mhpsubg - Metamath Proof Explorer

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Theorem mhpsubg 22068

Description: Homogeneous polynomials form a subgroup of the polynomials. (Contributed by SN, 25-Sep-2023.)

**Hypotheses**
Ref	Expression
mhpsubg.h	⊢ 𝐻 = (𝐼 mHomP 𝑅)
mhpsubg.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mhpsubg.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
mhpsubg.r	⊢ (𝜑 → 𝑅 ∈ Grp)
mhpsubg.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)

**Assertion**
Ref	Expression
mhpsubg	⊢ (𝜑 → (𝐻‘𝑁) ∈ (SubGrp‘𝑃))

Proof of Theorem mhpsubg Dummy variables 𝑥 𝑦 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mhpsubg.h	. . . . 5 ⊢ 𝐻 = (𝐼 mHomP 𝑅)
2		mhpsubg.p	. . . . 5 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
3		eqid 2731	. . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃)
4		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑥 ∈ (𝐻‘𝑁))
5	1, 2, 3, 4	mhpmpl 22059	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑥 ∈ (Base‘𝑃))
6	5	ex 412	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐻‘𝑁) → 𝑥 ∈ (Base‘𝑃)))
7	6	ssrdv 3935	. 2 ⊢ (𝜑 → (𝐻‘𝑁) ⊆ (Base‘𝑃))
8		eqid 2731	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
9		eqid 2731	. . . 4 ⊢ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
10		mhpsubg.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
11		mhpsubg.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
12		mhpsubg.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
13	1, 8, 9, 10, 11, 12	mhp0cl 22061	. . 3 ⊢ (𝜑 → ({ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} × {(0_g‘𝑅)}) ∈ (𝐻‘𝑁))
14	13	ne0d 4289	. 2 ⊢ (𝜑 → (𝐻‘𝑁) ≠ ∅)
15		eqid 2731	. . . . . 6 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
16	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑅 ∈ Grp)
17	16	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝑅 ∈ Grp)
18		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝑥 ∈ (𝐻‘𝑁))
19		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝑦 ∈ (𝐻‘𝑁))
20	1, 2, 15, 17, 18, 19	mhpaddcl 22066	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → (𝑥(+_g‘𝑃)𝑦) ∈ (𝐻‘𝑁))
21	20	ralrimiva 3124	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → ∀𝑦 ∈ (𝐻‘𝑁)(𝑥(+_g‘𝑃)𝑦) ∈ (𝐻‘𝑁))
22		eqid 2731	. . . . 5 ⊢ (inv_g‘𝑃) = (inv_g‘𝑃)
23	1, 2, 22, 16, 4	mhpinvcl 22067	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → ((inv_g‘𝑃)‘𝑥) ∈ (𝐻‘𝑁))
24	21, 23	jca 511	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → (∀𝑦 ∈ (𝐻‘𝑁)(𝑥(+_g‘𝑃)𝑦) ∈ (𝐻‘𝑁) ∧ ((inv_g‘𝑃)‘𝑥) ∈ (𝐻‘𝑁)))
25	24	ralrimiva 3124	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝐻‘𝑁)(∀𝑦 ∈ (𝐻‘𝑁)(𝑥(+_g‘𝑃)𝑦) ∈ (𝐻‘𝑁) ∧ ((inv_g‘𝑃)‘𝑥) ∈ (𝐻‘𝑁)))
26	2	mplgrp 21954	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp)
27	10, 11, 26	syl2anc 584	. . 3 ⊢ (𝜑 → 𝑃 ∈ Grp)
28	3, 15, 22	issubg2 19054	. . 3 ⊢ (𝑃 ∈ Grp → ((𝐻‘𝑁) ∈ (SubGrp‘𝑃) ↔ ((𝐻‘𝑁) ⊆ (Base‘𝑃) ∧ (𝐻‘𝑁) ≠ ∅ ∧ ∀𝑥 ∈ (𝐻‘𝑁)(∀𝑦 ∈ (𝐻‘𝑁)(𝑥(+_g‘𝑃)𝑦) ∈ (𝐻‘𝑁) ∧ ((inv_g‘𝑃)‘𝑥) ∈ (𝐻‘𝑁)))))
29	27, 28	syl 17	. 2 ⊢ (𝜑 → ((𝐻‘𝑁) ∈ (SubGrp‘𝑃) ↔ ((𝐻‘𝑁) ⊆ (Base‘𝑃) ∧ (𝐻‘𝑁) ≠ ∅ ∧ ∀𝑥 ∈ (𝐻‘𝑁)(∀𝑦 ∈ (𝐻‘𝑁)(𝑥(+_g‘𝑃)𝑦) ∈ (𝐻‘𝑁) ∧ ((inv_g‘𝑃)‘𝑥) ∈ (𝐻‘𝑁)))))
30	7, 14, 25, 29	mpbir3and 1343	1 ⊢ (𝜑 → (𝐻‘𝑁) ∈ (SubGrp‘𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 {crab 3395 ⊆ wss 3897 ∅c0 4280 {csn 4573 × cxp 5612 ^◡ccnv 5613 “ cima 5617 ‘cfv 6481 (class class class)co 7346 ↑_m cmap 8750 Fincfn 8869 ℕcn 12125 ℕ₀cn0 12381 Basecbs 17120 +_gcplusg 17161 0_gc0g 17343 Grpcgrp 18846 inv_gcminusg 18847 SubGrpcsubg 19033 mPoly cmpl 21843 mHomP cmhp 22044

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-map 8752 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-fz 13408 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-hom 17185 df-cco 17186 df-0g 17345 df-prds 17351 df-pws 17353 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-grp 18849 df-minusg 18850 df-subg 19036 df-psr 21846 df-mpl 21848 df-mhp 22051

This theorem is referenced by: mhplss 22070

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