|   | Mathbox for Steven Nguyen | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > mplmapghm | Structured version Visualization version GIF version | ||
| Description: The function 𝐻 mapping polynomials 𝑝 to their coefficient given a bag of variables 𝐹 is a group homomorphism. (Contributed by SN, 15-Mar-2025.) | 
| Ref | Expression | 
|---|---|
| mplmapghm.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) | 
| mplmapghm.b | ⊢ 𝐵 = (Base‘𝑃) | 
| mplmapghm.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | 
| mplmapghm.h | ⊢ 𝐻 = (𝑝 ∈ 𝐵 ↦ (𝑝‘𝐹)) | 
| mplmapghm.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) | 
| mplmapghm.r | ⊢ (𝜑 → 𝑅 ∈ Grp) | 
| mplmapghm.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) | 
| Ref | Expression | 
|---|---|
| mplmapghm | ⊢ (𝜑 → 𝐻 ∈ (𝑃 GrpHom 𝑅)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mplmapghm.b | . 2 ⊢ 𝐵 = (Base‘𝑃) | |
| 2 | eqid 2736 | . 2 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 3 | eqid 2736 | . 2 ⊢ (+g‘𝑃) = (+g‘𝑃) | |
| 4 | eqid 2736 | . 2 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
| 5 | mplmapghm.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
| 6 | mplmapghm.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
| 7 | mplmapghm.p | . . . 4 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
| 8 | 7 | mplgrp 22038 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp) | 
| 9 | 5, 6, 8 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝑃 ∈ Grp) | 
| 10 | mplmapghm.d | . . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
| 11 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝑝 ∈ 𝐵) | |
| 12 | 7, 2, 1, 10, 11 | mplelf 22019 | . . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝑝:𝐷⟶(Base‘𝑅)) | 
| 13 | mplmapghm.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
| 14 | 13 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ 𝐷) | 
| 15 | 12, 14 | ffvelcdmd 7104 | . . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝑝‘𝐹) ∈ (Base‘𝑅)) | 
| 16 | mplmapghm.h | . . 3 ⊢ 𝐻 = (𝑝 ∈ 𝐵 ↦ (𝑝‘𝐹)) | |
| 17 | 15, 16 | fmptd 7133 | . 2 ⊢ (𝜑 → 𝐻:𝐵⟶(Base‘𝑅)) | 
| 18 | simprl 770 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑞 ∈ 𝐵) | |
| 19 | simprr 772 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑟 ∈ 𝐵) | |
| 20 | 7, 1, 4, 3, 18, 19 | mpladd 22030 | . . . . 5 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑞(+g‘𝑃)𝑟) = (𝑞 ∘f (+g‘𝑅)𝑟)) | 
| 21 | 20 | fveq1d 6907 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑞(+g‘𝑃)𝑟)‘𝐹) = ((𝑞 ∘f (+g‘𝑅)𝑟)‘𝐹)) | 
| 22 | 7, 2, 1, 10, 18 | mplelf 22019 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑞:𝐷⟶(Base‘𝑅)) | 
| 23 | 22 | ffnd 6736 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑞 Fn 𝐷) | 
| 24 | 7, 2, 1, 10, 19 | mplelf 22019 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑟:𝐷⟶(Base‘𝑅)) | 
| 25 | 24 | ffnd 6736 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑟 Fn 𝐷) | 
| 26 | ovex 7465 | . . . . . . . 8 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
| 27 | 10, 26 | rabex2 5340 | . . . . . . 7 ⊢ 𝐷 ∈ V | 
| 28 | 27 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝐷 ∈ V) | 
| 29 | inidm 4226 | . . . . . 6 ⊢ (𝐷 ∩ 𝐷) = 𝐷 | |
| 30 | eqidd 2737 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝐹 ∈ 𝐷) → (𝑞‘𝐹) = (𝑞‘𝐹)) | |
| 31 | eqidd 2737 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝐹 ∈ 𝐷) → (𝑟‘𝐹) = (𝑟‘𝐹)) | |
| 32 | 23, 25, 28, 28, 29, 30, 31 | ofval 7709 | . . . . 5 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝐹 ∈ 𝐷) → ((𝑞 ∘f (+g‘𝑅)𝑟)‘𝐹) = ((𝑞‘𝐹)(+g‘𝑅)(𝑟‘𝐹))) | 
| 33 | 13, 32 | mpidan 689 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑞 ∘f (+g‘𝑅)𝑟)‘𝐹) = ((𝑞‘𝐹)(+g‘𝑅)(𝑟‘𝐹))) | 
| 34 | 21, 33 | eqtrd 2776 | . . 3 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑞(+g‘𝑃)𝑟)‘𝐹) = ((𝑞‘𝐹)(+g‘𝑅)(𝑟‘𝐹))) | 
| 35 | fveq1 6904 | . . . 4 ⊢ (𝑝 = (𝑞(+g‘𝑃)𝑟) → (𝑝‘𝐹) = ((𝑞(+g‘𝑃)𝑟)‘𝐹)) | |
| 36 | 9 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑃 ∈ Grp) | 
| 37 | 1, 3, 36, 18, 19 | grpcld 18966 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑞(+g‘𝑃)𝑟) ∈ 𝐵) | 
| 38 | fvexd 6920 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑞(+g‘𝑃)𝑟)‘𝐹) ∈ V) | |
| 39 | 16, 35, 37, 38 | fvmptd3 7038 | . . 3 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐻‘(𝑞(+g‘𝑃)𝑟)) = ((𝑞(+g‘𝑃)𝑟)‘𝐹)) | 
| 40 | fveq1 6904 | . . . . 5 ⊢ (𝑝 = 𝑞 → (𝑝‘𝐹) = (𝑞‘𝐹)) | |
| 41 | fvexd 6920 | . . . . 5 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑞‘𝐹) ∈ V) | |
| 42 | 16, 40, 18, 41 | fvmptd3 7038 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐻‘𝑞) = (𝑞‘𝐹)) | 
| 43 | fveq1 6904 | . . . . 5 ⊢ (𝑝 = 𝑟 → (𝑝‘𝐹) = (𝑟‘𝐹)) | |
| 44 | fvexd 6920 | . . . . 5 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑟‘𝐹) ∈ V) | |
| 45 | 16, 43, 19, 44 | fvmptd3 7038 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐻‘𝑟) = (𝑟‘𝐹)) | 
| 46 | 42, 45 | oveq12d 7450 | . . 3 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝐻‘𝑞)(+g‘𝑅)(𝐻‘𝑟)) = ((𝑞‘𝐹)(+g‘𝑅)(𝑟‘𝐹))) | 
| 47 | 34, 39, 46 | 3eqtr4d 2786 | . 2 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐻‘(𝑞(+g‘𝑃)𝑟)) = ((𝐻‘𝑞)(+g‘𝑅)(𝐻‘𝑟))) | 
| 48 | 1, 2, 3, 4, 9, 6, 17, 47 | isghmd 19244 | 1 ⊢ (𝜑 → 𝐻 ∈ (𝑃 GrpHom 𝑅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {crab 3435 Vcvv 3479 ↦ cmpt 5224 ◡ccnv 5683 “ cima 5687 ‘cfv 6560 (class class class)co 7432 ∘f cof 7696 ↑m cmap 8867 Fincfn 8986 ℕcn 12267 ℕ0cn0 12528 Basecbs 17248 +gcplusg 17298 Grpcgrp 18952 GrpHom cghm 19231 mPoly cmpl 21927 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-sup 9483 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17487 df-prds 17493 df-pws 17495 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-minusg 18956 df-subg 19142 df-ghm 19232 df-psr 21930 df-mpl 21932 | 
| This theorem is referenced by: selvvvval 42600 evlselv 42602 | 
| Copyright terms: Public domain | W3C validator |