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Mirrors > Home > MPE Home > Th. List > evlsgsummul | Structured version Visualization version GIF version |
Description: Polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by SN, 13-Feb-2024.) |
Ref | Expression |
---|---|
evlsgsummul.q | ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) |
evlsgsummul.w | ⊢ 𝑊 = (𝐼 mPoly 𝑈) |
evlsgsummul.g | ⊢ 𝐺 = (mulGrp‘𝑊) |
evlsgsummul.1 | ⊢ 1 = (1r‘𝑊) |
evlsgsummul.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evlsgsummul.p | ⊢ 𝑃 = (𝑆 ↑s (𝐾 ↑m 𝐼)) |
evlsgsummul.h | ⊢ 𝐻 = (mulGrp‘𝑃) |
evlsgsummul.k | ⊢ 𝐾 = (Base‘𝑆) |
evlsgsummul.b | ⊢ 𝐵 = (Base‘𝑊) |
evlsgsummul.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
evlsgsummul.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evlsgsummul.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evlsgsummul.y | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) |
evlsgsummul.n | ⊢ (𝜑 → 𝑁 ⊆ ℕ0) |
evlsgsummul.f | ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 ) |
Ref | Expression |
---|---|
evlsgsummul | ⊢ (𝜑 → (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlsgsummul.g | . . . 4 ⊢ 𝐺 = (mulGrp‘𝑊) | |
2 | evlsgsummul.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 1, 2 | mgpbas 19240 | . . 3 ⊢ 𝐵 = (Base‘𝐺) |
4 | evlsgsummul.1 | . . . 4 ⊢ 1 = (1r‘𝑊) | |
5 | 1, 4 | ringidval 19248 | . . 3 ⊢ 1 = (0g‘𝐺) |
6 | evlsgsummul.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
7 | evlsgsummul.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
8 | evlsgsummul.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
9 | evlsgsummul.u | . . . . . . 7 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
10 | 9 | subrgcrng 19534 | . . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing) |
11 | 7, 8, 10 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ CRing) |
12 | evlsgsummul.w | . . . . . 6 ⊢ 𝑊 = (𝐼 mPoly 𝑈) | |
13 | 12 | mplcrng 20229 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑈 ∈ CRing) → 𝑊 ∈ CRing) |
14 | 6, 11, 13 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ CRing) |
15 | 1 | crngmgp 19300 | . . . 4 ⊢ (𝑊 ∈ CRing → 𝐺 ∈ CMnd) |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
17 | crngring 19303 | . . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
18 | 7, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring) |
19 | ovex 7182 | . . . . 5 ⊢ (𝐾 ↑m 𝐼) ∈ V | |
20 | 18, 19 | jctir 523 | . . . 4 ⊢ (𝜑 → (𝑆 ∈ Ring ∧ (𝐾 ↑m 𝐼) ∈ V)) |
21 | evlsgsummul.p | . . . . 5 ⊢ 𝑃 = (𝑆 ↑s (𝐾 ↑m 𝐼)) | |
22 | 21 | pwsring 19360 | . . . 4 ⊢ ((𝑆 ∈ Ring ∧ (𝐾 ↑m 𝐼) ∈ V) → 𝑃 ∈ Ring) |
23 | evlsgsummul.h | . . . . 5 ⊢ 𝐻 = (mulGrp‘𝑃) | |
24 | 23 | ringmgp 19298 | . . . 4 ⊢ (𝑃 ∈ Ring → 𝐻 ∈ Mnd) |
25 | 20, 22, 24 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
26 | nn0ex 11897 | . . . . 5 ⊢ ℕ0 ∈ V | |
27 | 26 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
28 | evlsgsummul.n | . . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ0) | |
29 | 27, 28 | ssexd 5221 | . . 3 ⊢ (𝜑 → 𝑁 ∈ V) |
30 | evlsgsummul.q | . . . . . 6 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) | |
31 | evlsgsummul.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑆) | |
32 | 30, 12, 9, 21, 31 | evlsrhm 20296 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
33 | 6, 7, 8, 32 | syl3anc 1366 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
34 | 1, 23 | rhmmhm 19469 | . . . 4 ⊢ (𝑄 ∈ (𝑊 RingHom 𝑃) → 𝑄 ∈ (𝐺 MndHom 𝐻)) |
35 | 33, 34 | syl 17 | . . 3 ⊢ (𝜑 → 𝑄 ∈ (𝐺 MndHom 𝐻)) |
36 | evlsgsummul.y | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) | |
37 | evlsgsummul.f | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 ) | |
38 | 3, 5, 16, 25, 29, 35, 36, 37 | gsummptmhm 19055 | . 2 ⊢ (𝜑 → (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌)))) |
39 | 38 | eqcomd 2826 | 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 1536 ∈ wcel 2113 Vcvv 3491 ⊆ wss 3929 class class class wbr 5059 ↦ cmpt 5139 ‘cfv 6348 (class class class)co 7149 ↑m cmap 8399 finSupp cfsupp 8826 ℕ0cn0 11891 Basecbs 16478 ↾s cress 16479 Σg cgsu 16709 ↑s cpws 16715 Mndcmnd 17906 MndHom cmhm 17949 CMndccmn 18901 mulGrpcmgp 19234 1rcur 19246 Ringcrg 19292 CRingccrg 19293 RingHom crh 19459 SubRingcsubrg 19526 mPoly cmpl 20128 evalSub ces 20279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-ofr 7403 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-er 8282 df-map 8401 df-pm 8402 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-sup 8899 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 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 12890 df-fzo 13031 df-seq 13367 df-hash 13688 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-sca 16576 df-vsca 16577 df-ip 16578 df-tset 16579 df-ple 16580 df-ds 16582 df-hom 16584 df-cco 16585 df-0g 16710 df-gsum 16711 df-prds 16716 df-pws 16718 df-mre 16852 df-mrc 16853 df-acs 16855 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-mhm 17951 df-submnd 17952 df-grp 18101 df-minusg 18102 df-sbg 18103 df-mulg 18220 df-subg 18271 df-ghm 18351 df-cntz 18442 df-cmn 18903 df-abl 18904 df-mgp 19235 df-ur 19247 df-srg 19251 df-ring 19294 df-cring 19295 df-rnghom 19462 df-subrg 19528 df-lmod 19631 df-lss 19699 df-lsp 19739 df-assa 20080 df-asp 20081 df-ascl 20082 df-psr 20131 df-mvr 20132 df-mpl 20133 df-evls 20281 |
This theorem is referenced by: (None) |
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