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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 2724 | . 2 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2724 | . 2 ⊢ (+g‘𝑃) = (+g‘𝑃) | |
4 | eqid 2724 | . 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 21877 | . . 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 21858 | . . . 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 7077 | . . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝑝‘𝐹) ∈ (Base‘𝑅)) |
16 | mplmapghm.h | . . 3 ⊢ 𝐻 = (𝑝 ∈ 𝐵 ↦ (𝑝‘𝐹)) | |
17 | 15, 16 | fmptd 7105 | . 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 21869 | . . . . 5 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑞(+g‘𝑃)𝑟) = (𝑞 ∘f (+g‘𝑅)𝑟)) |
21 | 20 | fveq1d 6883 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑞(+g‘𝑃)𝑟)‘𝐹) = ((𝑞 ∘f (+g‘𝑅)𝑟)‘𝐹)) |
22 | 7, 2, 1, 10, 18 | mplelf 21858 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑞:𝐷⟶(Base‘𝑅)) |
23 | 22 | ffnd 6708 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑞 Fn 𝐷) |
24 | 7, 2, 1, 10, 19 | mplelf 21858 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑟:𝐷⟶(Base‘𝑅)) |
25 | 24 | ffnd 6708 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝑟 Fn 𝐷) |
26 | ovex 7434 | . . . . . . . 8 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
27 | 10, 26 | rabex2 5324 | . . . . . . 7 ⊢ 𝐷 ∈ V |
28 | 27 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → 𝐷 ∈ V) |
29 | inidm 4210 | . . . . . 6 ⊢ (𝐷 ∩ 𝐷) = 𝐷 | |
30 | eqidd 2725 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝐹 ∈ 𝐷) → (𝑞‘𝐹) = (𝑞‘𝐹)) | |
31 | eqidd 2725 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) ∧ 𝐹 ∈ 𝐷) → (𝑟‘𝐹) = (𝑟‘𝐹)) | |
32 | 23, 25, 28, 28, 29, 30, 31 | ofval 7674 | . . . . 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 2764 | . . 3 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑞(+g‘𝑃)𝑟)‘𝐹) = ((𝑞‘𝐹)(+g‘𝑅)(𝑟‘𝐹))) |
35 | fveq1 6880 | . . . 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 18864 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑞(+g‘𝑃)𝑟) ∈ 𝐵) |
38 | fvexd 6896 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝑞(+g‘𝑃)𝑟)‘𝐹) ∈ V) | |
39 | 16, 35, 37, 38 | fvmptd3 7011 | . . 3 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐻‘(𝑞(+g‘𝑃)𝑟)) = ((𝑞(+g‘𝑃)𝑟)‘𝐹)) |
40 | fveq1 6880 | . . . . 5 ⊢ (𝑝 = 𝑞 → (𝑝‘𝐹) = (𝑞‘𝐹)) | |
41 | fvexd 6896 | . . . . 5 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑞‘𝐹) ∈ V) | |
42 | 16, 40, 18, 41 | fvmptd3 7011 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐻‘𝑞) = (𝑞‘𝐹)) |
43 | fveq1 6880 | . . . . 5 ⊢ (𝑝 = 𝑟 → (𝑝‘𝐹) = (𝑟‘𝐹)) | |
44 | fvexd 6896 | . . . . 5 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑟‘𝐹) ∈ V) | |
45 | 16, 43, 19, 44 | fvmptd3 7011 | . . . 4 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝐻‘𝑟) = (𝑟‘𝐹)) |
46 | 42, 45 | oveq12d 7419 | . . 3 ⊢ ((𝜑 ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → ((𝐻‘𝑞)(+g‘𝑅)(𝐻‘𝑟)) = ((𝑞‘𝐹)(+g‘𝑅)(𝑟‘𝐹))) |
47 | 34, 39, 46 | 3eqtr4d 2774 | . 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 19135 | 1 ⊢ (𝜑 → 𝐻 ∈ (𝑃 GrpHom 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3424 Vcvv 3466 ↦ cmpt 5221 ◡ccnv 5665 “ cima 5669 ‘cfv 6533 (class class class)co 7401 ∘f cof 7661 ↑m cmap 8815 Fincfn 8934 ℕcn 12208 ℕ0cn0 12468 Basecbs 17140 +gcplusg 17193 Grpcgrp 18850 GrpHom cghm 19123 mPoly cmpl 21759 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-sup 9432 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-hom 17217 df-cco 17218 df-0g 17383 df-prds 17389 df-pws 17391 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-grp 18853 df-minusg 18854 df-subg 19035 df-ghm 19124 df-psr 21762 df-mpl 21764 |
This theorem is referenced by: selvvvval 41612 evlselv 41614 |
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