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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 2737 | . . . . 5 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
| 2 | coe1add.y | . . . . . 6 ⊢ 𝑌 = (Poly1‘𝑅) | |
| 3 | coe1add.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑌) | |
| 4 | 2, 3 | ply1bas 22196 | . . . . 5 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅)) |
| 5 | coe1add.q | . . . . 5 ⊢ + = (+g‘𝑅) | |
| 6 | coe1add.p | . . . . . 6 ⊢ ✚ = (+g‘𝑌) | |
| 7 | 2, 1, 6 | ply1plusg 22225 | . . . . 5 ⊢ ✚ = (+g‘(1o mPoly 𝑅)) |
| 8 | simp2 1138 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵) | |
| 9 | simp3 1139 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵) | |
| 10 | 1, 4, 5, 7, 8, 9 | mpladd 22029 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ✚ 𝐺) = (𝐹 ∘f + 𝐺)) |
| 11 | 10 | coeq1d 5872 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐹 ✚ 𝐺) ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎}))) = ((𝐹 ∘f + 𝐺) ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
| 12 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 13 | 2, 3, 12 | ply1basf 22204 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑m 1o)⟶(Base‘𝑅)) |
| 14 | 13 | ffnd 6737 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → 𝐹 Fn (ℕ0 ↑m 1o)) |
| 15 | 14 | 3ad2ant2 1135 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 Fn (ℕ0 ↑m 1o)) |
| 16 | 2, 3, 12 | ply1basf 22204 | . . . . . 6 ⊢ (𝐺 ∈ 𝐵 → 𝐺:(ℕ0 ↑m 1o)⟶(Base‘𝑅)) |
| 17 | 16 | ffnd 6737 | . . . . 5 ⊢ (𝐺 ∈ 𝐵 → 𝐺 Fn (ℕ0 ↑m 1o)) |
| 18 | 17 | 3ad2ant3 1136 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 Fn (ℕ0 ↑m 1o)) |
| 19 | df1o2 8513 | . . . . . 6 ⊢ 1o = {∅} | |
| 20 | nn0ex 12532 | . . . . . 6 ⊢ ℕ0 ∈ V | |
| 21 | 0ex 5307 | . . . . . 6 ⊢ ∅ ∈ V | |
| 22 | eqid 2737 | . . . . . 6 ⊢ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})) = (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})) | |
| 23 | 19, 20, 21, 22 | mapsnf1o3 8935 | . . . . 5 ⊢ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})):ℕ0–1-1-onto→(ℕ0 ↑m 1o) |
| 24 | f1of 6848 | . . . . 5 ⊢ ((𝑎 ∈ ℕ0 ↦ (1o × {𝑎})):ℕ0–1-1-onto→(ℕ0 ↑m 1o) → (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})):ℕ0⟶(ℕ0 ↑m 1o)) | |
| 25 | 23, 24 | mp1i 13 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})):ℕ0⟶(ℕ0 ↑m 1o)) |
| 26 | ovexd 7466 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (ℕ0 ↑m 1o) ∈ V) | |
| 27 | 20 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ℕ0 ∈ V) |
| 28 | inidm 4227 | . . . 4 ⊢ ((ℕ0 ↑m 1o) ∩ (ℕ0 ↑m 1o)) = (ℕ0 ↑m 1o) | |
| 29 | 15, 18, 25, 26, 26, 27, 28 | ofco 7722 | . . 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}))))) |
| 30 | 11, 29 | eqtrd 2777 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐹 ✚ 𝐺) ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎}))) = ((𝐹 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎}))) ∘f + (𝐺 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎}))))) |
| 31 | 2 | ply1ring 22249 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ Ring) |
| 32 | 3, 6 | ringacl 20275 | . . . 4 ⊢ ((𝑌 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ✚ 𝐺) ∈ 𝐵) |
| 33 | 31, 32 | syl3an1 1164 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ✚ 𝐺) ∈ 𝐵) |
| 34 | eqid 2737 | . . . 4 ⊢ (coe1‘(𝐹 ✚ 𝐺)) = (coe1‘(𝐹 ✚ 𝐺)) | |
| 35 | 34, 3, 2, 22 | coe1fval2 22212 | . . 3 ⊢ ((𝐹 ✚ 𝐺) ∈ 𝐵 → (coe1‘(𝐹 ✚ 𝐺)) = ((𝐹 ✚ 𝐺) ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
| 36 | 33, 35 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘(𝐹 ✚ 𝐺)) = ((𝐹 ✚ 𝐺) ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
| 37 | eqid 2737 | . . . . 5 ⊢ (coe1‘𝐹) = (coe1‘𝐹) | |
| 38 | 37, 3, 2, 22 | coe1fval2 22212 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → (coe1‘𝐹) = (𝐹 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
| 39 | 38 | 3ad2ant2 1135 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘𝐹) = (𝐹 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
| 40 | eqid 2737 | . . . . 5 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
| 41 | 40, 3, 2, 22 | coe1fval2 22212 | . . . 4 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺) = (𝐺 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
| 42 | 41 | 3ad2ant3 1136 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘𝐺) = (𝐺 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
| 43 | 39, 42 | oveq12d 7449 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((coe1‘𝐹) ∘f + (coe1‘𝐺)) = ((𝐹 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎}))) ∘f + (𝐺 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎}))))) |
| 44 | 30, 36, 43 | 3eqtr4d 2787 | 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 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 {csn 4626 ↦ cmpt 5225 × cxp 5683 ∘ ccom 5689 Fn wfn 6556 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 1oc1o 8499 ↑m cmap 8866 ℕ0cn0 12526 Basecbs 17247 +gcplusg 17297 Ringcrg 20230 mPoly cmpl 21926 Poly1cpl1 22178 coe1cco1 22179 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-subrng 20546 df-subrg 20570 df-psr 21929 df-mpl 21931 df-opsr 21933 df-psr1 22181 df-ply1 22183 df-coe1 22184 |
| This theorem is referenced by: coe1addfv 22268 |
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