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Mirrors > Home > MPE Home > Th. List > evl1gsummul | Structured version Visualization version GIF version |
Description: Univariate polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by AV, 15-Sep-2019.) |
Ref | Expression |
---|---|
evl1gsumadd.q | ⊢ 𝑄 = (eval1‘𝑅) |
evl1gsumadd.k | ⊢ 𝐾 = (Base‘𝑅) |
evl1gsumadd.w | ⊢ 𝑊 = (Poly1‘𝑅) |
evl1gsumadd.p | ⊢ 𝑃 = (𝑅 ↑s 𝐾) |
evl1gsumadd.b | ⊢ 𝐵 = (Base‘𝑊) |
evl1gsumadd.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
evl1gsumadd.y | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) |
evl1gsumadd.n | ⊢ (𝜑 → 𝑁 ⊆ ℕ0) |
evl1gsummul.1 | ⊢ 1 = (1r‘𝑊) |
evl1gsummul.g | ⊢ 𝐺 = (mulGrp‘𝑊) |
evl1gsummul.h | ⊢ 𝐻 = (mulGrp‘𝑃) |
evl1gsummul.f | ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 ) |
Ref | Expression |
---|---|
evl1gsummul | ⊢ (𝜑 → (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1gsummul.g | . . . 4 ⊢ 𝐺 = (mulGrp‘𝑊) | |
2 | evl1gsumadd.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 1, 2 | mgpbas 19238 | . . 3 ⊢ 𝐵 = (Base‘𝐺) |
4 | evl1gsummul.1 | . . . 4 ⊢ 1 = (1r‘𝑊) | |
5 | 1, 4 | ringidval 19246 | . . 3 ⊢ 1 = (0g‘𝐺) |
6 | evl1gsumadd.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
7 | evl1gsumadd.w | . . . . 5 ⊢ 𝑊 = (Poly1‘𝑅) | |
8 | 7 | ply1crng 20827 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑊 ∈ CRing) |
9 | 1 | crngmgp 19298 | . . . 4 ⊢ (𝑊 ∈ CRing → 𝐺 ∈ CMnd) |
10 | 6, 8, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
11 | crngring 19302 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
12 | 6, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
13 | evl1gsumadd.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑅) | |
14 | 13 | fvexi 6659 | . . . . 5 ⊢ 𝐾 ∈ V |
15 | 12, 14 | jctir 524 | . . . 4 ⊢ (𝜑 → (𝑅 ∈ Ring ∧ 𝐾 ∈ V)) |
16 | evl1gsumadd.p | . . . . 5 ⊢ 𝑃 = (𝑅 ↑s 𝐾) | |
17 | 16 | pwsring 19361 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ V) → 𝑃 ∈ Ring) |
18 | evl1gsummul.h | . . . . 5 ⊢ 𝐻 = (mulGrp‘𝑃) | |
19 | 18 | ringmgp 19296 | . . . 4 ⊢ (𝑃 ∈ Ring → 𝐻 ∈ Mnd) |
20 | 15, 17, 19 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
21 | nn0ex 11891 | . . . . 5 ⊢ ℕ0 ∈ V | |
22 | 21 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
23 | evl1gsumadd.n | . . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ0) | |
24 | 22, 23 | ssexd 5192 | . . 3 ⊢ (𝜑 → 𝑁 ∈ V) |
25 | evl1gsumadd.q | . . . . 5 ⊢ 𝑄 = (eval1‘𝑅) | |
26 | 25, 7, 16, 13 | evl1rhm 20956 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
27 | 1, 18 | rhmmhm 19470 | . . . 4 ⊢ (𝑄 ∈ (𝑊 RingHom 𝑃) → 𝑄 ∈ (𝐺 MndHom 𝐻)) |
28 | 6, 26, 27 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑄 ∈ (𝐺 MndHom 𝐻)) |
29 | evl1gsumadd.y | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) | |
30 | evl1gsummul.f | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 ) | |
31 | 3, 5, 10, 20, 24, 28, 29, 30 | gsummptmhm 19053 | . 2 ⊢ (𝜑 → (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌)))) |
32 | 31 | eqcomd 2804 | 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 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 finSupp cfsupp 8817 ℕ0cn0 11885 Basecbs 16475 Σg cgsu 16706 ↑s cpws 16712 Mndcmnd 17903 MndHom cmhm 17946 CMndccmn 18898 mulGrpcmgp 19232 1rcur 19244 Ringcrg 19290 CRingccrg 19291 RingHom crh 19460 Poly1cpl1 20806 eval1ce1 20938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-srg 19249 df-ring 19292 df-cring 19293 df-rnghom 19463 df-subrg 19526 df-lmod 19629 df-lss 19697 df-lsp 19737 df-assa 20542 df-asp 20543 df-ascl 20544 df-psr 20594 df-mvr 20595 df-mpl 20596 df-opsr 20598 df-evls 20745 df-evl 20746 df-psr1 20809 df-ply1 20811 df-evl1 20940 |
This theorem is referenced by: (None) |
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