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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 19724 | . . 3 ⊢ 𝐵 = (Base‘𝐺) |
4 | evls1gsummul.1 | . . . 4 ⊢ 1 = (1r‘𝑊) | |
5 | 1, 4 | ringidval 19737 | . . 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 20026 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing) |
10 | 6, 7, 9 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ CRing) |
11 | evls1gsummul.w | . . . . 5 ⊢ 𝑊 = (Poly1‘𝑈) | |
12 | 11 | ply1crng 21367 | . . . 4 ⊢ (𝑈 ∈ CRing → 𝑊 ∈ CRing) |
13 | 1 | crngmgp 19789 | . . . 4 ⊢ (𝑊 ∈ CRing → 𝐺 ∈ CMnd) |
14 | 10, 12, 13 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
15 | crngring 19793 | . . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
16 | 6, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring) |
17 | evls1gsummul.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑆) | |
18 | 17 | fvexi 6785 | . . . . 5 ⊢ 𝐾 ∈ V |
19 | 16, 18 | jctir 521 | . . . 4 ⊢ (𝜑 → (𝑆 ∈ Ring ∧ 𝐾 ∈ V)) |
20 | evls1gsummul.p | . . . . 5 ⊢ 𝑃 = (𝑆 ↑s 𝐾) | |
21 | 20 | pwsring 19852 | . . . 4 ⊢ ((𝑆 ∈ Ring ∧ 𝐾 ∈ V) → 𝑃 ∈ Ring) |
22 | evls1gsummul.h | . . . . 5 ⊢ 𝐻 = (mulGrp‘𝑃) | |
23 | 22 | ringmgp 19787 | . . . 4 ⊢ (𝑃 ∈ Ring → 𝐻 ∈ Mnd) |
24 | 19, 21, 23 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
25 | nn0ex 12239 | . . . . 5 ⊢ ℕ0 ∈ V | |
26 | 25 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
27 | evls1gsummul.n | . . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ0) | |
28 | 26, 27 | ssexd 5252 | . . 3 ⊢ (𝜑 → 𝑁 ∈ V) |
29 | evls1gsummul.q | . . . . . 6 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
30 | 29, 17, 20, 8, 11 | evls1rhm 21486 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
31 | 6, 7, 30 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
32 | 1, 22 | rhmmhm 19964 | . . . 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 19539 | . 2 ⊢ (𝜑 → (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌)))) |
37 | 36 | eqcomd 2746 | 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 396 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ⊆ wss 3892 class class class wbr 5079 ↦ cmpt 5162 ‘cfv 6432 (class class class)co 7271 finSupp cfsupp 9106 ℕ0cn0 12233 Basecbs 16910 ↾s cress 16939 Σg cgsu 17149 ↑s cpws 17155 Mndcmnd 18383 MndHom cmhm 18426 CMndccmn 19384 mulGrpcmgp 19718 1rcur 19735 Ringcrg 19781 CRingccrg 19782 RingHom crh 19954 SubRingcsubrg 20018 Poly1cpl1 21346 evalSub1 ces1 21477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-ofr 7528 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-pm 8601 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-sup 9179 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-fz 13239 df-fzo 13382 df-seq 13720 df-hash 14043 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-hom 16984 df-cco 16985 df-0g 17150 df-gsum 17151 df-prds 17156 df-pws 17158 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-mulg 18699 df-subg 18750 df-ghm 18830 df-cntz 18921 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-srg 19740 df-ring 19783 df-cring 19784 df-rnghom 19957 df-subrg 20020 df-lmod 20123 df-lss 20192 df-lsp 20232 df-assa 21058 df-asp 21059 df-ascl 21060 df-psr 21110 df-mvr 21111 df-mpl 21112 df-opsr 21114 df-evls 21280 df-psr1 21349 df-ply1 21351 df-evls1 21479 |
This theorem is referenced by: (None) |
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