
25	22, 23, 24	3pm3.2i 1342	. . . 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 8983	. . . 4 ⊢ {𝑋} ∈ Fin
28	27	a1i 11	. . 3 ⊢ (𝜑 → {𝑋} ∈ Fin)
29		eldifsni 4726	. . . . . . 7 ⊢ (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦 ≠ 𝑋)
30	29	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦 ≠ 𝑋)
31	30	neneqd 2936	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋)
32	31	iffalsed 4468	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 )
33	14	a1i 11	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
34	32, 33	suppss2 8143	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋})
35		suppssfifsupp 9286	. . 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 838	. 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 21968	. 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 585	1 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1088 = wceq 1543 ∈ wcel 2115 ≠ wne 2931 {crab 3388 Vcvv 3428 ∖ cdif 3883 ⊆ wss 3886 ifcif 4457 {csn 4558 class class class wbr 5075 ↦ cmpt 5156 ^◡ccnv 5620 “ cima 5624 Fun wfun 6482 ⟶wf 6484 ‘cfv 6488 (class class class)co 7359 supp csupp 8103 ↑_m cmap 8766 Fincfn 8886 finSupp cfsupp 9267 ℕcn 12168 ℕ₀cn0 12431 Basecbs 17173 0_gc0g 17396 1_rcur 20156 Ringcrg 20208 mPwSer cmps 21882 mPoly cmpl 21884

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 1970 ax-7 2011 ax-8 2117 ax-9 2125 ax-10 2148 ax-11 2164 ax-12 2185 ax-ext 2708 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7681 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 850 df-3or 1089 df-3an 1090 df-tru 1546 df-fal 1556 df-ex 1783 df-nf 1787 df-sb 2070 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3061 df-rmo 3341 df-reu 3342 df-rab 3389 df-v 3430 df-sbc 3727 df-csb 3835 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3906 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-of 7623 df-om 7810 df-1st 7934 df-2nd 7935 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-uz 12783 df-fz 13456 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-sca 17230 df-vsca 17231 df-tset 17233 df-0g 17398 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-grp 18906 df-mgp 20116 df-ur 20157 df-ring 20210 df-psr 21887 df-mpl 21889

This theorem is referenced by: mplmonmul 22015 mplcoe1 22016 mplbas2 22021 mplmon2 22040 mplmon2cl 22047 mplmon2mul 22048 selvvvval 22121

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