mplmon - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > mplmon		Structured version Visualization version GIF version

Theorem mplmon 21973

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 2728	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		mplmon.o	. . . . . . . . 9 ⊢ 1 = (1_r‘𝑅)
4	2, 3	ringidcl 20202	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
5		mplmon.z	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑅)
6	2, 5	ring0cl 20203	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
7	4, 6	ifcld 4575	. . . . . . 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 7125	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
11		fvex 6910	. . . . 5 ⊢ (Base‘𝑅) ∈ V
12		mplmon.d	. . . . . 6 ⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
13		ovex 7453	. . . . . 6 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
14	12, 13	rabex2 5336	. . . . 5 ⊢ 𝐷 ∈ V
15	11, 14	elmap 8890	. . . 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 233	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ ((Base‘𝑅) ↑_m 𝐷))
17		eqid 2728	. . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
18		eqid 2728	. . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
19		mplmon.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
20	17, 2, 12, 18, 19	psrbas 21878	. . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑_m 𝐷))
21	16, 20	eleqtrrd 2832	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅)))
22	14	mptex 7235	. . . . 5 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ V
23		funmpt 6591	. . . . 5 ⊢ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))
24	5	fvexi 6911	. . . . 5 ⊢ 0 ∈ V
25	22, 23, 24	3pm3.2i 1337	. . . 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 9069	. . . 4 ⊢ {𝑋} ∈ Fin
28	27	a1i 11	. . 3 ⊢ (𝜑 → {𝑋} ∈ Fin)
29		eldifsni 4794	. . . . . . 7 ⊢ (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦 ≠ 𝑋)
30	29	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦 ≠ 𝑋)
31	30	neneqd 2942	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋)
32	31	iffalsed 4540	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 )
33	14	a1i 11	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
34	32, 33	suppss2 8206	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋})
35		suppssfifsupp 9404	. . 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 836	. 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 21933	. 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 582	1 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 {crab 3429 Vcvv 3471 ∖ cdif 3944 ⊆ wss 3947 ifcif 4529 {csn 4629 class class class wbr 5148 ↦ cmpt 5231 ^◡ccnv 5677 “ cima 5681 Fun wfun 6542 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 supp csupp 8165 ↑_m cmap 8845 Fincfn 8964 finSupp cfsupp 9386 ℕcn 12243 ℕ₀cn0 12503 Basecbs 17180 0_gc0g 17421 1_rcur 20121 Ringcrg 20173 mPwSer cmps 21837 mPoly cmpl 21839

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-tset 17252 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-mgp 20075 df-ur 20122 df-ring 20175 df-psr 21842 df-mpl 21844

This theorem is referenced by: mplmonmul 21974 mplcoe1 21975 mplbas2 21980 mplmon2 22005 mplmon2cl 22012 mplmon2mul 22013 selvvvval 41818

W3C validator