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Mirrors > Home > MPE Home > Th. List > coe1tmmul2fv | Structured version Visualization version GIF version |
Description: Function value of a right-multiplication by a term in the shifted domain. (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) |
coe1tmmul.b | ⊢ 𝐵 = (Base‘𝑃) |
coe1tmmul.t | ⊢ ∙ = (.r‘𝑃) |
coe1tmmul.u | ⊢ × = (.r‘𝑅) |
coe1tmmul.a | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
coe1tmmul.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
coe1tmmul.c | ⊢ (𝜑 → 𝐶 ∈ 𝐾) |
coe1tmmul.d | ⊢ (𝜑 → 𝐷 ∈ ℕ0) |
coe1tmmul2fv.y | ⊢ (𝜑 → 𝑌 ∈ ℕ0) |
Ref | Expression |
---|---|
coe1tmmul2fv | ⊢ (𝜑 → ((coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋))))‘(𝐷 + 𝑌)) = (((coe1‘𝐴)‘𝑌) × 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coe1tm.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
2 | coe1tm.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
3 | coe1tm.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | coe1tm.x | . . . 4 ⊢ 𝑋 = (var1‘𝑅) | |
5 | coe1tm.m | . . . 4 ⊢ · = ( ·𝑠 ‘𝑃) | |
6 | coe1tm.n | . . . 4 ⊢ 𝑁 = (mulGrp‘𝑃) | |
7 | coe1tm.e | . . . 4 ⊢ ↑ = (.g‘𝑁) | |
8 | coe1tmmul.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
9 | coe1tmmul.t | . . . 4 ⊢ ∙ = (.r‘𝑃) | |
10 | coe1tmmul.u | . . . 4 ⊢ × = (.r‘𝑅) | |
11 | coe1tmmul.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
12 | coe1tmmul.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
13 | coe1tmmul.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
14 | coe1tmmul.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℕ0) | |
15 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 | coe1tmmul2 20446 | . . 3 ⊢ (𝜑 → (coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 ))) |
16 | 15 | fveq1d 6674 | . 2 ⊢ (𝜑 → ((coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋))))‘(𝐷 + 𝑌)) = ((𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 ))‘(𝐷 + 𝑌))) |
17 | coe1tmmul2fv.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℕ0) | |
18 | 14, 17 | nn0addcld 11962 | . . . 4 ⊢ (𝜑 → (𝐷 + 𝑌) ∈ ℕ0) |
19 | breq2 5072 | . . . . . 6 ⊢ (𝑥 = (𝐷 + 𝑌) → (𝐷 ≤ 𝑥 ↔ 𝐷 ≤ (𝐷 + 𝑌))) | |
20 | fvoveq1 7181 | . . . . . . 7 ⊢ (𝑥 = (𝐷 + 𝑌) → ((coe1‘𝐴)‘(𝑥 − 𝐷)) = ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷))) | |
21 | 20 | oveq1d 7173 | . . . . . 6 ⊢ (𝑥 = (𝐷 + 𝑌) → (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶) = (((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)) × 𝐶)) |
22 | 19, 21 | ifbieq1d 4492 | . . . . 5 ⊢ (𝑥 = (𝐷 + 𝑌) → if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 ) = if(𝐷 ≤ (𝐷 + 𝑌), (((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)) × 𝐶), 0 )) |
23 | eqid 2823 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )) | |
24 | ovex 7191 | . . . . . 6 ⊢ (((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)) × 𝐶) ∈ V | |
25 | 1 | fvexi 6686 | . . . . . 6 ⊢ 0 ∈ V |
26 | 24, 25 | ifex 4517 | . . . . 5 ⊢ if(𝐷 ≤ (𝐷 + 𝑌), (((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)) × 𝐶), 0 ) ∈ V |
27 | 22, 23, 26 | fvmpt 6770 | . . . 4 ⊢ ((𝐷 + 𝑌) ∈ ℕ0 → ((𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 ))‘(𝐷 + 𝑌)) = if(𝐷 ≤ (𝐷 + 𝑌), (((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)) × 𝐶), 0 )) |
28 | 18, 27 | syl 17 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 ))‘(𝐷 + 𝑌)) = if(𝐷 ≤ (𝐷 + 𝑌), (((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)) × 𝐶), 0 )) |
29 | 14 | nn0red 11959 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
30 | nn0addge1 11946 | . . . . 5 ⊢ ((𝐷 ∈ ℝ ∧ 𝑌 ∈ ℕ0) → 𝐷 ≤ (𝐷 + 𝑌)) | |
31 | 29, 17, 30 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝐷 ≤ (𝐷 + 𝑌)) |
32 | 31 | iftrued 4477 | . . 3 ⊢ (𝜑 → if(𝐷 ≤ (𝐷 + 𝑌), (((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)) × 𝐶), 0 ) = (((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)) × 𝐶)) |
33 | 14 | nn0cnd 11960 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
34 | 17 | nn0cnd 11960 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ℂ) |
35 | 33, 34 | pncan2d 11001 | . . . . 5 ⊢ (𝜑 → ((𝐷 + 𝑌) − 𝐷) = 𝑌) |
36 | 35 | fveq2d 6676 | . . . 4 ⊢ (𝜑 → ((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)) = ((coe1‘𝐴)‘𝑌)) |
37 | 36 | oveq1d 7173 | . . 3 ⊢ (𝜑 → (((coe1‘𝐴)‘((𝐷 + 𝑌) − 𝐷)) × 𝐶) = (((coe1‘𝐴)‘𝑌) × 𝐶)) |
38 | 28, 32, 37 | 3eqtrd 2862 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 ))‘(𝐷 + 𝑌)) = (((coe1‘𝐴)‘𝑌) × 𝐶)) |
39 | 16, 38 | eqtrd 2858 | 1 ⊢ (𝜑 → ((coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋))))‘(𝐷 + 𝑌)) = (((coe1‘𝐴)‘𝑌) × 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ifcif 4469 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 + caddc 10542 ≤ cle 10678 − cmin 10872 ℕ0cn0 11900 Basecbs 16485 .rcmulr 16568 ·𝑠 cvsca 16571 0gc0g 16715 .gcmg 18226 mulGrpcmgp 19241 Ringcrg 19299 var1cv1 20346 Poly1cpl1 20347 coe1cco1 20348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-tset 16586 df-ple 16587 df-0g 16717 df-gsum 16718 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-subrg 19535 df-lmod 19638 df-lss 19706 df-psr 20138 df-mvr 20139 df-mpl 20140 df-opsr 20142 df-psr1 20350 df-vr1 20351 df-ply1 20352 df-coe1 20353 |
This theorem is referenced by: ply1divex 24732 |
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