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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 2778 | . . . . 5 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
2 | coe1add.y | . . . . . 6 ⊢ 𝑌 = (Poly1‘𝑅) | |
3 | eqid 2778 | . . . . . 6 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
4 | coe1add.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑌) | |
5 | 2, 3, 4 | ply1bas 19965 | . . . . 5 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅)) |
6 | coe1add.q | . . . . 5 ⊢ + = (+g‘𝑅) | |
7 | coe1add.p | . . . . . 6 ⊢ ✚ = (+g‘𝑌) | |
8 | 2, 1, 7 | ply1plusg 19995 | . . . . 5 ⊢ ✚ = (+g‘(1o mPoly 𝑅)) |
9 | simp2 1128 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵) | |
10 | simp3 1129 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵) | |
11 | 1, 5, 6, 8, 9, 10 | mpladd 19843 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ✚ 𝐺) = (𝐹 ∘𝑓 + 𝐺)) |
12 | 11 | coeq1d 5531 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐹 ✚ 𝐺) ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎}))) = ((𝐹 ∘𝑓 + 𝐺) ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
13 | eqid 2778 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
14 | 2, 4, 13 | ply1basf 19972 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑𝑚 1o)⟶(Base‘𝑅)) |
15 | 14 | ffnd 6294 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → 𝐹 Fn (ℕ0 ↑𝑚 1o)) |
16 | 15 | 3ad2ant2 1125 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 Fn (ℕ0 ↑𝑚 1o)) |
17 | 2, 4, 13 | ply1basf 19972 | . . . . . 6 ⊢ (𝐺 ∈ 𝐵 → 𝐺:(ℕ0 ↑𝑚 1o)⟶(Base‘𝑅)) |
18 | 17 | ffnd 6294 | . . . . 5 ⊢ (𝐺 ∈ 𝐵 → 𝐺 Fn (ℕ0 ↑𝑚 1o)) |
19 | 18 | 3ad2ant3 1126 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 Fn (ℕ0 ↑𝑚 1o)) |
20 | df1o2 7858 | . . . . . 6 ⊢ 1o = {∅} | |
21 | nn0ex 11653 | . . . . . 6 ⊢ ℕ0 ∈ V | |
22 | 0ex 5028 | . . . . . 6 ⊢ ∅ ∈ V | |
23 | eqid 2778 | . . . . . 6 ⊢ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})) = (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})) | |
24 | 20, 21, 22, 23 | mapsnf1o3 8194 | . . . . 5 ⊢ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})):ℕ0–1-1-onto→(ℕ0 ↑𝑚 1o) |
25 | f1of 6393 | . . . . 5 ⊢ ((𝑎 ∈ ℕ0 ↦ (1o × {𝑎})):ℕ0–1-1-onto→(ℕ0 ↑𝑚 1o) → (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})):ℕ0⟶(ℕ0 ↑𝑚 1o)) | |
26 | 24, 25 | mp1i 13 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})):ℕ0⟶(ℕ0 ↑𝑚 1o)) |
27 | ovexd 6958 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (ℕ0 ↑𝑚 1o) ∈ V) | |
28 | 21 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ℕ0 ∈ V) |
29 | inidm 4043 | . . . 4 ⊢ ((ℕ0 ↑𝑚 1o) ∩ (ℕ0 ↑𝑚 1o)) = (ℕ0 ↑𝑚 1o) | |
30 | 16, 19, 26, 27, 27, 28, 29 | ofco 7196 | . . 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}))))) |
31 | 12, 30 | eqtrd 2814 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐹 ✚ 𝐺) ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎}))) = ((𝐹 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎}))) ∘𝑓 + (𝐺 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎}))))) |
32 | 2 | ply1ring 20018 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ Ring) |
33 | 4, 7 | ringacl 18969 | . . . 4 ⊢ ((𝑌 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ✚ 𝐺) ∈ 𝐵) |
34 | 32, 33 | syl3an1 1163 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ✚ 𝐺) ∈ 𝐵) |
35 | eqid 2778 | . . . 4 ⊢ (coe1‘(𝐹 ✚ 𝐺)) = (coe1‘(𝐹 ✚ 𝐺)) | |
36 | 35, 4, 2, 23 | coe1fval2 19980 | . . 3 ⊢ ((𝐹 ✚ 𝐺) ∈ 𝐵 → (coe1‘(𝐹 ✚ 𝐺)) = ((𝐹 ✚ 𝐺) ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
37 | 34, 36 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘(𝐹 ✚ 𝐺)) = ((𝐹 ✚ 𝐺) ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
38 | eqid 2778 | . . . . 5 ⊢ (coe1‘𝐹) = (coe1‘𝐹) | |
39 | 38, 4, 2, 23 | coe1fval2 19980 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → (coe1‘𝐹) = (𝐹 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
40 | 39 | 3ad2ant2 1125 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘𝐹) = (𝐹 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
41 | eqid 2778 | . . . . 5 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
42 | 41, 4, 2, 23 | coe1fval2 19980 | . . . 4 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺) = (𝐺 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
43 | 42 | 3ad2ant3 1126 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘𝐺) = (𝐺 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
44 | 40, 43 | oveq12d 6942 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((coe1‘𝐹) ∘𝑓 + (coe1‘𝐺)) = ((𝐹 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎}))) ∘𝑓 + (𝐺 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎}))))) |
45 | 31, 37, 44 | 3eqtr4d 2824 | 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 1071 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ∅c0 4141 {csn 4398 ↦ cmpt 4967 × cxp 5355 ∘ ccom 5361 Fn wfn 6132 ⟶wf 6133 –1-1-onto→wf1o 6136 ‘cfv 6137 (class class class)co 6924 ∘𝑓 cof 7174 1oc1o 7838 ↑𝑚 cmap 8142 ℕ0cn0 11646 Basecbs 16259 +gcplusg 16342 Ringcrg 18938 mPoly cmpl 19754 PwSer1cps1 19945 Poly1cpl1 19947 coe1cco1 19948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-ofr 7177 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-fz 12648 df-fzo 12789 df-seq 13124 df-hash 13440 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-sca 16358 df-vsca 16359 df-tset 16361 df-ple 16362 df-0g 16492 df-gsum 16493 df-mre 16636 df-mrc 16637 df-acs 16639 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-mhm 17725 df-submnd 17726 df-grp 17816 df-minusg 17817 df-mulg 17932 df-subg 17979 df-ghm 18046 df-cntz 18137 df-cmn 18585 df-abl 18586 df-mgp 18881 df-ur 18893 df-ring 18940 df-subrg 19174 df-psr 19757 df-mpl 19759 df-opsr 19761 df-psr1 19950 df-ply1 19952 df-coe1 19953 |
This theorem is referenced by: coe1addfv 20035 |
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