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Mirrors > Home > MPE Home > Th. List > coe1tmfv2 | Structured version Visualization version GIF version |
Description: Zero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.) |
Ref | Expression |
---|---|
coe1tm.z | ⊢ 0 = (0g‘𝑅) |
coe1tm.k | ⊢ 𝐾 = (Base‘𝑅) |
coe1tm.p | ⊢ 𝑃 = (Poly1‘𝑅) |
coe1tm.x | ⊢ 𝑋 = (var1‘𝑅) |
coe1tm.m | ⊢ · = ( ·𝑠 ‘𝑃) |
coe1tm.n | ⊢ 𝑁 = (mulGrp‘𝑃) |
coe1tm.e | ⊢ ↑ = (.g‘𝑁) |
coe1tmfv2.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
coe1tmfv2.c | ⊢ (𝜑 → 𝐶 ∈ 𝐾) |
coe1tmfv2.d | ⊢ (𝜑 → 𝐷 ∈ ℕ0) |
coe1tmfv2.f | ⊢ (𝜑 → 𝐹 ∈ ℕ0) |
coe1tmfv2.q | ⊢ (𝜑 → 𝐷 ≠ 𝐹) |
Ref | Expression |
---|---|
coe1tmfv2 | ⊢ (𝜑 → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐹) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coe1tmfv2.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | coe1tmfv2.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
3 | coe1tmfv2.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℕ0) | |
4 | coe1tm.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
5 | coe1tm.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
6 | coe1tm.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
7 | coe1tm.x | . . . . 5 ⊢ 𝑋 = (var1‘𝑅) | |
8 | coe1tm.m | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑃) | |
9 | coe1tm.n | . . . . 5 ⊢ 𝑁 = (mulGrp‘𝑃) | |
10 | coe1tm.e | . . . . 5 ⊢ ↑ = (.g‘𝑁) | |
11 | 4, 5, 6, 7, 8, 9, 10 | coe1tm 20369 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 ))) |
12 | 1, 2, 3, 11 | syl3anc 1363 | . . 3 ⊢ (𝜑 → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 ))) |
13 | 12 | fveq1d 6665 | . 2 ⊢ (𝜑 → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐹) = ((𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 ))‘𝐹)) |
14 | eqid 2818 | . . 3 ⊢ (𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 )) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 )) | |
15 | eqeq1 2822 | . . . 4 ⊢ (𝑥 = 𝐹 → (𝑥 = 𝐷 ↔ 𝐹 = 𝐷)) | |
16 | 15 | ifbid 4485 | . . 3 ⊢ (𝑥 = 𝐹 → if(𝑥 = 𝐷, 𝐶, 0 ) = if(𝐹 = 𝐷, 𝐶, 0 )) |
17 | coe1tmfv2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ ℕ0) | |
18 | 5, 4 | ring0cl 19248 | . . . . 5 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾) |
19 | 1, 18 | syl 17 | . . . 4 ⊢ (𝜑 → 0 ∈ 𝐾) |
20 | 2, 19 | ifcld 4508 | . . 3 ⊢ (𝜑 → if(𝐹 = 𝐷, 𝐶, 0 ) ∈ 𝐾) |
21 | 14, 16, 17, 20 | fvmptd3 6783 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 ))‘𝐹) = if(𝐹 = 𝐷, 𝐶, 0 )) |
22 | coe1tmfv2.q | . . . . 5 ⊢ (𝜑 → 𝐷 ≠ 𝐹) | |
23 | 22 | necomd 3068 | . . . 4 ⊢ (𝜑 → 𝐹 ≠ 𝐷) |
24 | 23 | neneqd 3018 | . . 3 ⊢ (𝜑 → ¬ 𝐹 = 𝐷) |
25 | 24 | iffalsed 4474 | . 2 ⊢ (𝜑 → if(𝐹 = 𝐷, 𝐶, 0 ) = 0 ) |
26 | 13, 21, 25 | 3eqtrd 2857 | 1 ⊢ (𝜑 → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐹) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ifcif 4463 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 ℕ0cn0 11885 Basecbs 16471 ·𝑠 cvsca 16557 0gc0g 16701 .gcmg 18162 mulGrpcmgp 19168 Ringcrg 19226 var1cv1 20272 Poly1cpl1 20273 coe1cco1 20274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-ofr 7399 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-tset 16572 df-ple 16573 df-0g 16703 df-gsum 16704 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-ghm 18294 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-subrg 19462 df-lmod 19565 df-lss 19633 df-psr 20064 df-mvr 20065 df-mpl 20066 df-opsr 20068 df-psr1 20276 df-vr1 20277 df-ply1 20278 df-coe1 20279 |
This theorem is referenced by: coe1tmmul2 20372 coe1tmmul 20373 deg1tmle 24638 |
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