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Mirrors > Home > MPE Home > Th. List > coe1add | Structured version Visualization version GIF version |
Description: The coefficient vector of an addition. (Contributed by Stefan O'Rear, 24-Mar-2015.) |
Ref | Expression |
---|---|
coe1add.y | ⊢ 𝑌 = (Poly1‘𝑅) |
coe1add.b | ⊢ 𝐵 = (Base‘𝑌) |
coe1add.p | ⊢ ✚ = (+g‘𝑌) |
coe1add.q | ⊢ + = (+g‘𝑅) |
Ref | Expression |
---|---|
coe1add | ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘(𝐹 ✚ 𝐺)) = ((coe1‘𝐹) ∘𝑓 + (coe1‘𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2651 | . . . . 5 ⊢ (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅) | |
2 | coe1add.y | . . . . . 6 ⊢ 𝑌 = (Poly1‘𝑅) | |
3 | eqid 2651 | . . . . . 6 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
4 | coe1add.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑌) | |
5 | 2, 3, 4 | ply1bas 19613 | . . . . 5 ⊢ 𝐵 = (Base‘(1𝑜 mPoly 𝑅)) |
6 | coe1add.q | . . . . 5 ⊢ + = (+g‘𝑅) | |
7 | coe1add.p | . . . . . 6 ⊢ ✚ = (+g‘𝑌) | |
8 | 2, 1, 7 | ply1plusg 19643 | . . . . 5 ⊢ ✚ = (+g‘(1𝑜 mPoly 𝑅)) |
9 | simp2 1082 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵) | |
10 | simp3 1083 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵) | |
11 | 1, 5, 6, 8, 9, 10 | mpladd 19490 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ✚ 𝐺) = (𝐹 ∘𝑓 + 𝐺)) |
12 | 11 | coeq1d 5316 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐹 ✚ 𝐺) ∘ (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎}))) = ((𝐹 ∘𝑓 + 𝐺) ∘ (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎})))) |
13 | eqid 2651 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
14 | 2, 4, 13 | ply1basf 19620 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑𝑚 1𝑜)⟶(Base‘𝑅)) |
15 | ffn 6083 | . . . . . 6 ⊢ (𝐹:(ℕ0 ↑𝑚 1𝑜)⟶(Base‘𝑅) → 𝐹 Fn (ℕ0 ↑𝑚 1𝑜)) | |
16 | 14, 15 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → 𝐹 Fn (ℕ0 ↑𝑚 1𝑜)) |
17 | 16 | 3ad2ant2 1103 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 Fn (ℕ0 ↑𝑚 1𝑜)) |
18 | 2, 4, 13 | ply1basf 19620 | . . . . . 6 ⊢ (𝐺 ∈ 𝐵 → 𝐺:(ℕ0 ↑𝑚 1𝑜)⟶(Base‘𝑅)) |
19 | ffn 6083 | . . . . . 6 ⊢ (𝐺:(ℕ0 ↑𝑚 1𝑜)⟶(Base‘𝑅) → 𝐺 Fn (ℕ0 ↑𝑚 1𝑜)) | |
20 | 18, 19 | syl 17 | . . . . 5 ⊢ (𝐺 ∈ 𝐵 → 𝐺 Fn (ℕ0 ↑𝑚 1𝑜)) |
21 | 20 | 3ad2ant3 1104 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 Fn (ℕ0 ↑𝑚 1𝑜)) |
22 | df1o2 7617 | . . . . . 6 ⊢ 1𝑜 = {∅} | |
23 | nn0ex 11336 | . . . . . 6 ⊢ ℕ0 ∈ V | |
24 | 0ex 4823 | . . . . . 6 ⊢ ∅ ∈ V | |
25 | eqid 2651 | . . . . . 6 ⊢ (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎})) = (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎})) | |
26 | 22, 23, 24, 25 | mapsnf1o3 7948 | . . . . 5 ⊢ (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎})):ℕ0–1-1-onto→(ℕ0 ↑𝑚 1𝑜) |
27 | f1of 6175 | . . . . 5 ⊢ ((𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎})):ℕ0–1-1-onto→(ℕ0 ↑𝑚 1𝑜) → (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎})):ℕ0⟶(ℕ0 ↑𝑚 1𝑜)) | |
28 | 26, 27 | mp1i 13 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎})):ℕ0⟶(ℕ0 ↑𝑚 1𝑜)) |
29 | ovexd 6720 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (ℕ0 ↑𝑚 1𝑜) ∈ V) | |
30 | 23 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ℕ0 ∈ V) |
31 | inidm 3855 | . . . 4 ⊢ ((ℕ0 ↑𝑚 1𝑜) ∩ (ℕ0 ↑𝑚 1𝑜)) = (ℕ0 ↑𝑚 1𝑜) | |
32 | 17, 21, 28, 29, 29, 30, 31 | ofco 6959 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐹 ∘𝑓 + 𝐺) ∘ (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎}))) = ((𝐹 ∘ (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎}))) ∘𝑓 + (𝐺 ∘ (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎}))))) |
33 | 12, 32 | eqtrd 2685 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐹 ✚ 𝐺) ∘ (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎}))) = ((𝐹 ∘ (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎}))) ∘𝑓 + (𝐺 ∘ (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎}))))) |
34 | 2 | ply1ring 19666 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ Ring) |
35 | 4, 7 | ringacl 18624 | . . . 4 ⊢ ((𝑌 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ✚ 𝐺) ∈ 𝐵) |
36 | 34, 35 | syl3an1 1399 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ✚ 𝐺) ∈ 𝐵) |
37 | eqid 2651 | . . . 4 ⊢ (coe1‘(𝐹 ✚ 𝐺)) = (coe1‘(𝐹 ✚ 𝐺)) | |
38 | 37, 4, 2, 25 | coe1fval2 19628 | . . 3 ⊢ ((𝐹 ✚ 𝐺) ∈ 𝐵 → (coe1‘(𝐹 ✚ 𝐺)) = ((𝐹 ✚ 𝐺) ∘ (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎})))) |
39 | 36, 38 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘(𝐹 ✚ 𝐺)) = ((𝐹 ✚ 𝐺) ∘ (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎})))) |
40 | eqid 2651 | . . . . 5 ⊢ (coe1‘𝐹) = (coe1‘𝐹) | |
41 | 40, 4, 2, 25 | coe1fval2 19628 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → (coe1‘𝐹) = (𝐹 ∘ (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎})))) |
42 | 41 | 3ad2ant2 1103 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘𝐹) = (𝐹 ∘ (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎})))) |
43 | eqid 2651 | . . . . 5 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
44 | 43, 4, 2, 25 | coe1fval2 19628 | . . . 4 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺) = (𝐺 ∘ (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎})))) |
45 | 44 | 3ad2ant3 1104 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘𝐺) = (𝐺 ∘ (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎})))) |
46 | 42, 45 | oveq12d 6708 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((coe1‘𝐹) ∘𝑓 + (coe1‘𝐺)) = ((𝐹 ∘ (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎}))) ∘𝑓 + (𝐺 ∘ (𝑎 ∈ ℕ0 ↦ (1𝑜 × {𝑎}))))) |
47 | 33, 39, 46 | 3eqtr4d 2695 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘(𝐹 ✚ 𝐺)) = ((coe1‘𝐹) ∘𝑓 + (coe1‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ∅c0 3948 {csn 4210 ↦ cmpt 4762 × cxp 5141 ∘ ccom 5147 Fn wfn 5921 ⟶wf 5922 –1-1-onto→wf1o 5925 ‘cfv 5926 (class class class)co 6690 ∘𝑓 cof 6937 1𝑜c1o 7598 ↑𝑚 cmap 7899 ℕ0cn0 11330 Basecbs 15904 +gcplusg 15988 Ringcrg 18593 mPoly cmpl 19401 PwSer1cps1 19593 Poly1cpl1 19595 coe1cco1 19596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-ofr 6940 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-oi 8456 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-fz 12365 df-fzo 12505 df-seq 12842 df-hash 13158 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-tset 16007 df-ple 16008 df-0g 16149 df-gsum 16150 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-mhm 17382 df-submnd 17383 df-grp 17472 df-minusg 17473 df-mulg 17588 df-subg 17638 df-ghm 17705 df-cntz 17796 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-subrg 18826 df-psr 19404 df-mpl 19406 df-opsr 19408 df-psr1 19598 df-ply1 19600 df-coe1 19601 |
This theorem is referenced by: coe1addfv 19683 |
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