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Mirrors > Home > MPE Home > Th. List > coe1z | Structured version Visualization version GIF version |
Description: The coefficient vector of 0. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
Ref | Expression |
---|---|
coe1z.p | ⊢ 𝑃 = (Poly1‘𝑅) |
coe1z.z | ⊢ 0 = (0g‘𝑃) |
coe1z.y | ⊢ 𝑌 = (0g‘𝑅) |
Ref | Expression |
---|---|
coe1z | ⊢ (𝑅 ∈ Ring → (coe1‘ 0 ) = (ℕ0 × {𝑌})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconst6g 6767 | . . . . 5 ⊢ (𝑎 ∈ ℕ0 → (1o × {𝑎}):1o⟶ℕ0) | |
2 | 1 | adantl 482 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ ℕ0) → (1o × {𝑎}):1o⟶ℕ0) |
3 | nn0ex 12460 | . . . . 5 ⊢ ℕ0 ∈ V | |
4 | 1oex 8458 | . . . . 5 ⊢ 1o ∈ V | |
5 | 3, 4 | elmap 8848 | . . . 4 ⊢ ((1o × {𝑎}) ∈ (ℕ0 ↑m 1o) ↔ (1o × {𝑎}):1o⟶ℕ0) |
6 | 2, 5 | sylibr 233 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ ℕ0) → (1o × {𝑎}) ∈ (ℕ0 ↑m 1o)) |
7 | eqidd 2732 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})) = (𝑎 ∈ ℕ0 ↦ (1o × {𝑎}))) | |
8 | eqid 2731 | . . . . 5 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
9 | psr1baslem 21638 | . . . . 5 ⊢ (ℕ0 ↑m 1o) = {𝑐 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑐 “ ℕ) ∈ Fin} | |
10 | coe1z.y | . . . . 5 ⊢ 𝑌 = (0g‘𝑅) | |
11 | coe1z.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
12 | coe1z.z | . . . . . 6 ⊢ 0 = (0g‘𝑃) | |
13 | 8, 11, 12 | ply1mpl0 21708 | . . . . 5 ⊢ 0 = (0g‘(1o mPoly 𝑅)) |
14 | 1on 8460 | . . . . . 6 ⊢ 1o ∈ On | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝑅 ∈ Ring → 1o ∈ On) |
16 | ringgrp 20019 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
17 | 8, 9, 10, 13, 15, 16 | mpl0 21494 | . . . 4 ⊢ (𝑅 ∈ Ring → 0 = ((ℕ0 ↑m 1o) × {𝑌})) |
18 | fconstmpt 5730 | . . . 4 ⊢ ((ℕ0 ↑m 1o) × {𝑌}) = (𝑏 ∈ (ℕ0 ↑m 1o) ↦ 𝑌) | |
19 | 17, 18 | eqtrdi 2787 | . . 3 ⊢ (𝑅 ∈ Ring → 0 = (𝑏 ∈ (ℕ0 ↑m 1o) ↦ 𝑌)) |
20 | eqidd 2732 | . . 3 ⊢ (𝑏 = (1o × {𝑎}) → 𝑌 = 𝑌) | |
21 | 6, 7, 19, 20 | fmptco 7111 | . 2 ⊢ (𝑅 ∈ Ring → ( 0 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎}))) = (𝑎 ∈ ℕ0 ↦ 𝑌)) |
22 | 11 | ply1ring 21701 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
23 | eqid 2731 | . . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
24 | 23, 12 | ring0cl 20041 | . . 3 ⊢ (𝑃 ∈ Ring → 0 ∈ (Base‘𝑃)) |
25 | eqid 2731 | . . . 4 ⊢ (coe1‘ 0 ) = (coe1‘ 0 ) | |
26 | eqid 2731 | . . . 4 ⊢ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})) = (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})) | |
27 | 25, 23, 11, 26 | coe1fval2 21663 | . . 3 ⊢ ( 0 ∈ (Base‘𝑃) → (coe1‘ 0 ) = ( 0 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
28 | 22, 24, 27 | 3syl 18 | . 2 ⊢ (𝑅 ∈ Ring → (coe1‘ 0 ) = ( 0 ∘ (𝑎 ∈ ℕ0 ↦ (1o × {𝑎})))) |
29 | fconstmpt 5730 | . . 3 ⊢ (ℕ0 × {𝑌}) = (𝑎 ∈ ℕ0 ↦ 𝑌) | |
30 | 29 | a1i 11 | . 2 ⊢ (𝑅 ∈ Ring → (ℕ0 × {𝑌}) = (𝑎 ∈ ℕ0 ↦ 𝑌)) |
31 | 21, 28, 30 | 3eqtr4d 2781 | 1 ⊢ (𝑅 ∈ Ring → (coe1‘ 0 ) = (ℕ0 × {𝑌})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {csn 4622 ↦ cmpt 5224 × cxp 5667 ∘ ccom 5673 Oncon0 6353 ⟶wf 6528 ‘cfv 6532 (class class class)co 7393 1oc1o 8441 ↑m cmap 8803 ℕ0cn0 12454 Basecbs 17126 0gc0g 17367 Ringcrg 20014 mPoly cmpl 21390 Poly1cpl1 21630 coe1cco1 21631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-of 7653 df-ofr 7654 df-om 7839 df-1st 7957 df-2nd 7958 df-supp 8129 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-pm 8806 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fsupp 9345 df-sup 9419 df-oi 9487 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-fz 13467 df-fzo 13610 df-seq 13949 df-hash 14273 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-hom 17203 df-cco 17204 df-0g 17369 df-gsum 17370 df-prds 17375 df-pws 17377 df-mre 17512 df-mrc 17513 df-acs 17515 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-mhm 18647 df-submnd 18648 df-grp 18797 df-minusg 18798 df-mulg 18923 df-subg 18975 df-ghm 19056 df-cntz 19147 df-cmn 19614 df-abl 19615 df-mgp 19947 df-ur 19964 df-ring 20016 df-subrg 20310 df-psr 21393 df-mpl 21395 df-opsr 21397 df-psr1 21633 df-ply1 21635 df-coe1 21636 |
This theorem is referenced by: coe1fzgsumd 21755 decpmatid 22201 pmatcollpwscmatlem1 22220 fta1blem 25615 ply1gsumz 32507 hbtlem2 41635 |
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