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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 22360. (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 22367 | . . . 4 ⊢ 𝑄 = (𝑅 evalSub1 𝐾) |
| 4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑄 = (𝑅 evalSub1 𝐾)) |
| 5 | 4 | fveq1d 6858 | . 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 17256 | . . . . . . . 8 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐾) = 𝑅) |
| 9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑅 ↾s 𝐾) = 𝑅) |
| 10 | 9 | eqcomd 2762 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (𝑅 ↾s 𝐾)) |
| 11 | 10 | fveq2d 6860 | . . . . 5 ⊢ (𝜑 → (Poly1‘𝑅) = (Poly1‘(𝑅 ↾s 𝐾))) |
| 12 | 6, 11 | eqtrid 2803 | . . . 4 ⊢ (𝜑 → 𝑊 = (Poly1‘(𝑅 ↾s 𝐾))) |
| 13 | 12 | fvoveq1d 7407 | . . 3 ⊢ (𝜑 → ((𝑅 evalSub1 𝐾)‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = ((𝑅 evalSub1 𝐾)‘((Poly1‘(𝑅 ↾s 𝐾)) Σg (𝑥 ∈ 𝑁 ↦ 𝑌)))) |
| 14 | eqid 2756 | . . . 4 ⊢ (𝑅 evalSub1 𝐾) = (𝑅 evalSub1 𝐾) | |
| 15 | eqid 2756 | . . . 4 ⊢ (Poly1‘(𝑅 ↾s 𝐾)) = (Poly1‘(𝑅 ↾s 𝐾)) | |
| 16 | eqid 2756 | . . . 4 ⊢ (0g‘(Poly1‘(𝑅 ↾s 𝐾))) = (0g‘(Poly1‘(𝑅 ↾s 𝐾))) | |
| 17 | eqid 2756 | . . . 4 ⊢ (𝑅 ↾s 𝐾) = (𝑅 ↾s 𝐾) | |
| 18 | evl1gsumadd.p | . . . 4 ⊢ 𝑃 = (𝑅 ↑s 𝐾) | |
| 19 | eqid 2756 | . . . 4 ⊢ (Base‘(Poly1‘(𝑅 ↾s 𝐾))) = (Base‘(Poly1‘(𝑅 ↾s 𝐾))) | |
| 20 | crngring 20267 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
| 21 | 2 | subrgid 20595 | . . . . 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 6860 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (Base‘𝑊) = (Base‘(Poly1‘(𝑅 ↾s 𝐾)))) |
| 27 | 24, 26 | eqtrid 2803 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝐵 = (Base‘(Poly1‘(𝑅 ↾s 𝐾)))) |
| 28 | 23, 27 | eleqtrd 2858 | . . . 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 2762 | . . . . . . 7 ⊢ (𝜑 → (Poly1‘(𝑅 ↾s 𝐾)) = 𝑊) |
| 32 | 31 | fveq2d 6860 | . . . . . 6 ⊢ (𝜑 → (0g‘(Poly1‘(𝑅 ↾s 𝐾))) = (0g‘𝑊)) |
| 33 | evl1gsumadd.0 | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
| 34 | 32, 33 | eqtr4di 2809 | . . . . 5 ⊢ (𝜑 → (0g‘(Poly1‘(𝑅 ↾s 𝐾))) = 0 ) |
| 35 | 30, 34 | breqtrrd 5122 | . . . 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 22360 | . . 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 2791 | . 2 ⊢ (𝜑 → ((𝑅 evalSub1 𝐾)‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝑃 Σg (𝑥 ∈ 𝑁 ↦ ((𝑅 evalSub1 𝐾)‘𝑌)))) |
| 38 | 4 | fveq1d 6858 | . . . . 5 ⊢ (𝜑 → (𝑄‘𝑌) = ((𝑅 evalSub1 𝐾)‘𝑌)) |
| 39 | 38 | eqcomd 2762 | . . . 4 ⊢ (𝜑 → ((𝑅 evalSub1 𝐾)‘𝑌) = (𝑄‘𝑌)) |
| 40 | 39 | mpteq2dv 5188 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ ((𝑅 evalSub1 𝐾)‘𝑌)) = (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) |
| 41 | 40 | oveq2d 7401 | . 2 ⊢ (𝜑 → (𝑃 Σg (𝑥 ∈ 𝑁 ↦ ((𝑅 evalSub1 𝐾)‘𝑌))) = (𝑃 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌)))) |
| 42 | 5, 37, 41 | 3eqtrd 2795 | 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 1554 ∈ wcel 2136 ⊆ wss 3899 class class class wbr 5094 ↦ cmpt 5175 ‘cfv 6510 (class class class)co 7385 finSupp cfsupp 9297 ℕ0cn0 12471 Basecbs 17221 ↾s cress 17242 0gc0g 17444 Σg cgsu 17445 ↑s cpws 17451 Ringcrg 20255 CRingccrg 20256 SubRingcsubrg 20591 Poly1cpl1 22212 evalSub1 ces1 22349 eval1ce1 22350 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-rep 5221 ax-sep 5240 ax-nul 5250 ax-pow 5316 ax-pr 5384 ax-un 7707 ax-cnex 11119 ax-resscn 11120 ax-1cn 11121 ax-icn 11122 ax-addcl 11123 ax-addrcl 11124 ax-mulcl 11125 ax-mulrcl 11126 ax-mulcom 11127 ax-addass 11128 ax-mulass 11129 ax-distr 11130 ax-i2m1 11131 ax-1ne0 11132 ax-1rid 11133 ax-rnegex 11134 ax-rrecex 11135 ax-cnre 11136 ax-pre-lttri 11137 ax-pre-lttrn 11138 ax-pre-ltadd 11139 ax-pre-mulgt0 11140 |
| This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-nel 3056 df-ral 3071 df-rex 3081 df-rmo 3361 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4900 df-iun 4945 df-iin 4946 df-br 5095 df-opab 5157 df-mpt 5176 df-tr 5202 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-isom 6519 df-riota 7342 df-ov 7388 df-oprab 7389 df-mpo 7390 df-of 7649 df-ofr 7650 df-om 7836 df-1st 7959 df-2nd 7960 df-supp 8129 df-frecs 8250 df-wrecs 8281 df-recs 8330 df-rdg 8369 df-1o 8425 df-2o 8426 df-er 8666 df-map 8798 df-pm 8799 df-ixp 8869 df-en 8917 df-dom 8918 df-sdom 8919 df-fin 8920 df-fsupp 9298 df-sup 9378 df-oi 9448 df-card 9887 df-pnf 11208 df-mnf 11209 df-xr 11210 df-ltxr 11211 df-le 11212 df-sub 11406 df-neg 11407 df-nn 12201 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12472 df-z 12559 df-dec 12679 df-uz 12830 df-fz 13503 df-fzo 13650 df-seq 14005 df-hash 14334 df-struct 17159 df-sets 17176 df-slot 17194 df-ndx 17206 df-base 17222 df-ress 17243 df-plusg 17275 df-mulr 17276 df-sca 17278 df-vsca 17279 df-ip 17280 df-tset 17281 df-ple 17282 df-ds 17284 df-hom 17286 df-cco 17287 df-0g 17446 df-gsum 17447 df-prds 17452 df-pws 17454 df-mre 17590 df-mrc 17591 df-acs 17593 df-mgm 18650 df-sgrp 18729 df-mnd 18745 df-mhm 18793 df-submnd 18794 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19230 df-cntz 19333 df-cmn 19798 df-abl 19799 df-mgp 20163 df-rng 20175 df-ur 20204 df-srg 20209 df-ring 20257 df-cring 20258 df-rhm 20493 df-subrng 20568 df-subrg 20592 df-lmod 20902 df-lss 20972 df-lsp 21012 df-assa 21878 df-asp 21879 df-ascl 21880 df-psr 21934 df-mvr 21935 df-mpl 21936 df-opsr 21938 df-evls 22100 df-evl 22101 df-psr1 22215 df-ply1 22217 df-evls1 22351 df-evl1 22352 |
| This theorem is referenced by: (None) |
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