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Mirrors > Home > MPE Home > Th. List > coe1addfv | Structured version Visualization version GIF version |
Description: A particular coefficient of an addition. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
Ref | Expression |
---|---|
coe1add.y | ⊢ 𝑌 = (Poly1‘𝑅) |
coe1add.b | ⊢ 𝐵 = (Base‘𝑌) |
coe1add.p | ⊢ ✚ = (+g‘𝑌) |
coe1add.q | ⊢ + = (+g‘𝑅) |
Ref | Expression |
---|---|
coe1addfv | ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑋 ∈ ℕ0) → ((coe1‘(𝐹 ✚ 𝐺))‘𝑋) = (((coe1‘𝐹)‘𝑋) + ((coe1‘𝐺)‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coe1add.y | . . . . 5 ⊢ 𝑌 = (Poly1‘𝑅) | |
2 | coe1add.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑌) | |
3 | coe1add.p | . . . . 5 ⊢ ✚ = (+g‘𝑌) | |
4 | coe1add.q | . . . . 5 ⊢ + = (+g‘𝑅) | |
5 | 1, 2, 3, 4 | coe1add 19849 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘(𝐹 ✚ 𝐺)) = ((coe1‘𝐹) ∘𝑓 + (coe1‘𝐺))) |
6 | 5 | adantr 466 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑋 ∈ ℕ0) → (coe1‘(𝐹 ✚ 𝐺)) = ((coe1‘𝐹) ∘𝑓 + (coe1‘𝐺))) |
7 | 6 | fveq1d 6334 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑋 ∈ ℕ0) → ((coe1‘(𝐹 ✚ 𝐺))‘𝑋) = (((coe1‘𝐹) ∘𝑓 + (coe1‘𝐺))‘𝑋)) |
8 | eqid 2771 | . . . . . . 7 ⊢ (coe1‘𝐹) = (coe1‘𝐹) | |
9 | eqid 2771 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
10 | 8, 2, 1, 9 | coe1f 19796 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → (coe1‘𝐹):ℕ0⟶(Base‘𝑅)) |
11 | ffn 6185 | . . . . . 6 ⊢ ((coe1‘𝐹):ℕ0⟶(Base‘𝑅) → (coe1‘𝐹) Fn ℕ0) | |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → (coe1‘𝐹) Fn ℕ0) |
13 | 12 | 3ad2ant2 1128 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘𝐹) Fn ℕ0) |
14 | 13 | adantr 466 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑋 ∈ ℕ0) → (coe1‘𝐹) Fn ℕ0) |
15 | eqid 2771 | . . . . . . 7 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
16 | 15, 2, 1, 9 | coe1f 19796 | . . . . . 6 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
17 | ffn 6185 | . . . . . 6 ⊢ ((coe1‘𝐺):ℕ0⟶(Base‘𝑅) → (coe1‘𝐺) Fn ℕ0) | |
18 | 16, 17 | syl 17 | . . . . 5 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺) Fn ℕ0) |
19 | 18 | 3ad2ant3 1129 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘𝐺) Fn ℕ0) |
20 | 19 | adantr 466 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑋 ∈ ℕ0) → (coe1‘𝐺) Fn ℕ0) |
21 | nn0ex 11500 | . . . 4 ⊢ ℕ0 ∈ V | |
22 | 21 | a1i 11 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑋 ∈ ℕ0) → ℕ0 ∈ V) |
23 | simpr 471 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0) | |
24 | fnfvof 7058 | . . 3 ⊢ ((((coe1‘𝐹) Fn ℕ0 ∧ (coe1‘𝐺) Fn ℕ0) ∧ (ℕ0 ∈ V ∧ 𝑋 ∈ ℕ0)) → (((coe1‘𝐹) ∘𝑓 + (coe1‘𝐺))‘𝑋) = (((coe1‘𝐹)‘𝑋) + ((coe1‘𝐺)‘𝑋))) | |
25 | 14, 20, 22, 23, 24 | syl22anc 1477 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑋 ∈ ℕ0) → (((coe1‘𝐹) ∘𝑓 + (coe1‘𝐺))‘𝑋) = (((coe1‘𝐹)‘𝑋) + ((coe1‘𝐺)‘𝑋))) |
26 | 7, 25 | eqtrd 2805 | 1 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑋 ∈ ℕ0) → ((coe1‘(𝐹 ✚ 𝐺))‘𝑋) = (((coe1‘𝐹)‘𝑋) + ((coe1‘𝐺)‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 Vcvv 3351 Fn wfn 6026 ⟶wf 6027 ‘cfv 6031 (class class class)co 6793 ∘𝑓 cof 7042 ℕ0cn0 11494 Basecbs 16064 +gcplusg 16149 Ringcrg 18755 Poly1cpl1 19762 coe1cco1 19763 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-of 7044 df-ofr 7045 df-om 7213 df-1st 7315 df-2nd 7316 df-supp 7447 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-2o 7714 df-oadd 7717 df-er 7896 df-map 8011 df-pm 8012 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8432 df-oi 8571 df-card 8965 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-z 11580 df-dec 11696 df-uz 11889 df-fz 12534 df-fzo 12674 df-seq 13009 df-hash 13322 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-sca 16165 df-vsca 16166 df-tset 16168 df-ple 16169 df-0g 16310 df-gsum 16311 df-mre 16454 df-mrc 16455 df-acs 16457 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-mhm 17543 df-submnd 17544 df-grp 17633 df-minusg 17634 df-mulg 17749 df-subg 17799 df-ghm 17866 df-cntz 17957 df-cmn 18402 df-abl 18403 df-mgp 18698 df-ur 18710 df-ring 18757 df-subrg 18988 df-psr 19571 df-mpl 19573 df-opsr 19575 df-psr1 19765 df-ply1 19767 df-coe1 19768 |
This theorem is referenced by: coe1subfv 19851 coe1fzgsumdlem 19886 pm2mpghm 20841 deg1add 24083 hbtlem2 38220 |
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