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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 19239 | . . 3 ⊢ 𝐵 = (Base‘𝐺) |
4 | evls1gsummul.1 | . . . 4 ⊢ 1 = (1r‘𝑊) | |
5 | 1, 4 | ringidval 19247 | . . 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 19533 | . . . . 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 20360 | . . . 4 ⊢ (𝑈 ∈ CRing → 𝑊 ∈ CRing) |
13 | 1 | crngmgp 19299 | . . . 4 ⊢ (𝑊 ∈ CRing → 𝐺 ∈ CMnd) |
14 | 10, 12, 13 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
15 | crngring 19302 | . . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
16 | 6, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring) |
17 | evls1gsummul.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑆) | |
18 | 17 | fvexi 6678 | . . . . 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 19359 | . . . 4 ⊢ ((𝑆 ∈ Ring ∧ 𝐾 ∈ V) → 𝑃 ∈ Ring) |
22 | evls1gsummul.h | . . . . 5 ⊢ 𝐻 = (mulGrp‘𝑃) | |
23 | 22 | ringmgp 19297 | . . . 4 ⊢ (𝑃 ∈ Ring → 𝐻 ∈ Mnd) |
24 | 19, 21, 23 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
25 | nn0ex 11897 | . . . . 5 ⊢ ℕ0 ∈ V | |
26 | 25 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
27 | evls1gsummul.n | . . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ0) | |
28 | 26, 27 | ssexd 5220 | . . 3 ⊢ (𝜑 → 𝑁 ∈ V) |
29 | evls1gsummul.q | . . . . . 6 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
30 | 29, 17, 20, 8, 11 | evls1rhm 20479 | . . . . 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 19468 | . . . 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 19054 | . 2 ⊢ (𝜑 → (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌)))) |
37 | 36 | eqcomd 2827 | 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 1533 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 class class class wbr 5058 ↦ cmpt 5138 ‘cfv 6349 (class class class)co 7150 finSupp cfsupp 8827 ℕ0cn0 11891 Basecbs 16477 ↾s cress 16478 Σg cgsu 16708 ↑s cpws 16714 Mndcmnd 17905 MndHom cmhm 17948 CMndccmn 18900 mulGrpcmgp 19233 1rcur 19245 Ringcrg 19291 CRingccrg 19292 RingHom crh 19458 SubRingcsubrg 19525 Poly1cpl1 20339 evalSub1 ces1 20470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-hom 16583 df-cco 16584 df-0g 16709 df-gsum 16710 df-prds 16715 df-pws 16717 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-srg 19250 df-ring 19293 df-cring 19294 df-rnghom 19461 df-subrg 19527 df-lmod 19630 df-lss 19698 df-lsp 19738 df-assa 20079 df-asp 20080 df-ascl 20081 df-psr 20130 df-mvr 20131 df-mpl 20132 df-opsr 20134 df-evls 20280 df-psr1 20342 df-ply1 20344 df-evls1 20472 |
This theorem is referenced by: (None) |
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