![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 22344. (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 22351 | . . . 4 ⊢ 𝑄 = (𝑅 evalSub1 𝐾) |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑄 = (𝑅 evalSub1 𝐾)) |
5 | 4 | fveq1d 6909 | . 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 17290 | . . . . . . . 8 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐾) = 𝑅) |
9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑅 ↾s 𝐾) = 𝑅) |
10 | 9 | eqcomd 2741 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (𝑅 ↾s 𝐾)) |
11 | 10 | fveq2d 6911 | . . . . 5 ⊢ (𝜑 → (Poly1‘𝑅) = (Poly1‘(𝑅 ↾s 𝐾))) |
12 | 6, 11 | eqtrid 2787 | . . . 4 ⊢ (𝜑 → 𝑊 = (Poly1‘(𝑅 ↾s 𝐾))) |
13 | 12 | fvoveq1d 7453 | . . 3 ⊢ (𝜑 → ((𝑅 evalSub1 𝐾)‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = ((𝑅 evalSub1 𝐾)‘((Poly1‘(𝑅 ↾s 𝐾)) Σg (𝑥 ∈ 𝑁 ↦ 𝑌)))) |
14 | eqid 2735 | . . . 4 ⊢ (𝑅 evalSub1 𝐾) = (𝑅 evalSub1 𝐾) | |
15 | eqid 2735 | . . . 4 ⊢ (Poly1‘(𝑅 ↾s 𝐾)) = (Poly1‘(𝑅 ↾s 𝐾)) | |
16 | eqid 2735 | . . . 4 ⊢ (0g‘(Poly1‘(𝑅 ↾s 𝐾))) = (0g‘(Poly1‘(𝑅 ↾s 𝐾))) | |
17 | eqid 2735 | . . . 4 ⊢ (𝑅 ↾s 𝐾) = (𝑅 ↾s 𝐾) | |
18 | evl1gsumadd.p | . . . 4 ⊢ 𝑃 = (𝑅 ↑s 𝐾) | |
19 | eqid 2735 | . . . 4 ⊢ (Base‘(Poly1‘(𝑅 ↾s 𝐾))) = (Base‘(Poly1‘(𝑅 ↾s 𝐾))) | |
20 | crngring 20263 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
21 | 2 | subrgid 20590 | . . . . 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 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑊 = (Poly1‘(𝑅 ↾s 𝐾))) |
26 | 25 | fveq2d 6911 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (Base‘𝑊) = (Base‘(Poly1‘(𝑅 ↾s 𝐾)))) |
27 | 24, 26 | eqtrid 2787 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝐵 = (Base‘(Poly1‘(𝑅 ↾s 𝐾)))) |
28 | 23, 27 | eleqtrd 2841 | . . . 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 2741 | . . . . . . 7 ⊢ (𝜑 → (Poly1‘(𝑅 ↾s 𝐾)) = 𝑊) |
32 | 31 | fveq2d 6911 | . . . . . 6 ⊢ (𝜑 → (0g‘(Poly1‘(𝑅 ↾s 𝐾))) = (0g‘𝑊)) |
33 | evl1gsumadd.0 | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
34 | 32, 33 | eqtr4di 2793 | . . . . 5 ⊢ (𝜑 → (0g‘(Poly1‘(𝑅 ↾s 𝐾))) = 0 ) |
35 | 30, 34 | breqtrrd 5176 | . . . 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 22344 | . . 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 2775 | . 2 ⊢ (𝜑 → ((𝑅 evalSub1 𝐾)‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝑃 Σg (𝑥 ∈ 𝑁 ↦ ((𝑅 evalSub1 𝐾)‘𝑌)))) |
38 | 4 | fveq1d 6909 | . . . . 5 ⊢ (𝜑 → (𝑄‘𝑌) = ((𝑅 evalSub1 𝐾)‘𝑌)) |
39 | 38 | eqcomd 2741 | . . . 4 ⊢ (𝜑 → ((𝑅 evalSub1 𝐾)‘𝑌) = (𝑄‘𝑌)) |
40 | 39 | mpteq2dv 5250 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ ((𝑅 evalSub1 𝐾)‘𝑌)) = (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) |
41 | 40 | oveq2d 7447 | . 2 ⊢ (𝜑 → (𝑃 Σg (𝑥 ∈ 𝑁 ↦ ((𝑅 evalSub1 𝐾)‘𝑌))) = (𝑃 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌)))) |
42 | 5, 37, 41 | 3eqtrd 2779 | 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 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 finSupp cfsupp 9399 ℕ0cn0 12524 Basecbs 17245 ↾s cress 17274 0gc0g 17486 Σg cgsu 17487 ↑s cpws 17493 Ringcrg 20251 CRingccrg 20252 SubRingcsubrg 20586 Poly1cpl1 22194 evalSub1 ces1 22333 eval1ce1 22334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 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 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-srg 20205 df-ring 20253 df-cring 20254 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-assa 21891 df-asp 21892 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949 df-opsr 21951 df-evls 22116 df-evl 22117 df-psr1 22197 df-ply1 22199 df-evls1 22335 df-evl1 22336 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |