Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > evls1gsummul | Structured version Visualization version GIF version |
Description: Univariate polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by AV, 14-Sep-2019.) |
Ref | Expression |
---|---|
evls1gsummul.q | ⊢ 𝑄 = (𝑆 evalSub1 𝑅) |
evls1gsummul.k | ⊢ 𝐾 = (Base‘𝑆) |
evls1gsummul.w | ⊢ 𝑊 = (Poly1‘𝑈) |
evls1gsummul.g | ⊢ 𝐺 = (mulGrp‘𝑊) |
evls1gsummul.1 | ⊢ 1 = (1r‘𝑊) |
evls1gsummul.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evls1gsummul.p | ⊢ 𝑃 = (𝑆 ↑s 𝐾) |
evls1gsummul.h | ⊢ 𝐻 = (mulGrp‘𝑃) |
evls1gsummul.b | ⊢ 𝐵 = (Base‘𝑊) |
evls1gsummul.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evls1gsummul.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evls1gsummul.y | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) |
evls1gsummul.n | ⊢ (𝜑 → 𝑁 ⊆ ℕ0) |
evls1gsummul.f | ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 ) |
Ref | Expression |
---|---|
evls1gsummul | ⊢ (𝜑 → (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evls1gsummul.g | . . . 4 ⊢ 𝐺 = (mulGrp‘𝑊) | |
2 | evls1gsummul.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 1, 2 | mgpbas 19247 | . . 3 ⊢ 𝐵 = (Base‘𝐺) |
4 | evls1gsummul.1 | . . . 4 ⊢ 1 = (1r‘𝑊) | |
5 | 1, 4 | ringidval 19255 | . . 3 ⊢ 1 = (0g‘𝐺) |
6 | evls1gsummul.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
7 | evls1gsummul.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
8 | evls1gsummul.u | . . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
9 | 8 | subrgcrng 19541 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing) |
10 | 6, 7, 9 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ CRing) |
11 | evls1gsummul.w | . . . . 5 ⊢ 𝑊 = (Poly1‘𝑈) | |
12 | 11 | ply1crng 20368 | . . . 4 ⊢ (𝑈 ∈ CRing → 𝑊 ∈ CRing) |
13 | 1 | crngmgp 19307 | . . . 4 ⊢ (𝑊 ∈ CRing → 𝐺 ∈ CMnd) |
14 | 10, 12, 13 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
15 | crngring 19310 | . . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
16 | 6, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring) |
17 | evls1gsummul.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑆) | |
18 | 17 | fvexi 6686 | . . . . 5 ⊢ 𝐾 ∈ V |
19 | 16, 18 | jctir 523 | . . . 4 ⊢ (𝜑 → (𝑆 ∈ Ring ∧ 𝐾 ∈ V)) |
20 | evls1gsummul.p | . . . . 5 ⊢ 𝑃 = (𝑆 ↑s 𝐾) | |
21 | 20 | pwsring 19367 | . . . 4 ⊢ ((𝑆 ∈ Ring ∧ 𝐾 ∈ V) → 𝑃 ∈ Ring) |
22 | evls1gsummul.h | . . . . 5 ⊢ 𝐻 = (mulGrp‘𝑃) | |
23 | 22 | ringmgp 19305 | . . . 4 ⊢ (𝑃 ∈ Ring → 𝐻 ∈ Mnd) |
24 | 19, 21, 23 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
25 | nn0ex 11906 | . . . . 5 ⊢ ℕ0 ∈ V | |
26 | 25 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
27 | evls1gsummul.n | . . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ0) | |
28 | 26, 27 | ssexd 5230 | . . 3 ⊢ (𝜑 → 𝑁 ∈ V) |
29 | evls1gsummul.q | . . . . . 6 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
30 | 29, 17, 20, 8, 11 | evls1rhm 20487 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
31 | 6, 7, 30 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
32 | 1, 22 | rhmmhm 19476 | . . . 4 ⊢ (𝑄 ∈ (𝑊 RingHom 𝑃) → 𝑄 ∈ (𝐺 MndHom 𝐻)) |
33 | 31, 32 | syl 17 | . . 3 ⊢ (𝜑 → 𝑄 ∈ (𝐺 MndHom 𝐻)) |
34 | evls1gsummul.y | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) | |
35 | evls1gsummul.f | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 ) | |
36 | 3, 5, 14, 24, 28, 33, 34, 35 | gsummptmhm 19062 | . 2 ⊢ (𝜑 → (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌)))) |
37 | 36 | eqcomd 2829 | 1 ⊢ (𝜑 → (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 finSupp cfsupp 8835 ℕ0cn0 11900 Basecbs 16485 ↾s cress 16486 Σg cgsu 16716 ↑s cpws 16722 Mndcmnd 17913 MndHom cmhm 17956 CMndccmn 18908 mulGrpcmgp 19241 1rcur 19253 Ringcrg 19299 CRingccrg 19300 RingHom crh 19466 SubRingcsubrg 19533 Poly1cpl1 20347 evalSub1 ces1 20478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-hom 16591 df-cco 16592 df-0g 16717 df-gsum 16718 df-prds 16723 df-pws 16725 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-srg 19258 df-ring 19301 df-cring 19302 df-rnghom 19469 df-subrg 19535 df-lmod 19638 df-lss 19706 df-lsp 19746 df-assa 20087 df-asp 20088 df-ascl 20089 df-psr 20138 df-mvr 20139 df-mpl 20140 df-opsr 20142 df-evls 20288 df-psr1 20350 df-ply1 20352 df-evls1 20480 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |