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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 18606 | . . . . 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 19439 | . . . 4 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring) |
9 | ringcmn 18404 | . . . 4 ⊢ (𝑊 ∈ Ring → 𝑊 ∈ CMnd) | |
10 | 6, 8, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑊 ∈ CMnd) |
11 | evls1gsumadd.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
12 | crngring 18381 | . . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring) |
14 | evls1gsumadd.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑆) | |
15 | fvex 6113 | . . . . . 6 ⊢ (Base‘𝑆) ∈ V | |
16 | 14, 15 | eqeltri 2684 | . . . . 5 ⊢ 𝐾 ∈ V |
17 | 13, 16 | jctir 559 | . . . 4 ⊢ (𝜑 → (𝑆 ∈ Ring ∧ 𝐾 ∈ V)) |
18 | evls1gsumadd.p | . . . . 5 ⊢ 𝑃 = (𝑆 ↑s 𝐾) | |
19 | 18 | pwsring 18438 | . . . 4 ⊢ ((𝑆 ∈ Ring ∧ 𝐾 ∈ V) → 𝑃 ∈ Ring) |
20 | ringmnd 18379 | . . . 4 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Mnd) | |
21 | 17, 19, 20 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Mnd) |
22 | nn0ex 11175 | . . . . 5 ⊢ ℕ0 ∈ V | |
23 | 22 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
24 | evls1gsumadd.n | . . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ0) | |
25 | 23, 24 | ssexd 4733 | . . 3 ⊢ (𝜑 → 𝑁 ∈ V) |
26 | evls1gsumadd.q | . . . . . 6 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
27 | 26, 14, 18, 4, 7 | evls1rhm 19508 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
28 | 11, 3, 27 | syl2anc 691 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
29 | rhmghm 18548 | . . . 4 ⊢ (𝑄 ∈ (𝑊 RingHom 𝑃) → 𝑄 ∈ (𝑊 GrpHom 𝑃)) | |
30 | ghmmhm 17493 | . . . 4 ⊢ (𝑄 ∈ (𝑊 GrpHom 𝑃) → 𝑄 ∈ (𝑊 MndHom 𝑃)) | |
31 | 28, 29, 30 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑄 ∈ (𝑊 MndHom 𝑃)) |
32 | evls1gsumadd.y | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) | |
33 | evls1gsumadd.f | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 0 ) | |
34 | 1, 2, 10, 21, 25, 31, 32, 33 | gsummptmhm 18163 | . 2 ⊢ (𝜑 → (𝑃 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌)))) |
35 | 34 | eqcomd 2616 | 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 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 finSupp cfsupp 8158 ℕ0cn0 11169 Basecbs 15695 ↾s cress 15696 0gc0g 15923 Σg cgsu 15924 ↑s cpws 15930 Mndcmnd 17117 MndHom cmhm 17156 GrpHom cghm 17480 CMndccmn 18016 Ringcrg 18370 CRingccrg 18371 RingHom crh 18535 SubRingcsubrg 18599 Poly1cpl1 19368 evalSub1 ces1 19499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-ofr 6796 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-0g 15925 df-gsum 15926 df-prds 15931 df-pws 15933 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-srg 18329 df-ring 18372 df-cring 18373 df-rnghom 18538 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-assa 19133 df-asp 19134 df-ascl 19135 df-psr 19177 df-mvr 19178 df-mpl 19179 df-opsr 19181 df-evls 19327 df-psr1 19371 df-ply1 19373 df-evls1 19501 |
This theorem is referenced by: evl1gsumadd 19543 |
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