![]() |
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 2725 | . 2 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2725 | . 2 ⊢ (+g‘𝑃) = (+g‘𝑃) | |
4 | eqid 2725 | . 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 21966 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp) |
9 | 5, 6, 8 | syl2anc 582 | . 2 ⊢ (𝜑 → 𝑃 ∈ Grp) |
10 | mplmapghm.d | . . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
11 | simpr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝑝 ∈ 𝐵) | |
12 | 7, 2, 1, 10, 11 | mplelf 21947 | . . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝑝:𝐷⟶(Base‘𝑅)) |
13 | mplmapghm.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
14 | 13 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ 𝐷) |
15 | 12, 14 | ffvelcdmd 7090 | . . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝑝‘𝐹) ∈ (Base‘𝑅)) |
16 | mplmapghm.h | . . 3 ⊢ 𝐻 = (𝑝 ∈ 𝐵 ↦ (𝑝‘𝐹)) | |
17 | 15, 16 | fmptd 7119 | . 2 ⊢ (𝜑 → 𝐻:𝐵⟶(Base‘𝑅)) |
18 | simprl 769 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑞 ∈ 𝐵) | |
19 | simprr 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑟 ∈ 𝐵) | |
20 | 7, 1, 4, 3, 18, 19 | mpladd 21958 | . . . . 5 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑞(+g‘𝑃)𝑟) = (𝑞 ∘f (+g‘𝑅)𝑟)) |
21 | 20 | fveq1d 6894 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑞(+g‘𝑃)𝑟)‘𝐹) = ((𝑞 ∘f (+g‘𝑅)𝑟)‘𝐹)) |
22 | 7, 2, 1, 10, 18 | mplelf 21947 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑞:𝐷⟶(Base‘𝑅)) |
23 | 22 | ffnd 6718 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑞 Fn 𝐷) |
24 | 7, 2, 1, 10, 19 | mplelf 21947 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑟:𝐷⟶(Base‘𝑅)) |
25 | 24 | ffnd 6718 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑟 Fn 𝐷) |
26 | ovex 7449 | . . . . . . . 8 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
27 | 10, 26 | rabex2 5331 | . . . . . . 7 ⊢ 𝐷 ∈ V |
28 | 27 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝐷 ∈ V) |
29 | inidm 4213 | . . . . . 6 ⊢ (𝐷 ∩ 𝐷) = 𝐷 | |
30 | eqidd 2726 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝐹 ∈ 𝐷) → (𝑞‘𝐹) = (𝑞‘𝐹)) | |
31 | eqidd 2726 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝐹 ∈ 𝐷) → (𝑟‘𝐹) = (𝑟‘𝐹)) | |
32 | 23, 25, 28, 28, 29, 30, 31 | ofval 7693 | . . . . 5 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝐹 ∈ 𝐷) → ((𝑞 ∘f (+g‘𝑅)𝑟)‘𝐹) = ((𝑞‘𝐹)(+g‘𝑅)(𝑟‘𝐹))) |
33 | 13, 32 | mpidan 687 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑞 ∘f (+g‘𝑅)𝑟)‘𝐹) = ((𝑞‘𝐹)(+g‘𝑅)(𝑟‘𝐹))) |
34 | 21, 33 | eqtrd 2765 | . . 3 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑞(+g‘𝑃)𝑟)‘𝐹) = ((𝑞‘𝐹)(+g‘𝑅)(𝑟‘𝐹))) |
35 | fveq1 6891 | . . . 4 ⊢ (𝑝 = (𝑞(+g‘𝑃)𝑟) → (𝑝‘𝐹) = ((𝑞(+g‘𝑃)𝑟)‘𝐹)) | |
36 | 9 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑃 ∈ Grp) |
37 | 1, 3, 36, 18, 19 | grpcld 18908 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑞(+g‘𝑃)𝑟) ∈ 𝐵) |
38 | fvexd 6907 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑞(+g‘𝑃)𝑟)‘𝐹) ∈ V) | |
39 | 16, 35, 37, 38 | fvmptd3 7023 | . . 3 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐻‘(𝑞(+g‘𝑃)𝑟)) = ((𝑞(+g‘𝑃)𝑟)‘𝐹)) |
40 | fveq1 6891 | . . . . 5 ⊢ (𝑝 = 𝑞 → (𝑝‘𝐹) = (𝑞‘𝐹)) | |
41 | fvexd 6907 | . . . . 5 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑞‘𝐹) ∈ V) | |
42 | 16, 40, 18, 41 | fvmptd3 7023 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐻‘𝑞) = (𝑞‘𝐹)) |
43 | fveq1 6891 | . . . . 5 ⊢ (𝑝 = 𝑟 → (𝑝‘𝐹) = (𝑟‘𝐹)) | |
44 | fvexd 6907 | . . . . 5 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑟‘𝐹) ∈ V) | |
45 | 16, 43, 19, 44 | fvmptd3 7023 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐻‘𝑟) = (𝑟‘𝐹)) |
46 | 42, 45 | oveq12d 7434 | . . 3 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝐻‘𝑞)(+g‘𝑅)(𝐻‘𝑟)) = ((𝑞‘𝐹)(+g‘𝑅)(𝑟‘𝐹))) |
47 | 34, 39, 46 | 3eqtr4d 2775 | . 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 19183 | 1 ⊢ (𝜑 → 𝐻 ∈ (𝑃 GrpHom 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {crab 3419 Vcvv 3463 ↦ cmpt 5226 ◡ccnv 5671 “ cima 5675 ‘cfv 6543 (class class class)co 7416 ∘f cof 7680 ↑m cmap 8843 Fincfn 8962 ℕcn 12242 ℕ0cn0 12502 Basecbs 17179 +gcplusg 17232 Grpcgrp 18894 GrpHom cghm 19171 mPoly cmpl 21843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-hom 17256 df-cco 17257 df-0g 17422 df-prds 17428 df-pws 17430 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-subg 19082 df-ghm 19172 df-psr 21846 df-mpl 21848 |
This theorem is referenced by: selvvvval 41883 evlselv 41885 |
Copyright terms: Public domain | W3C validator |