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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 20043 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 ))) |
12 | 1, 2, 3, 11 | syl3anc 1439 | . . 3 ⊢ (𝜑 → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 ))) |
13 | 12 | fveq1d 6450 | . 2 ⊢ (𝜑 → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐹) = ((𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 ))‘𝐹)) |
14 | eqid 2778 | . . 3 ⊢ (𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 )) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 )) | |
15 | eqeq1 2782 | . . . 4 ⊢ (𝑥 = 𝐹 → (𝑥 = 𝐷 ↔ 𝐹 = 𝐷)) | |
16 | 15 | ifbid 4329 | . . 3 ⊢ (𝑥 = 𝐹 → if(𝑥 = 𝐷, 𝐶, 0 ) = if(𝐹 = 𝐷, 𝐶, 0 )) |
17 | coe1tmfv2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ ℕ0) | |
18 | 5, 4 | ring0cl 18960 | . . . . 5 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾) |
19 | 1, 18 | syl 17 | . . . 4 ⊢ (𝜑 → 0 ∈ 𝐾) |
20 | 2, 19 | ifcld 4352 | . . 3 ⊢ (𝜑 → if(𝐹 = 𝐷, 𝐶, 0 ) ∈ 𝐾) |
21 | 14, 16, 17, 20 | fvmptd3 6566 | . 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 3024 | . . . 4 ⊢ (𝜑 → 𝐹 ≠ 𝐷) |
24 | 23 | neneqd 2974 | . . 3 ⊢ (𝜑 → ¬ 𝐹 = 𝐷) |
25 | 24 | iffalsed 4318 | . 2 ⊢ (𝜑 → if(𝐹 = 𝐷, 𝐶, 0 ) = 0 ) |
26 | 13, 21, 25 | 3eqtrd 2818 | 1 ⊢ (𝜑 → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐹) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ifcif 4307 ↦ cmpt 4967 ‘cfv 6137 (class class class)co 6924 ℕ0cn0 11646 Basecbs 16259 ·𝑠 cvsca 16346 0gc0g 16490 .gcmg 17931 mulGrpcmgp 18880 Ringcrg 18938 var1cv1 19946 Poly1cpl1 19947 coe1cco1 19948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-ofr 7177 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-fz 12648 df-fzo 12789 df-seq 13124 df-hash 13440 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-sca 16358 df-vsca 16359 df-tset 16361 df-ple 16362 df-0g 16492 df-gsum 16493 df-mre 16636 df-mrc 16637 df-acs 16639 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-mhm 17725 df-submnd 17726 df-grp 17816 df-minusg 17817 df-sbg 17818 df-mulg 17932 df-subg 17979 df-ghm 18046 df-cntz 18137 df-cmn 18585 df-abl 18586 df-mgp 18881 df-ur 18893 df-ring 18940 df-subrg 19174 df-lmod 19261 df-lss 19329 df-psr 19757 df-mvr 19758 df-mpl 19759 df-opsr 19761 df-psr1 19950 df-vr1 19951 df-ply1 19952 df-coe1 19953 |
This theorem is referenced by: coe1tmmul2 20046 coe1tmmul 20047 deg1tmle 24318 |
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