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Mirrors > Home > MPE Home > Th. List > evl1gsumadd | Structured version Visualization version GIF version |
Description: Univariate polynomial evaluation maps (additive) group sums to group sums. Remark: the proof would be shorter if the theorem is proved directly instead of using evls1gsumadd 20489. (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) |
evl1gsumadd.0 | ⊢ 0 = (0g‘𝑊) |
evl1gsumadd.f | ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 0 ) |
Ref | Expression |
---|---|
evl1gsumadd | ⊢ (𝜑 → (𝑄‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝑃 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1gsumadd.q | . . . . 5 ⊢ 𝑄 = (eval1‘𝑅) | |
2 | evl1gsumadd.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
3 | 1, 2 | evl1fval1 20496 | . . . 4 ⊢ 𝑄 = (𝑅 evalSub1 𝐾) |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑄 = (𝑅 evalSub1 𝐾)) |
5 | 4 | fveq1d 6674 | . 2 ⊢ (𝜑 → (𝑄‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = ((𝑅 evalSub1 𝐾)‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌)))) |
6 | evl1gsumadd.w | . . . . 5 ⊢ 𝑊 = (Poly1‘𝑅) | |
7 | evl1gsumadd.r | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
8 | 2 | ressid 16561 | . . . . . . . 8 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐾) = 𝑅) |
9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑅 ↾s 𝐾) = 𝑅) |
10 | 9 | eqcomd 2829 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (𝑅 ↾s 𝐾)) |
11 | 10 | fveq2d 6676 | . . . . 5 ⊢ (𝜑 → (Poly1‘𝑅) = (Poly1‘(𝑅 ↾s 𝐾))) |
12 | 6, 11 | syl5eq 2870 | . . . 4 ⊢ (𝜑 → 𝑊 = (Poly1‘(𝑅 ↾s 𝐾))) |
13 | 12 | fvoveq1d 7180 | . . 3 ⊢ (𝜑 → ((𝑅 evalSub1 𝐾)‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = ((𝑅 evalSub1 𝐾)‘((Poly1‘(𝑅 ↾s 𝐾)) Σg (𝑥 ∈ 𝑁 ↦ 𝑌)))) |
14 | eqid 2823 | . . . 4 ⊢ (𝑅 evalSub1 𝐾) = (𝑅 evalSub1 𝐾) | |
15 | eqid 2823 | . . . 4 ⊢ (Poly1‘(𝑅 ↾s 𝐾)) = (Poly1‘(𝑅 ↾s 𝐾)) | |
16 | eqid 2823 | . . . 4 ⊢ (0g‘(Poly1‘(𝑅 ↾s 𝐾))) = (0g‘(Poly1‘(𝑅 ↾s 𝐾))) | |
17 | eqid 2823 | . . . 4 ⊢ (𝑅 ↾s 𝐾) = (𝑅 ↾s 𝐾) | |
18 | evl1gsumadd.p | . . . 4 ⊢ 𝑃 = (𝑅 ↑s 𝐾) | |
19 | eqid 2823 | . . . 4 ⊢ (Base‘(Poly1‘(𝑅 ↾s 𝐾))) = (Base‘(Poly1‘(𝑅 ↾s 𝐾))) | |
20 | crngring 19310 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
21 | 2 | subrgid 19539 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝐾 ∈ (SubRing‘𝑅)) |
22 | 7, 20, 21 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (SubRing‘𝑅)) |
23 | evl1gsumadd.y | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) | |
24 | evl1gsumadd.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑊) | |
25 | 12 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑊 = (Poly1‘(𝑅 ↾s 𝐾))) |
26 | 25 | fveq2d 6676 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (Base‘𝑊) = (Base‘(Poly1‘(𝑅 ↾s 𝐾)))) |
27 | 24, 26 | syl5eq 2870 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝐵 = (Base‘(Poly1‘(𝑅 ↾s 𝐾)))) |
28 | 23, 27 | eleqtrd 2917 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ (Base‘(Poly1‘(𝑅 ↾s 𝐾)))) |
29 | evl1gsumadd.n | . . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ0) | |
30 | evl1gsumadd.f | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 0 ) | |
31 | 12 | eqcomd 2829 | . . . . . . 7 ⊢ (𝜑 → (Poly1‘(𝑅 ↾s 𝐾)) = 𝑊) |
32 | 31 | fveq2d 6676 | . . . . . 6 ⊢ (𝜑 → (0g‘(Poly1‘(𝑅 ↾s 𝐾))) = (0g‘𝑊)) |
33 | evl1gsumadd.0 | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
34 | 32, 33 | syl6eqr 2876 | . . . . 5 ⊢ (𝜑 → (0g‘(Poly1‘(𝑅 ↾s 𝐾))) = 0 ) |
35 | 30, 34 | breqtrrd 5096 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp (0g‘(Poly1‘(𝑅 ↾s 𝐾)))) |
36 | 14, 2, 15, 16, 17, 18, 19, 7, 22, 28, 29, 35 | evls1gsumadd 20489 | . . 3 ⊢ (𝜑 → ((𝑅 evalSub1 𝐾)‘((Poly1‘(𝑅 ↾s 𝐾)) Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝑃 Σg (𝑥 ∈ 𝑁 ↦ ((𝑅 evalSub1 𝐾)‘𝑌)))) |
37 | 13, 36 | eqtrd 2858 | . 2 ⊢ (𝜑 → ((𝑅 evalSub1 𝐾)‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝑃 Σg (𝑥 ∈ 𝑁 ↦ ((𝑅 evalSub1 𝐾)‘𝑌)))) |
38 | 4 | fveq1d 6674 | . . . . 5 ⊢ (𝜑 → (𝑄‘𝑌) = ((𝑅 evalSub1 𝐾)‘𝑌)) |
39 | 38 | eqcomd 2829 | . . . 4 ⊢ (𝜑 → ((𝑅 evalSub1 𝐾)‘𝑌) = (𝑄‘𝑌)) |
40 | 39 | mpteq2dv 5164 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ ((𝑅 evalSub1 𝐾)‘𝑌)) = (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) |
41 | 40 | oveq2d 7174 | . 2 ⊢ (𝜑 → (𝑃 Σg (𝑥 ∈ 𝑁 ↦ ((𝑅 evalSub1 𝐾)‘𝑌))) = (𝑃 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌)))) |
42 | 5, 37, 41 | 3eqtrd 2862 | 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 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 finSupp cfsupp 8835 ℕ0cn0 11900 Basecbs 16485 ↾s cress 16486 0gc0g 16715 Σg cgsu 16716 ↑s cpws 16722 Ringcrg 19299 CRingccrg 19300 SubRingcsubrg 19533 Poly1cpl1 20347 evalSub1 ces1 20478 eval1ce1 20479 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-hom 16591 df-cco 16592 df-0g 16717 df-gsum 16718 df-prds 16723 df-pws 16725 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-srg 19258 df-ring 19301 df-cring 19302 df-rnghom 19469 df-subrg 19535 df-lmod 19638 df-lss 19706 df-lsp 19746 df-assa 20087 df-asp 20088 df-ascl 20089 df-psr 20138 df-mvr 20139 df-mpl 20140 df-opsr 20142 df-evls 20288 df-evl 20289 df-psr1 20350 df-ply1 20352 df-evls1 20480 df-evl1 20481 |
This theorem is referenced by: (None) |
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