mhpsubg - Metamath Proof Explorer

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Theorem mhpsubg 21915

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 2732	. . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃)
4		mhpsubg.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
5	4	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝐼 ∈ 𝑉)
6		mhpsubg.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
7	6	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑅 ∈ Grp)
8		mhpsubg.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
9	8	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑁 ∈ ℕ₀)
10		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑥 ∈ (𝐻‘𝑁))
11	1, 2, 3, 5, 7, 9, 10	mhpmpl 21906	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑥 ∈ (Base‘𝑃))
12	11	ex 413	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐻‘𝑁) → 𝑥 ∈ (Base‘𝑃)))
13	12	ssrdv 3988	. 2 ⊢ (𝜑 → (𝐻‘𝑁) ⊆ (Base‘𝑃))
14		eqid 2732	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
15		eqid 2732	. . . 4 ⊢ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
16	1, 14, 15, 4, 6, 8	mhp0cl 21908	. . 3 ⊢ (𝜑 → ({ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} × {(0_g‘𝑅)}) ∈ (𝐻‘𝑁))
17	16	ne0d 4335	. 2 ⊢ (𝜑 → (𝐻‘𝑁) ≠ ∅)
18		eqid 2732	. . . . . 6 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
19	5	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝐼 ∈ 𝑉)
20	7	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝑅 ∈ Grp)
21	9	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝑁 ∈ ℕ₀)
22		simplr 767	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝑥 ∈ (𝐻‘𝑁))
23		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝑦 ∈ (𝐻‘𝑁))
24	1, 2, 18, 19, 20, 21, 22, 23	mhpaddcl 21913	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → (𝑥(+_g‘𝑃)𝑦) ∈ (𝐻‘𝑁))
25	24	ralrimiva 3146	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → ∀𝑦 ∈ (𝐻‘𝑁)(𝑥(+_g‘𝑃)𝑦) ∈ (𝐻‘𝑁))
26		eqid 2732	. . . . 5 ⊢ (inv_g‘𝑃) = (inv_g‘𝑃)
27	1, 2, 26, 5, 7, 9, 10	mhpinvcl 21914	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → ((inv_g‘𝑃)‘𝑥) ∈ (𝐻‘𝑁))
28	25, 27	jca 512	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → (∀𝑦 ∈ (𝐻‘𝑁)(𝑥(+_g‘𝑃)𝑦) ∈ (𝐻‘𝑁) ∧ ((inv_g‘𝑃)‘𝑥) ∈ (𝐻‘𝑁)))
29	28	ralrimiva 3146	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝐻‘𝑁)(∀𝑦 ∈ (𝐻‘𝑁)(𝑥(+_g‘𝑃)𝑦) ∈ (𝐻‘𝑁) ∧ ((inv_g‘𝑃)‘𝑥) ∈ (𝐻‘𝑁)))
30	2	mplgrp 21795	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp)
31	4, 6, 30	syl2anc 584	. . 3 ⊢ (𝜑 → 𝑃 ∈ Grp)
32	3, 18, 26	issubg2 19057	. . 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)))))
33	31, 32	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)))))
34	13, 17, 29, 33	mpbir3and 1342	1 ⊢ (𝜑 → (𝐻‘𝑁) ∈ (SubGrp‘𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 {crab 3432 ⊆ wss 3948 ∅c0 4322 {csn 4628 × cxp 5674 ^◡ccnv 5675 “ cima 5679 ‘cfv 6543 (class class class)co 7411 ↑_m cmap 8822 Fincfn 8941 ℕcn 12216 ℕ₀cn0 12476 Basecbs 17148 +_gcplusg 17201 0_gc0g 17389 Grpcgrp 18855 inv_gcminusg 18856 SubGrpcsubg 19036 mPoly cmpl 21678 mHomP cmhp 21891

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-hom 17225 df-cco 17226 df-0g 17391 df-prds 17397 df-pws 17399 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-subg 19039 df-psr 21681 df-mpl 21683 df-mhp 21895

This theorem is referenced by: mhplss 21917

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