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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 2820 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
4 | mhpsubg.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
5 | 4 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝐼 ∈ 𝑉) |
6 | mhpsubg.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
7 | 6 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑅 ∈ Grp) |
8 | mhpsubg.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
9 | 8 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑁 ∈ ℕ0) |
10 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑥 ∈ (𝐻‘𝑁)) | |
11 | 1, 2, 3, 5, 7, 9, 10 | mhpmpl 20328 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑥 ∈ (Base‘𝑃)) |
12 | 11 | ex 415 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐻‘𝑁) → 𝑥 ∈ (Base‘𝑃))) |
13 | 12 | ssrdv 3966 | . 2 ⊢ (𝜑 → (𝐻‘𝑁) ⊆ (Base‘𝑃)) |
14 | eqid 2820 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
15 | eqid 2820 | . . . 4 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
16 | 1, 14, 15, 4, 6, 8 | mhp0cl 20330 | . . 3 ⊢ (𝜑 → ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝑅)}) ∈ (𝐻‘𝑁)) |
17 | 16 | ne0d 4294 | . 2 ⊢ (𝜑 → (𝐻‘𝑁) ≠ ∅) |
18 | eqid 2820 | . . . . . 6 ⊢ (+g‘𝑃) = (+g‘𝑃) | |
19 | 5 | adantr 483 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝐼 ∈ 𝑉) |
20 | 7 | adantr 483 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝑅 ∈ Grp) |
21 | 9 | adantr 483 | . . . . . 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 487 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝑦 ∈ (𝐻‘𝑁)) | |
24 | 1, 2, 18, 19, 20, 21, 22, 23 | mhpaddcl 20331 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) ∧ 𝑦 ∈ (𝐻‘𝑁)) → (𝑥(+g‘𝑃)𝑦) ∈ (𝐻‘𝑁)) |
25 | 24 | ralrimiva 3181 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → ∀𝑦 ∈ (𝐻‘𝑁)(𝑥(+g‘𝑃)𝑦) ∈ (𝐻‘𝑁)) |
26 | eqid 2820 | . . . . 5 ⊢ (invg‘𝑃) = (invg‘𝑃) | |
27 | 1, 2, 26, 5, 7, 9, 10 | mhpinvcl 20332 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → ((invg‘𝑃)‘𝑥) ∈ (𝐻‘𝑁)) |
28 | 25, 27 | jca 514 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → (∀𝑦 ∈ (𝐻‘𝑁)(𝑥(+g‘𝑃)𝑦) ∈ (𝐻‘𝑁) ∧ ((invg‘𝑃)‘𝑥) ∈ (𝐻‘𝑁))) |
29 | 28 | ralrimiva 3181 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝐻‘𝑁)(∀𝑦 ∈ (𝐻‘𝑁)(𝑥(+g‘𝑃)𝑦) ∈ (𝐻‘𝑁) ∧ ((invg‘𝑃)‘𝑥) ∈ (𝐻‘𝑁))) |
30 | 2 | mplgrp 20223 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp) |
31 | 4, 6, 30 | syl2anc 586 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Grp) |
32 | 3, 18, 26 | issubg2 18287 | . . 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 1337 | 1 ⊢ (𝜑 → (𝐻‘𝑁) ∈ (SubGrp‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ∀wral 3137 {crab 3141 ⊆ wss 3929 ∅c0 4284 {csn 4560 × cxp 5546 ◡ccnv 5547 “ cima 5551 ‘cfv 6348 (class class class)co 7149 ↑m cmap 8399 Fincfn 8502 ℕcn 11631 ℕ0cn0 11891 Basecbs 16476 +gcplusg 16558 0gc0g 16706 Grpcgrp 18096 invgcminusg 18097 SubGrpcsubg 18266 mPoly cmpl 20126 mHomP cmhp 20315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12890 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-mulr 16572 df-sca 16574 df-vsca 16575 df-tset 16577 df-0g 16708 df-mgm 17845 df-sgrp 17894 df-mnd 17905 df-grp 18099 df-minusg 18100 df-subg 18269 df-psr 20129 df-mpl 20131 df-mhp 20319 |
This theorem is referenced by: mhplss 20335 |
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