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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 21209 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘(𝐹 ✚ 𝐺)) = ((coe1‘𝐹) ∘f + (coe1‘𝐺))) |
6 | 5 | adantr 484 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑋 ∈ ℕ0) → (coe1‘(𝐹 ✚ 𝐺)) = ((coe1‘𝐹) ∘f + (coe1‘𝐺))) |
7 | 6 | fveq1d 6737 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑋 ∈ ℕ0) → ((coe1‘(𝐹 ✚ 𝐺))‘𝑋) = (((coe1‘𝐹) ∘f + (coe1‘𝐺))‘𝑋)) |
8 | eqid 2738 | . . . . . . 7 ⊢ (coe1‘𝐹) = (coe1‘𝐹) | |
9 | eqid 2738 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
10 | 8, 2, 1, 9 | coe1f 21156 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → (coe1‘𝐹):ℕ0⟶(Base‘𝑅)) |
11 | 10 | ffnd 6564 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → (coe1‘𝐹) Fn ℕ0) |
12 | 11 | 3ad2ant2 1136 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘𝐹) Fn ℕ0) |
13 | 12 | adantr 484 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑋 ∈ ℕ0) → (coe1‘𝐹) Fn ℕ0) |
14 | eqid 2738 | . . . . . . 7 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
15 | 14, 2, 1, 9 | coe1f 21156 | . . . . . 6 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
16 | 15 | ffnd 6564 | . . . . 5 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺) Fn ℕ0) |
17 | 16 | 3ad2ant3 1137 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘𝐺) Fn ℕ0) |
18 | 17 | adantr 484 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑋 ∈ ℕ0) → (coe1‘𝐺) Fn ℕ0) |
19 | nn0ex 12120 | . . . 4 ⊢ ℕ0 ∈ V | |
20 | 19 | a1i 11 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑋 ∈ ℕ0) → ℕ0 ∈ V) |
21 | simpr 488 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0) | |
22 | fnfvof 7503 | . . 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))) | |
23 | 13, 18, 20, 21, 22 | syl22anc 839 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑋 ∈ ℕ0) → (((coe1‘𝐹) ∘f + (coe1‘𝐺))‘𝑋) = (((coe1‘𝐹)‘𝑋) + ((coe1‘𝐺)‘𝑋))) |
24 | 7, 23 | eqtrd 2778 | 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 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 Vcvv 3420 Fn wfn 6392 ‘cfv 6397 (class class class)co 7231 ∘f cof 7485 ℕ0cn0 12114 Basecbs 16784 +gcplusg 16826 Ringcrg 19586 Poly1cpl1 21122 coe1cco1 21123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-int 4874 df-iun 4920 df-iin 4921 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-se 5524 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-isom 6406 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-of 7487 df-ofr 7488 df-om 7663 df-1st 7779 df-2nd 7780 df-supp 7924 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-1o 8222 df-er 8411 df-map 8530 df-pm 8531 df-ixp 8599 df-en 8647 df-dom 8648 df-sdom 8649 df-fin 8650 df-fsupp 9010 df-oi 9150 df-card 9579 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-nn 11855 df-2 11917 df-3 11918 df-4 11919 df-5 11920 df-6 11921 df-7 11922 df-8 11923 df-9 11924 df-n0 12115 df-z 12201 df-dec 12318 df-uz 12463 df-fz 13120 df-fzo 13263 df-seq 13599 df-hash 13921 df-struct 16724 df-sets 16741 df-slot 16759 df-ndx 16769 df-base 16785 df-ress 16809 df-plusg 16839 df-mulr 16840 df-sca 16842 df-vsca 16843 df-tset 16845 df-ple 16846 df-0g 16970 df-gsum 16971 df-mre 17113 df-mrc 17114 df-acs 17116 df-mgm 18138 df-sgrp 18187 df-mnd 18198 df-mhm 18242 df-submnd 18243 df-grp 18392 df-minusg 18393 df-mulg 18513 df-subg 18564 df-ghm 18644 df-cntz 18735 df-cmn 19196 df-abl 19197 df-mgp 19529 df-ur 19541 df-ring 19588 df-subrg 19822 df-psr 20892 df-mpl 20894 df-opsr 20896 df-psr1 21125 df-ply1 21127 df-coe1 21128 |
This theorem is referenced by: coe1subfv 21211 coe1fzgsumdlem 21246 pm2mpghm 21737 deg1add 25025 hbtlem2 40680 |
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