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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 2823 | . . . . 5 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
2 | coe1add.y | . . . . . 6 ⊢ 𝑌 = (Poly1‘𝑅) | |
3 | eqid 2823 | . . . . . 6 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
4 | coe1add.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑌) | |
5 | 2, 3, 4 | ply1bas 20365 | . . . . 5 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅)) |
6 | coe1add.q | . . . . 5 ⊢ + = (+g‘𝑅) | |
7 | coe1add.p | . . . . . 6 ⊢ ✚ = (+g‘𝑌) | |
8 | 2, 1, 7 | ply1plusg 20395 | . . . . 5 ⊢ ✚ = (+g‘(1o mPoly 𝑅)) |
9 | simp2 1133 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵) | |
10 | simp3 1134 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵) | |
11 | 1, 5, 6, 8, 9, 10 | mpladd 20224 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ✚ 𝐺) = (𝐹 ∘f + 𝐺)) |
12 | 11 | coeq1d 5734 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐹 ✚ 𝐺) ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎}))) = ((𝐹 ∘f + 𝐺) ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
13 | eqid 2823 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
14 | 2, 4, 13 | ply1basf 20372 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑m 1o)⟶(Base‘𝑅)) |
15 | 14 | ffnd 6517 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → 𝐹 Fn (ℕ0 ↑m 1o)) |
16 | 15 | 3ad2ant2 1130 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 Fn (ℕ0 ↑m 1o)) |
17 | 2, 4, 13 | ply1basf 20372 | . . . . . 6 ⊢ (𝐺 ∈ 𝐵 → 𝐺:(ℕ0 ↑m 1o)⟶(Base‘𝑅)) |
18 | 17 | ffnd 6517 | . . . . 5 ⊢ (𝐺 ∈ 𝐵 → 𝐺 Fn (ℕ0 ↑m 1o)) |
19 | 18 | 3ad2ant3 1131 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 Fn (ℕ0 ↑m 1o)) |
20 | df1o2 8118 | . . . . . 6 ⊢ 1o = {∅} | |
21 | nn0ex 11906 | . . . . . 6 ⊢ ℕ0 ∈ V | |
22 | 0ex 5213 | . . . . . 6 ⊢ ∅ ∈ V | |
23 | eqid 2823 | . . . . . 6 ⊢ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})) = (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})) | |
24 | 20, 21, 22, 23 | mapsnf1o3 8461 | . . . . 5 ⊢ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})):ℕ0–1-1-onto→(ℕ0 ↑m 1o) |
25 | f1of 6617 | . . . . 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 7193 | . . . 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 4197 | . . . 4 ⊢ ((ℕ0 ↑m 1o) ∩ (ℕ0 ↑m 1o)) = (ℕ0 ↑m 1o) | |
30 | 16, 19, 26, 27, 27, 28, 29 | ofco 7431 | . . 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 2858 | . 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 20418 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ Ring) |
33 | 4, 7 | ringacl 19330 | . . . 4 ⊢ ((𝑌 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ✚ 𝐺) ∈ 𝐵) |
34 | 32, 33 | syl3an1 1159 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ✚ 𝐺) ∈ 𝐵) |
35 | eqid 2823 | . . . 4 ⊢ (coe1‘(𝐹 ✚ 𝐺)) = (coe1‘(𝐹 ✚ 𝐺)) | |
36 | 35, 4, 2, 23 | coe1fval2 20380 | . . 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 2823 | . . . . 5 ⊢ (coe1‘𝐹) = (coe1‘𝐹) | |
39 | 38, 4, 2, 23 | coe1fval2 20380 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → (coe1‘𝐹) = (𝐹 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
40 | 39 | 3ad2ant2 1130 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘𝐹) = (𝐹 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
41 | eqid 2823 | . . . . 5 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
42 | 41, 4, 2, 23 | coe1fval2 20380 | . . . 4 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺) = (𝐺 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
43 | 42 | 3ad2ant3 1131 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘𝐺) = (𝐺 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
44 | 40, 43 | oveq12d 7176 | . 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 2868 | 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 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 {csn 4569 ↦ cmpt 5148 × cxp 5555 ∘ ccom 5561 Fn wfn 6352 ⟶wf 6353 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 ∘f cof 7409 1oc1o 8097 ↑m cmap 8408 ℕ0cn0 11900 Basecbs 16485 +gcplusg 16567 Ringcrg 19299 mPoly cmpl 20135 PwSer1cps1 20345 Poly1cpl1 20347 coe1cco1 20348 |
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-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-tset 16586 df-ple 16587 df-0g 16717 df-gsum 16718 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-mulg 18227 df-subg 18278 df-ghm 18358 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-subrg 19535 df-psr 20138 df-mpl 20140 df-opsr 20142 df-psr1 20350 df-ply1 20352 df-coe1 20353 |
This theorem is referenced by: coe1addfv 20435 |
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