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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 2726 | . 2 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2726 | . 2 ⊢ (+g‘𝑃) = (+g‘𝑃) | |
4 | eqid 2726 | . 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 21918 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp) |
9 | 5, 6, 8 | syl2anc 583 | . 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 21899 | . . . 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 7081 | . . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝑝‘𝐹) ∈ (Base‘𝑅)) |
16 | mplmapghm.h | . . 3 ⊢ 𝐻 = (𝑝 ∈ 𝐵 ↦ (𝑝‘𝐹)) | |
17 | 15, 16 | fmptd 7109 | . 2 ⊢ (𝜑 → 𝐻:𝐵⟶(Base‘𝑅)) |
18 | simprl 768 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑞 ∈ 𝐵) | |
19 | simprr 770 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑟 ∈ 𝐵) | |
20 | 7, 1, 4, 3, 18, 19 | mpladd 21910 | . . . . 5 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑞(+g‘𝑃)𝑟) = (𝑞 ∘f (+g‘𝑅)𝑟)) |
21 | 20 | fveq1d 6887 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑞(+g‘𝑃)𝑟)‘𝐹) = ((𝑞 ∘f (+g‘𝑅)𝑟)‘𝐹)) |
22 | 7, 2, 1, 10, 18 | mplelf 21899 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑞:𝐷⟶(Base‘𝑅)) |
23 | 22 | ffnd 6712 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑞 Fn 𝐷) |
24 | 7, 2, 1, 10, 19 | mplelf 21899 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑟:𝐷⟶(Base‘𝑅)) |
25 | 24 | ffnd 6712 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑟 Fn 𝐷) |
26 | ovex 7438 | . . . . . . . 8 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
27 | 10, 26 | rabex2 5327 | . . . . . . 7 ⊢ 𝐷 ∈ V |
28 | 27 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝐷 ∈ V) |
29 | inidm 4213 | . . . . . 6 ⊢ (𝐷 ∩ 𝐷) = 𝐷 | |
30 | eqidd 2727 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝐹 ∈ 𝐷) → (𝑞‘𝐹) = (𝑞‘𝐹)) | |
31 | eqidd 2727 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝐹 ∈ 𝐷) → (𝑟‘𝐹) = (𝑟‘𝐹)) | |
32 | 23, 25, 28, 28, 29, 30, 31 | ofval 7678 | . . . . 5 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝐹 ∈ 𝐷) → ((𝑞 ∘f (+g‘𝑅)𝑟)‘𝐹) = ((𝑞‘𝐹)(+g‘𝑅)(𝑟‘𝐹))) |
33 | 13, 32 | mpidan 686 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑞 ∘f (+g‘𝑅)𝑟)‘𝐹) = ((𝑞‘𝐹)(+g‘𝑅)(𝑟‘𝐹))) |
34 | 21, 33 | eqtrd 2766 | . . 3 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑞(+g‘𝑃)𝑟)‘𝐹) = ((𝑞‘𝐹)(+g‘𝑅)(𝑟‘𝐹))) |
35 | fveq1 6884 | . . . 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 18877 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑞(+g‘𝑃)𝑟) ∈ 𝐵) |
38 | fvexd 6900 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑞(+g‘𝑃)𝑟)‘𝐹) ∈ V) | |
39 | 16, 35, 37, 38 | fvmptd3 7015 | . . 3 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐻‘(𝑞(+g‘𝑃)𝑟)) = ((𝑞(+g‘𝑃)𝑟)‘𝐹)) |
40 | fveq1 6884 | . . . . 5 ⊢ (𝑝 = 𝑞 → (𝑝‘𝐹) = (𝑞‘𝐹)) | |
41 | fvexd 6900 | . . . . 5 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑞‘𝐹) ∈ V) | |
42 | 16, 40, 18, 41 | fvmptd3 7015 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐻‘𝑞) = (𝑞‘𝐹)) |
43 | fveq1 6884 | . . . . 5 ⊢ (𝑝 = 𝑟 → (𝑝‘𝐹) = (𝑟‘𝐹)) | |
44 | fvexd 6900 | . . . . 5 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑟‘𝐹) ∈ V) | |
45 | 16, 43, 19, 44 | fvmptd3 7015 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐻‘𝑟) = (𝑟‘𝐹)) |
46 | 42, 45 | oveq12d 7423 | . . 3 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝐻‘𝑞)(+g‘𝑅)(𝐻‘𝑟)) = ((𝑞‘𝐹)(+g‘𝑅)(𝑟‘𝐹))) |
47 | 34, 39, 46 | 3eqtr4d 2776 | . 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 19150 | 1 ⊢ (𝜑 → 𝐻 ∈ (𝑃 GrpHom 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3426 Vcvv 3468 ↦ cmpt 5224 ◡ccnv 5668 “ cima 5672 ‘cfv 6537 (class class class)co 7405 ∘f cof 7665 ↑m cmap 8822 Fincfn 8941 ℕcn 12216 ℕ0cn0 12476 Basecbs 17153 +gcplusg 17206 Grpcgrp 18863 GrpHom cghm 19138 mPoly cmpl 21800 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 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 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 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 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-hom 17230 df-cco 17231 df-0g 17396 df-prds 17402 df-pws 17404 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-subg 19050 df-ghm 19139 df-psr 21803 df-mpl 21805 |
This theorem is referenced by: selvvvval 41714 evlselv 41716 |
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