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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.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
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 6900 | . . . 4 ⊢ (𝜑 → (Base‘𝑆) ∈ V) | |
2 | ovex 7438 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
3 | 2 | rabex 5325 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V) |
5 | mhmcompl.h | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆)) | |
6 | eqid 2726 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | eqid 2726 | . . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
8 | 6, 7 | mhmf 18719 | . . . . . 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 | eqid 2726 | . . . . . 6 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
13 | mhmcompl.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
14 | 10, 6, 11, 12, 13 | mplelf 21899 | . . . . 5 ⊢ (𝜑 → 𝐹:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
15 | 9, 14 | fcod 6737 | . . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝐹):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑆)) |
16 | 1, 4, 15 | elmapdd 8837 | . . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ ((Base‘𝑆) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
17 | eqid 2726 | . . . 4 ⊢ (𝐼 mPwSer 𝑆) = (𝐼 mPwSer 𝑆) | |
18 | eqid 2726 | . . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑆)) = (Base‘(𝐼 mPwSer 𝑆)) | |
19 | mhmcompl.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
20 | 17, 7, 12, 18, 19 | psrbas 21838 | . . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑆)) = ((Base‘𝑆) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
21 | 16, 20 | eleqtrrd 2830 | . 2 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ (Base‘(𝐼 mPwSer 𝑆))) |
22 | fvexd 6900 | . . 3 ⊢ (𝜑 → (0g‘𝑆) ∈ V) | |
23 | mhmrcl1 18717 | . . . . 5 ⊢ (𝐻 ∈ (𝑅 MndHom 𝑆) → 𝑅 ∈ Mnd) | |
24 | 5, 23 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Mnd) |
25 | eqid 2726 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
26 | 6, 25 | mndidcl 18682 | . . . 4 ⊢ (𝑅 ∈ Mnd → (0g‘𝑅) ∈ (Base‘𝑅)) |
27 | 24, 26 | syl 17 | . . 3 ⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅)) |
28 | ssidd 4000 | . . 3 ⊢ (𝜑 → (Base‘𝑅) ⊆ (Base‘𝑅)) | |
29 | fvexd 6900 | . . 3 ⊢ (𝜑 → (Base‘𝑅) ∈ V) | |
30 | 10, 11, 25, 13, 24 | mplelsfi 21896 | . . 3 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝑅)) |
31 | eqid 2726 | . . . . 5 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
32 | 25, 31 | mhm0 18724 | . . . 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, 14, 9, 28, 4, 29, 30, 33 | fsuppcor 9401 | . 2 ⊢ (𝜑 → (𝐻 ∘ 𝐹) finSupp (0g‘𝑆)) |
35 | mhmcompl.q | . . 3 ⊢ 𝑄 = (𝐼 mPoly 𝑆) | |
36 | mhmcompl.c | . . 3 ⊢ 𝐶 = (Base‘𝑄) | |
37 | 35, 17, 18, 31, 36 | mplelbas 21892 | . 2 ⊢ ((𝐻 ∘ 𝐹) ∈ 𝐶 ↔ ((𝐻 ∘ 𝐹) ∈ (Base‘(𝐼 mPwSer 𝑆)) ∧ (𝐻 ∘ 𝐹) finSupp (0g‘𝑆))) |
38 | 21, 34, 37 | sylanbrc 582 | 1 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3426 Vcvv 3468 class class class wbr 5141 ◡ccnv 5668 “ cima 5672 ∘ ccom 5673 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ↑m cmap 8822 Fincfn 8941 finSupp cfsupp 9363 ℕcn 12216 ℕ0cn0 12476 Basecbs 17153 0gc0g 17394 Mndcmnd 18667 MndHom cmhm 18711 mPwSer cmps 21798 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-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 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-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-tset 17225 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-psr 21803 df-mpl 21805 |
This theorem is referenced by: mhmcoaddmpl 41680 rhmcomulmpl 41681 rhmmpl 41682 selvcllem4 41710 selvvvval 41714 selvadd 41717 selvmul 41718 |
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