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Mirrors > Home > MPE Home > Th. List > mhpsubg | Structured version Visualization version GIF version |
Description: Homogeneous polynomials form a subgroup of the polynomials. (Contributed by SN, 25-Sep-2023.) |
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) |
Ref | Expression |
---|---|
mhpsubg | ⊢ (𝜑 → (𝐻‘𝑁) ∈ (SubGrp‘𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mhpsubg.h | . . . . 5 ⊢ 𝐻 = (𝐼 mHomP 𝑅) | |
2 | mhpsubg.p | . . . . 5 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
3 | eqid 2737 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
4 | mhpsubg.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
5 | 4 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝐼 ∈ 𝑉) |
6 | mhpsubg.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
7 | 6 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑅 ∈ Grp) |
8 | mhpsubg.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
9 | 8 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑁 ∈ ℕ0) |
10 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑥 ∈ (𝐻‘𝑁)) | |
11 | 1, 2, 3, 5, 7, 9, 10 | mhpmpl 21089 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑥 ∈ (Base‘𝑃)) |
12 | 11 | ex 416 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐻‘𝑁) → 𝑥 ∈ (Base‘𝑃))) |
13 | 12 | ssrdv 3912 | . 2 ⊢ (𝜑 → (𝐻‘𝑁) ⊆ (Base‘𝑃)) |
14 | eqid 2737 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
15 | eqid 2737 | . . . 4 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
16 | 1, 14, 15, 4, 6, 8 | mhp0cl 21091 | . . 3 ⊢ (𝜑 → ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝑅)}) ∈ (𝐻‘𝑁)) |
17 | 16 | ne0d 4255 | . 2 ⊢ (𝜑 → (𝐻‘𝑁) ≠ ∅) |
18 | eqid 2737 | . . . . . 6 ⊢ (+g‘𝑃) = (+g‘𝑃) | |
19 | 5 | adantr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝐼 ∈ 𝑉) |
20 | 7 | adantr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝑅 ∈ Grp) |
21 | 9 | adantr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝑁 ∈ ℕ0) |
22 | simplr 769 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝑥 ∈ (𝐻‘𝑁)) | |
23 | simpr 488 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝑦 ∈ (𝐻‘𝑁)) | |
24 | 1, 2, 18, 19, 20, 21, 22, 23 | mhpaddcl 21096 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → (𝑥(+g‘𝑃)𝑦) ∈ (𝐻‘𝑁)) |
25 | 24 | ralrimiva 3105 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → ∀𝑦 ∈ (𝐻‘𝑁)(𝑥(+g‘𝑃)𝑦) ∈ (𝐻‘𝑁)) |
26 | eqid 2737 | . . . . 5 ⊢ (invg‘𝑃) = (invg‘𝑃) | |
27 | 1, 2, 26, 5, 7, 9, 10 | mhpinvcl 21097 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → ((invg‘𝑃)‘𝑥) ∈ (𝐻‘𝑁)) |
28 | 25, 27 | jca 515 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → (∀𝑦 ∈ (𝐻‘𝑁)(𝑥(+g‘𝑃)𝑦) ∈ (𝐻‘𝑁) ∧ ((invg‘𝑃)‘𝑥) ∈ (𝐻‘𝑁))) |
29 | 28 | ralrimiva 3105 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝐻‘𝑁)(∀𝑦 ∈ (𝐻‘𝑁)(𝑥(+g‘𝑃)𝑦) ∈ (𝐻‘𝑁) ∧ ((invg‘𝑃)‘𝑥) ∈ (𝐻‘𝑁))) |
30 | 2 | mplgrp 20983 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp) |
31 | 4, 6, 30 | syl2anc 587 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Grp) |
32 | 3, 18, 26 | issubg2 18563 | . . 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) ∧ ((invg‘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) ∧ ((invg‘𝑃)‘𝑥) ∈ (𝐻‘𝑁))))) |
34 | 13, 17, 29, 33 | mpbir3and 1344 | 1 ⊢ (𝜑 → (𝐻‘𝑁) ∈ (SubGrp‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∀wral 3061 {crab 3065 ⊆ wss 3871 ∅c0 4242 {csn 4546 × cxp 5554 ◡ccnv 5555 “ cima 5559 ‘cfv 6385 (class class class)co 7218 ↑m cmap 8513 Fincfn 8631 ℕcn 11835 ℕ0cn0 12095 Basecbs 16765 +gcplusg 16807 0gc0g 16949 Grpcgrp 18370 invgcminusg 18371 SubGrpcsubg 18542 mPoly cmpl 20870 mHomP cmhp 21074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5184 ax-sep 5197 ax-nul 5204 ax-pow 5263 ax-pr 5327 ax-un 7528 ax-cnex 10790 ax-resscn 10791 ax-1cn 10792 ax-icn 10793 ax-addcl 10794 ax-addrcl 10795 ax-mulcl 10796 ax-mulrcl 10797 ax-mulcom 10798 ax-addass 10799 ax-mulass 10800 ax-distr 10801 ax-i2m1 10802 ax-1ne0 10803 ax-1rid 10804 ax-rnegex 10805 ax-rrecex 10806 ax-cnre 10807 ax-pre-lttri 10808 ax-pre-lttrn 10809 ax-pre-ltadd 10810 ax-pre-mulgt0 10811 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3415 df-sbc 3700 df-csb 3817 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-pss 3890 df-nul 4243 df-if 4445 df-pw 4520 df-sn 4547 df-pr 4549 df-tp 4551 df-op 4553 df-uni 4825 df-iun 4911 df-br 5059 df-opab 5121 df-mpt 5141 df-tr 5167 df-id 5460 df-eprel 5465 df-po 5473 df-so 5474 df-fr 5514 df-we 5516 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-pred 6165 df-ord 6221 df-on 6222 df-lim 6223 df-suc 6224 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-f1 6390 df-fo 6391 df-f1o 6392 df-fv 6393 df-riota 7175 df-ov 7221 df-oprab 7222 df-mpo 7223 df-of 7474 df-om 7650 df-1st 7766 df-2nd 7767 df-supp 7909 df-wrecs 8052 df-recs 8113 df-rdg 8151 df-1o 8207 df-er 8396 df-map 8515 df-en 8632 df-dom 8633 df-sdom 8634 df-fin 8635 df-fsupp 8991 df-pnf 10874 df-mnf 10875 df-xr 10876 df-ltxr 10877 df-le 10878 df-sub 11069 df-neg 11070 df-nn 11836 df-2 11898 df-3 11899 df-4 11900 df-5 11901 df-6 11902 df-7 11903 df-8 11904 df-9 11905 df-n0 12096 df-z 12182 df-uz 12444 df-fz 13101 df-struct 16705 df-sets 16722 df-slot 16740 df-ndx 16750 df-base 16766 df-ress 16790 df-plusg 16820 df-mulr 16821 df-sca 16823 df-vsca 16824 df-tset 16826 df-0g 16951 df-mgm 18119 df-sgrp 18168 df-mnd 18179 df-grp 18373 df-minusg 18374 df-subg 18545 df-psr 20873 df-mpl 20875 df-mhp 21078 |
This theorem is referenced by: mhplss 21100 |
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