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Mirrors > Home > MPE Home > Th. List > evls1gsumadd | Structured version Visualization version GIF version |
Description: Univariate polynomial evaluation maps (additive) group sums to group sums. (Contributed by AV, 14-Sep-2019.) |
Ref | Expression |
---|---|
evls1gsumadd.q | ⊢ 𝑄 = (𝑆 evalSub1 𝑅) |
evls1gsumadd.k | ⊢ 𝐾 = (Base‘𝑆) |
evls1gsumadd.w | ⊢ 𝑊 = (Poly1‘𝑈) |
evls1gsumadd.0 | ⊢ 0 = (0g‘𝑊) |
evls1gsumadd.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evls1gsumadd.p | ⊢ 𝑃 = (𝑆 ↑s 𝐾) |
evls1gsumadd.b | ⊢ 𝐵 = (Base‘𝑊) |
evls1gsumadd.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evls1gsumadd.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evls1gsumadd.y | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) |
evls1gsumadd.n | ⊢ (𝜑 → 𝑁 ⊆ ℕ0) |
evls1gsumadd.f | ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 0 ) |
Ref | Expression |
---|---|
evls1gsumadd | ⊢ (𝜑 → (𝑄‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝑃 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evls1gsumadd.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
2 | evls1gsumadd.0 | . . 3 ⊢ 0 = (0g‘𝑊) | |
3 | evls1gsumadd.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
4 | evls1gsumadd.u | . . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
5 | 4 | subrgring 19467 | . . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring) |
6 | 3, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ Ring) |
7 | evls1gsumadd.w | . . . . 5 ⊢ 𝑊 = (Poly1‘𝑈) | |
8 | 7 | ply1ring 20344 | . . . 4 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring) |
9 | ringcmn 19260 | . . . 4 ⊢ (𝑊 ∈ Ring → 𝑊 ∈ CMnd) | |
10 | 6, 8, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑊 ∈ CMnd) |
11 | evls1gsumadd.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
12 | crngring 19237 | . . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring) |
14 | evls1gsumadd.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑆) | |
15 | 14 | fvexi 6677 | . . . . 5 ⊢ 𝐾 ∈ V |
16 | 13, 15 | jctir 521 | . . . 4 ⊢ (𝜑 → (𝑆 ∈ Ring ∧ 𝐾 ∈ V)) |
17 | evls1gsumadd.p | . . . . 5 ⊢ 𝑃 = (𝑆 ↑s 𝐾) | |
18 | 17 | pwsring 19294 | . . . 4 ⊢ ((𝑆 ∈ Ring ∧ 𝐾 ∈ V) → 𝑃 ∈ Ring) |
19 | ringmnd 19235 | . . . 4 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Mnd) | |
20 | 16, 18, 19 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Mnd) |
21 | nn0ex 11891 | . . . . 5 ⊢ ℕ0 ∈ V | |
22 | 21 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
23 | evls1gsumadd.n | . . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ0) | |
24 | 22, 23 | ssexd 5219 | . . 3 ⊢ (𝜑 → 𝑁 ∈ V) |
25 | evls1gsumadd.q | . . . . . 6 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
26 | 25, 14, 17, 4, 7 | evls1rhm 20413 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
27 | 11, 3, 26 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
28 | rhmghm 19406 | . . . 4 ⊢ (𝑄 ∈ (𝑊 RingHom 𝑃) → 𝑄 ∈ (𝑊 GrpHom 𝑃)) | |
29 | ghmmhm 18306 | . . . 4 ⊢ (𝑄 ∈ (𝑊 GrpHom 𝑃) → 𝑄 ∈ (𝑊 MndHom 𝑃)) | |
30 | 27, 28, 29 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑄 ∈ (𝑊 MndHom 𝑃)) |
31 | evls1gsumadd.y | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) | |
32 | evls1gsumadd.f | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 0 ) | |
33 | 1, 2, 10, 20, 24, 30, 31, 32 | gsummptmhm 18989 | . 2 ⊢ (𝜑 → (𝑃 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌)))) |
34 | 33 | eqcomd 2824 | 1 ⊢ (𝜑 → (𝑄‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝑃 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 class class class wbr 5057 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 finSupp cfsupp 8821 ℕ0cn0 11885 Basecbs 16471 ↾s cress 16472 0gc0g 16701 Σg cgsu 16702 ↑s cpws 16708 Mndcmnd 17899 MndHom cmhm 17942 GrpHom cghm 18293 CMndccmn 18835 Ringcrg 19226 CRingccrg 19227 RingHom crh 19393 SubRingcsubrg 19460 Poly1cpl1 20273 evalSub1 ces1 20404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 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 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-ofr 7399 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 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 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-hom 16577 df-cco 16578 df-0g 16703 df-gsum 16704 df-prds 16709 df-pws 16711 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-ghm 18294 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-srg 19185 df-ring 19228 df-cring 19229 df-rnghom 19396 df-subrg 19462 df-lmod 19565 df-lss 19633 df-lsp 19673 df-assa 20013 df-asp 20014 df-ascl 20015 df-psr 20064 df-mvr 20065 df-mpl 20066 df-opsr 20068 df-evls 20214 df-psr1 20276 df-ply1 20278 df-evls1 20406 |
This theorem is referenced by: evl1gsumadd 20449 |
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