mhpsclcl - Metamath Proof Explorer

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Theorem mhpsclcl 22064

Description: A scalar (or constant) polynomial has degree 0. Compare deg1scl 26042. In other contexts, there may be an exception for the zero polynomial, but under df-mhp 22055 the zero polynomial can be any degree (see mhp0cl 22063) so there is no exception. (Contributed by SN, 25-May-2024.)

**Hypotheses**
Ref	Expression
mhpsclcl.h	⊢ 𝐻 = (𝐼 mHomP 𝑅)
mhpsclcl.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mhpsclcl.a	⊢ 𝐴 = (algSc‘𝑃)
mhpsclcl.k	⊢ 𝐾 = (Base‘𝑅)
mhpsclcl.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
mhpsclcl.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mhpsclcl.c	⊢ (𝜑 → 𝐶 ∈ 𝐾)

**Assertion**
Ref	Expression
mhpsclcl	⊢ (𝜑 → (𝐴‘𝐶) ∈ (𝐻‘0))

Proof of Theorem mhpsclcl Dummy variables 𝑑 𝑦 ℎ 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mhpsclcl.p	. . . . . . 7 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
2		eqid 2728	. . . . . . 7 ⊢ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
3		eqid 2728	. . . . . . 7 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
4		mhpsclcl.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝑅)
5		mhpsclcl.a	. . . . . . 7 ⊢ 𝐴 = (algSc‘𝑃)
6		mhpsclcl.i	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
7	6	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑉)
8		mhpsclcl.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
9	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
10		mhpsclcl.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
11	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝐶 ∈ 𝐾)
12	1, 2, 3, 4, 5, 7, 9, 11	mplascl 22001	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → (𝐴‘𝐶) = (𝑦 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝐼 × {0}), 𝐶, (0_g‘𝑅))))
13		eqeq1 2732	. . . . . . . 8 ⊢ (𝑦 = 𝑑 → (𝑦 = (𝐼 × {0}) ↔ 𝑑 = (𝐼 × {0})))
14	13	ifbid 4547	. . . . . . 7 ⊢ (𝑦 = 𝑑 → if(𝑦 = (𝐼 × {0}), 𝐶, (0_g‘𝑅)) = if(𝑑 = (𝐼 × {0}), 𝐶, (0_g‘𝑅)))
15	14	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) ∧ 𝑦 = 𝑑) → if(𝑦 = (𝐼 × {0}), 𝐶, (0_g‘𝑅)) = if(𝑑 = (𝐼 × {0}), 𝐶, (0_g‘𝑅)))
16		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin})
17		fvexd 6906	. . . . . . . 8 ⊢ (𝜑 → (0_g‘𝑅) ∈ V)
18	10, 17	ifexd 4572	. . . . . . 7 ⊢ (𝜑 → if(𝑑 = (𝐼 × {0}), 𝐶, (0_g‘𝑅)) ∈ V)
19	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → if(𝑑 = (𝐼 × {0}), 𝐶, (0_g‘𝑅)) ∈ V)
20	12, 15, 16, 19	fvmptd 7006	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → ((𝐴‘𝐶)‘𝑑) = if(𝑑 = (𝐼 × {0}), 𝐶, (0_g‘𝑅)))
21	20	neeq1d 2996	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → (((𝐴‘𝐶)‘𝑑) ≠ (0_g‘𝑅) ↔ if(𝑑 = (𝐼 × {0}), 𝐶, (0_g‘𝑅)) ≠ (0_g‘𝑅)))
22		iffalse 4533	. . . . . 6 ⊢ (¬ 𝑑 = (𝐼 × {0}) → if(𝑑 = (𝐼 × {0}), 𝐶, (0_g‘𝑅)) = (0_g‘𝑅))
23	22	necon1ai 2964	. . . . 5 ⊢ (if(𝑑 = (𝐼 × {0}), 𝐶, (0_g‘𝑅)) ≠ (0_g‘𝑅) → 𝑑 = (𝐼 × {0}))
24		fconstmpt 5734	. . . . . . . 8 ⊢ (𝐼 × {0}) = (𝑘 ∈ 𝐼 ↦ 0)
25	24	oveq2i 7425	. . . . . . 7 ⊢ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝐼 × {0})) = ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑘 ∈ 𝐼 ↦ 0))
26		nn0subm 21348	. . . . . . . . 9 ⊢ ℕ₀ ∈ (SubMnd‘ℂ_fld)
27		eqid 2728	. . . . . . . . . 10 ⊢ (ℂ_fld ↾_s ℕ₀) = (ℂ_fld ↾_s ℕ₀)
28	27	submmnd 18758	. . . . . . . . 9 ⊢ (ℕ₀ ∈ (SubMnd‘ℂ_fld) → (ℂ_fld ↾_s ℕ₀) ∈ Mnd)
29	26, 28	ax-mp 5	. . . . . . . 8 ⊢ (ℂ_fld ↾_s ℕ₀) ∈ Mnd
30		cnfld0 21313	. . . . . . . . . . 11 ⊢ 0 = (0_g‘ℂ_fld)
31	27, 30	subm0 18760	. . . . . . . . . 10 ⊢ (ℕ₀ ∈ (SubMnd‘ℂ_fld) → 0 = (0_g‘(ℂ_fld ↾_s ℕ₀)))
32	26, 31	ax-mp 5	. . . . . . . . 9 ⊢ 0 = (0_g‘(ℂ_fld ↾_s ℕ₀))
33	32	gsumz 18781	. . . . . . . 8 ⊢ (((ℂ_fld ↾_s ℕ₀) ∈ Mnd ∧ 𝐼 ∈ 𝑉) → ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑘 ∈ 𝐼 ↦ 0)) = 0)
34	29, 7, 33	sylancr 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑘 ∈ 𝐼 ↦ 0)) = 0)
35	25, 34	eqtrid 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → ((ℂ_fld ↾_s ℕ₀) Σ_g (𝐼 × {0})) = 0)
36		oveq2 7422	. . . . . . 7 ⊢ (𝑑 = (𝐼 × {0}) → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑑) = ((ℂ_fld ↾_s ℕ₀) Σ_g (𝐼 × {0})))
37	36	eqeq1d 2730	. . . . . 6 ⊢ (𝑑 = (𝐼 × {0}) → (((ℂ_fld ↾_s ℕ₀) Σ_g 𝑑) = 0 ↔ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝐼 × {0})) = 0))
38	35, 37	syl5ibrcom 246	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → (𝑑 = (𝐼 × {0}) → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑑) = 0))
39	23, 38	syl5 34	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → (if(𝑑 = (𝐼 × {0}), 𝐶, (0_g‘𝑅)) ≠ (0_g‘𝑅) → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑑) = 0))
40	21, 39	sylbid 239	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → (((𝐴‘𝐶)‘𝑑) ≠ (0_g‘𝑅) → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑑) = 0))
41	40	ralrimiva 3142	. 2 ⊢ (𝜑 → ∀𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} (((𝐴‘𝐶)‘𝑑) ≠ (0_g‘𝑅) → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑑) = 0))
42		mhpsclcl.h	. . 3 ⊢ 𝐻 = (𝐼 mHomP 𝑅)
43		eqid 2728	. . 3 ⊢ (Base‘𝑃) = (Base‘𝑃)
44		0nn0 12511	. . . 4 ⊢ 0 ∈ ℕ₀
45	44	a1i 11	. . 3 ⊢ (𝜑 → 0 ∈ ℕ₀)
46	1, 43, 4, 5, 6, 8	mplasclf 22002	. . . 4 ⊢ (𝜑 → 𝐴:𝐾⟶(Base‘𝑃))
47	46, 10	ffvelcdmd 7089	. . 3 ⊢ (𝜑 → (𝐴‘𝐶) ∈ (Base‘𝑃))
48	42, 1, 43, 3, 2, 6, 8, 45, 47	ismhp3 22060	. 2 ⊢ (𝜑 → ((𝐴‘𝐶) ∈ (𝐻‘0) ↔ ∀𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} (((𝐴‘𝐶)‘𝑑) ≠ (0_g‘𝑅) → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑑) = 0)))
49	41, 48	mpbird 257	1 ⊢ (𝜑 → (𝐴‘𝐶) ∈ (𝐻‘0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ∀wral 3057 {crab 3428 Vcvv 3470 ifcif 4524 {csn 4624 ↦ cmpt 5225 × cxp 5670 ^◡ccnv 5671 “ cima 5675 ‘cfv 6542 (class class class)co 7414 ↑_m cmap 8838 Fincfn 8957 0cc0 11132 ℕcn 12236 ℕ₀cn0 12496 Basecbs 17173 ↾_s cress 17202 0_gc0g 17414 Σ_g cgsu 17415 Mndcmnd 18687 SubMndcsubmnd 18732 Ringcrg 20166 ℂ_fldccnfld 21272 algSccascl 21779 mPoly cmpl 21832 mHomP cmhp 22048

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 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-addf 11211

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 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-sup 9459 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-fzo 13654 df-seq 13993 df-hash 14316 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-0g 17416 df-gsum 17417 df-prds 17422 df-pws 17424 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-mhm 18733 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-mulg 19017 df-subg 19071 df-ghm 19161 df-cntz 19261 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-subrng 20476 df-subrg 20501 df-lmod 20738 df-lss 20809 df-cnfld 21273 df-ascl 21782 df-psr 21835 df-mpl 21837 df-mhp 22055

This theorem is referenced by: mhppwdeg 22067

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