![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
4 | evls1gsumadd.u | . . . . 5 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
5 | 4 | subrgring 20558 | . . . 4 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring) |
6 | evls1gsumadd.w | . . . . 5 ⊢ 𝑊 = (Poly1‘𝑈) | |
7 | 6 | ply1ring 22237 | . . . 4 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring) |
8 | ringcmn 20261 | . . . 4 ⊢ (𝑊 ∈ Ring → 𝑊 ∈ CMnd) | |
9 | 3, 5, 7, 8 | 4syl 19 | . . 3 ⊢ (𝜑 → 𝑊 ∈ CMnd) |
10 | evls1gsumadd.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
11 | crngring 20228 | . . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring) |
13 | evls1gsumadd.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑆) | |
14 | 13 | fvexi 6915 | . . . . 5 ⊢ 𝐾 ∈ V |
15 | 12, 14 | jctir 519 | . . . 4 ⊢ (𝜑 → (𝑆 ∈ Ring ∧ 𝐾 ∈ V)) |
16 | evls1gsumadd.p | . . . . 5 ⊢ 𝑃 = (𝑆 ↑s 𝐾) | |
17 | 16 | pwsring 20303 | . . . 4 ⊢ ((𝑆 ∈ Ring ∧ 𝐾 ∈ V) → 𝑃 ∈ Ring) |
18 | ringmnd 20226 | . . . 4 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Mnd) | |
19 | 15, 17, 18 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Mnd) |
20 | nn0ex 12530 | . . . . 5 ⊢ ℕ0 ∈ V | |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
22 | evls1gsumadd.n | . . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ0) | |
23 | 21, 22 | ssexd 5329 | . . 3 ⊢ (𝜑 → 𝑁 ∈ V) |
24 | evls1gsumadd.q | . . . . . 6 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
25 | 24, 13, 16, 4, 6 | evls1rhm 22313 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
26 | 10, 3, 25 | syl2anc 582 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
27 | rhmghm 20466 | . . . 4 ⊢ (𝑄 ∈ (𝑊 RingHom 𝑃) → 𝑄 ∈ (𝑊 GrpHom 𝑃)) | |
28 | ghmmhm 19220 | . . . 4 ⊢ (𝑄 ∈ (𝑊 GrpHom 𝑃) → 𝑄 ∈ (𝑊 MndHom 𝑃)) | |
29 | 26, 27, 28 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑄 ∈ (𝑊 MndHom 𝑃)) |
30 | evls1gsumadd.y | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) | |
31 | evls1gsumadd.f | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 0 ) | |
32 | 1, 2, 9, 19, 23, 29, 30, 31 | gsummptmhm 19938 | . 2 ⊢ (𝜑 → (𝑃 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌)))) |
33 | 32 | eqcomd 2732 | 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 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ⊆ wss 3947 class class class wbr 5153 ↦ cmpt 5236 ‘cfv 6554 (class class class)co 7424 finSupp cfsupp 9405 ℕ0cn0 12524 Basecbs 17213 ↾s cress 17242 0gc0g 17454 Σg cgsu 17455 ↑s cpws 17461 Mndcmnd 18727 MndHom cmhm 18771 GrpHom cghm 19206 CMndccmn 19778 Ringcrg 20216 CRingccrg 20217 RingHom crh 20451 SubRingcsubrg 20551 Poly1cpl1 22166 evalSub1 ces1 22304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-ofr 7691 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-pm 8858 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-sup 9485 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-fz 13539 df-fzo 13682 df-seq 14022 df-hash 14348 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-hom 17290 df-cco 17291 df-0g 17456 df-gsum 17457 df-prds 17462 df-pws 17464 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-mhm 18773 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-mulg 19062 df-subg 19117 df-ghm 19207 df-cntz 19311 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-srg 20170 df-ring 20218 df-cring 20219 df-rhm 20454 df-subrng 20528 df-subrg 20553 df-lmod 20838 df-lss 20909 df-lsp 20949 df-assa 21851 df-asp 21852 df-ascl 21853 df-psr 21906 df-mvr 21907 df-mpl 21908 df-opsr 21910 df-evls 22087 df-psr1 22169 df-ply1 22171 df-evls1 22306 |
This theorem is referenced by: evl1gsumadd 22349 evls1fpws 22360 |
Copyright terms: Public domain | W3C validator |