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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 20038 | . . 3 ⊢ 𝐵 = (Base‘𝐺) |
4 | evl1gsummul.1 | . . . 4 ⊢ 1 = (1r‘𝑊) | |
5 | 1, 4 | ringidval 20081 | . . 3 ⊢ 1 = (0g‘𝐺) |
6 | evl1gsumadd.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
7 | evl1gsumadd.w | . . . . 5 ⊢ 𝑊 = (Poly1‘𝑅) | |
8 | 7 | ply1crng 21954 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑊 ∈ CRing) |
9 | 1 | crngmgp 20139 | . . . 4 ⊢ (𝑊 ∈ CRing → 𝐺 ∈ CMnd) |
10 | 6, 8, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
11 | crngring 20143 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
12 | 6, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
13 | evl1gsumadd.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑅) | |
14 | 13 | fvexi 6905 | . . . . 5 ⊢ 𝐾 ∈ V |
15 | 12, 14 | jctir 520 | . . . 4 ⊢ (𝜑 → (𝑅 ∈ Ring ∧ 𝐾 ∈ V)) |
16 | evl1gsumadd.p | . . . . 5 ⊢ 𝑃 = (𝑅 ↑s 𝐾) | |
17 | 16 | pwsring 20216 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ V) → 𝑃 ∈ Ring) |
18 | evl1gsummul.h | . . . . 5 ⊢ 𝐻 = (mulGrp‘𝑃) | |
19 | 18 | ringmgp 20137 | . . . 4 ⊢ (𝑃 ∈ Ring → 𝐻 ∈ Mnd) |
20 | 15, 17, 19 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
21 | nn0ex 12485 | . . . . 5 ⊢ ℕ0 ∈ V | |
22 | 21 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
23 | evl1gsumadd.n | . . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ0) | |
24 | 22, 23 | ssexd 5324 | . . 3 ⊢ (𝜑 → 𝑁 ∈ V) |
25 | evl1gsumadd.q | . . . . 5 ⊢ 𝑄 = (eval1‘𝑅) | |
26 | 25, 7, 16, 13 | evl1rhm 22084 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
27 | 1, 18 | rhmmhm 20374 | . . . 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 19853 | . 2 ⊢ (𝜑 → (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌)))) |
32 | 31 | eqcomd 2737 | 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 395 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ⊆ wss 3948 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7412 finSupp cfsupp 9367 ℕ0cn0 12479 Basecbs 17151 Σg cgsu 17393 ↑s cpws 17399 Mndcmnd 18662 MndHom cmhm 18706 CMndccmn 19693 mulGrpcmgp 20032 1rcur 20079 Ringcrg 20131 CRingccrg 20132 RingHom crh 20364 Poly1cpl1 21933 eval1ce1 22066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-ofr 7675 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-fzo 13635 df-seq 13974 df-hash 14298 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-hom 17228 df-cco 17229 df-0g 17394 df-gsum 17395 df-prds 17400 df-pws 17402 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-mhm 18708 df-submnd 18709 df-grp 18861 df-minusg 18862 df-sbg 18863 df-mulg 18991 df-subg 19043 df-ghm 19132 df-cntz 19226 df-cmn 19695 df-abl 19696 df-mgp 20033 df-rng 20051 df-ur 20080 df-srg 20085 df-ring 20133 df-cring 20134 df-rhm 20367 df-subrng 20438 df-subrg 20463 df-lmod 20620 df-lss 20691 df-lsp 20731 df-assa 21631 df-asp 21632 df-ascl 21633 df-psr 21685 df-mvr 21686 df-mpl 21687 df-opsr 21689 df-evls 21859 df-evl 21860 df-psr1 21936 df-ply1 21938 df-evl1 22068 |
This theorem is referenced by: (None) |
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