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Mirrors > Home > MPE Home > Th. List > Mathboxes > mhmcompl | Structured version Visualization version GIF version |
Description: The composition of a monoid homomorphism and a polynomial is a polynomial. (Contributed by SN, 7-Feb-2025.) |
Ref | Expression |
---|---|
mhmcompl.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mhmcompl.q | ⊢ 𝑄 = (𝐼 mPoly 𝑆) |
mhmcompl.b | ⊢ 𝐵 = (Base‘𝑃) |
mhmcompl.c | ⊢ 𝐶 = (Base‘𝑄) |
mhmcompl.h | ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆)) |
mhmcompl.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
Ref | Expression |
---|---|
mhmcompl | ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvexd 6917 | . . . 4 ⊢ (𝜑 → (Base‘𝑆) ∈ V) | |
2 | eqid 2728 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
3 | ovexd 7461 | . . . . 5 ⊢ (𝜑 → (ℕ0 ↑m 𝐼) ∈ V) | |
4 | 2, 3 | rabexd 5339 | . . . 4 ⊢ (𝜑 → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V) |
5 | mhmcompl.h | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆)) | |
6 | eqid 2728 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | eqid 2728 | . . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
8 | 6, 7 | mhmf 18755 | . . . . . 6 ⊢ (𝐻 ∈ (𝑅 MndHom 𝑆) → 𝐻:(Base‘𝑅)⟶(Base‘𝑆)) |
9 | 5, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐻:(Base‘𝑅)⟶(Base‘𝑆)) |
10 | mhmcompl.p | . . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
11 | mhmcompl.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑃) | |
12 | mhmcompl.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
13 | 10, 6, 11, 2, 12 | mplelf 21957 | . . . . 5 ⊢ (𝜑 → 𝐹:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
14 | 9, 13 | fcod 6754 | . . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝐹):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑆)) |
15 | 1, 4, 14 | elmapdd 8868 | . . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ ((Base‘𝑆) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
16 | eqid 2728 | . . . 4 ⊢ (𝐼 mPwSer 𝑆) = (𝐼 mPwSer 𝑆) | |
17 | eqid 2728 | . . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑆)) = (Base‘(𝐼 mPwSer 𝑆)) | |
18 | 10, 11 | mplrcl 21953 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → 𝐼 ∈ V) |
19 | 12, 18 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ V) |
20 | 16, 7, 2, 17, 19 | psrbas 21892 | . . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑆)) = ((Base‘𝑆) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
21 | 15, 20 | eleqtrrd 2832 | . 2 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ (Base‘(𝐼 mPwSer 𝑆))) |
22 | fvexd 6917 | . . 3 ⊢ (𝜑 → (0g‘𝑆) ∈ V) | |
23 | mhmrcl1 18753 | . . . . 5 ⊢ (𝐻 ∈ (𝑅 MndHom 𝑆) → 𝑅 ∈ Mnd) | |
24 | 5, 23 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Mnd) |
25 | eqid 2728 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
26 | 6, 25 | mndidcl 18718 | . . . 4 ⊢ (𝑅 ∈ Mnd → (0g‘𝑅) ∈ (Base‘𝑅)) |
27 | 24, 26 | syl 17 | . . 3 ⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅)) |
28 | ssidd 4005 | . . 3 ⊢ (𝜑 → (Base‘𝑅) ⊆ (Base‘𝑅)) | |
29 | fvexd 6917 | . . 3 ⊢ (𝜑 → (Base‘𝑅) ∈ V) | |
30 | 10, 11, 25, 12, 24 | mplelsfi 21954 | . . 3 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝑅)) |
31 | eqid 2728 | . . . . 5 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
32 | 25, 31 | mhm0 18760 | . . . 4 ⊢ (𝐻 ∈ (𝑅 MndHom 𝑆) → (𝐻‘(0g‘𝑅)) = (0g‘𝑆)) |
33 | 5, 32 | syl 17 | . . 3 ⊢ (𝜑 → (𝐻‘(0g‘𝑅)) = (0g‘𝑆)) |
34 | 22, 27, 13, 9, 28, 4, 29, 30, 33 | fsuppcor 9437 | . 2 ⊢ (𝜑 → (𝐻 ∘ 𝐹) finSupp (0g‘𝑆)) |
35 | mhmcompl.q | . . 3 ⊢ 𝑄 = (𝐼 mPoly 𝑆) | |
36 | mhmcompl.c | . . 3 ⊢ 𝐶 = (Base‘𝑄) | |
37 | 35, 16, 17, 31, 36 | mplelbas 21950 | . 2 ⊢ ((𝐻 ∘ 𝐹) ∈ 𝐶 ↔ ((𝐻 ∘ 𝐹) ∈ (Base‘(𝐼 mPwSer 𝑆)) ∧ (𝐻 ∘ 𝐹) finSupp (0g‘𝑆))) |
38 | 21, 34, 37 | sylanbrc 581 | 1 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3430 Vcvv 3473 class class class wbr 5152 ◡ccnv 5681 “ cima 5685 ∘ ccom 5686 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ↑m cmap 8853 Fincfn 8972 finSupp cfsupp 9395 ℕcn 12252 ℕ0cn0 12512 Basecbs 17189 0gc0g 17430 Mndcmnd 18703 MndHom cmhm 18747 mPwSer cmps 21851 mPoly cmpl 21853 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-uz 12863 df-fz 13527 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-sca 17258 df-vsca 17259 df-tset 17261 df-0g 17432 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 df-psr 21856 df-mpl 21858 |
This theorem is referenced by: mhmcoaddmpl 41826 rhmcomulmpl 41827 rhmmpl 41828 mhmcoply1 41830 selvcllem4 41863 selvvvval 41867 selvadd 41870 selvmul 41871 |
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