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Mirrors > Home > MPE Home > Th. List > mplmon2 | Structured version Visualization version GIF version |
Description: Express a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.) |
Ref | Expression |
---|---|
mplmon2.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplmon2.v | ⊢ · = ( ·𝑠 ‘𝑃) |
mplmon2.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
mplmon2.o | ⊢ 1 = (1r‘𝑅) |
mplmon2.z | ⊢ 0 = (0g‘𝑅) |
mplmon2.b | ⊢ 𝐵 = (Base‘𝑅) |
mplmon2.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
mplmon2.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mplmon2.k | ⊢ (𝜑 → 𝐾 ∈ 𝐷) |
mplmon2.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
mplmon2 | ⊢ (𝜑 → (𝑋 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mplmon2.p | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
2 | mplmon2.v | . . 3 ⊢ · = ( ·𝑠 ‘𝑃) | |
3 | mplmon2.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
4 | eqid 2622 | . . 3 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
5 | eqid 2622 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
6 | mplmon2.d | . . 3 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
7 | mplmon2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | mplmon2.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
9 | mplmon2.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
10 | mplmon2.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
11 | mplmon2.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
12 | mplmon2.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝐷) | |
13 | 1, 4, 8, 9, 6, 10, 11, 12 | mplmon 19463 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) ∈ (Base‘𝑃)) |
14 | 1, 2, 3, 4, 5, 6, 7, 13 | mplvsca 19447 | . 2 ⊢ (𝜑 → (𝑋 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = ((𝐷 × {𝑋}) ∘𝑓 (.r‘𝑅)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )))) |
15 | ovex 6678 | . . . . 5 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V | |
16 | 6, 15 | rabex2 4815 | . . . 4 ⊢ 𝐷 ∈ V |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
18 | 7 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝑋 ∈ 𝐵) |
19 | fvex 6201 | . . . . . 6 ⊢ (1r‘𝑅) ∈ V | |
20 | 9, 19 | eqeltri 2697 | . . . . 5 ⊢ 1 ∈ V |
21 | fvex 6201 | . . . . . 6 ⊢ (0g‘𝑅) ∈ V | |
22 | 8, 21 | eqeltri 2697 | . . . . 5 ⊢ 0 ∈ V |
23 | 20, 22 | ifex 4156 | . . . 4 ⊢ if(𝑦 = 𝐾, 1 , 0 ) ∈ V |
24 | 23 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → if(𝑦 = 𝐾, 1 , 0 ) ∈ V) |
25 | fconstmpt 5163 | . . . 4 ⊢ (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋) | |
26 | 25 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋)) |
27 | eqidd 2623 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) | |
28 | 17, 18, 24, 26, 27 | offval2 6914 | . 2 ⊢ (𝜑 → ((𝐷 × {𝑋}) ∘𝑓 (.r‘𝑅)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )))) |
29 | oveq2 6658 | . . . . 5 ⊢ ( 1 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(.r‘𝑅) 1 ) = (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) | |
30 | 29 | eqeq1d 2624 | . . . 4 ⊢ ( 1 = if(𝑦 = 𝐾, 1 , 0 ) → ((𝑋(.r‘𝑅) 1 ) = if(𝑦 = 𝐾, 𝑋, 0 ) ↔ (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )) = if(𝑦 = 𝐾, 𝑋, 0 ))) |
31 | oveq2 6658 | . . . . 5 ⊢ ( 0 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(.r‘𝑅) 0 ) = (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) | |
32 | 31 | eqeq1d 2624 | . . . 4 ⊢ ( 0 = if(𝑦 = 𝐾, 1 , 0 ) → ((𝑋(.r‘𝑅) 0 ) = if(𝑦 = 𝐾, 𝑋, 0 ) ↔ (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )) = if(𝑦 = 𝐾, 𝑋, 0 ))) |
33 | 3, 5, 9 | ringridm 18572 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋(.r‘𝑅) 1 ) = 𝑋) |
34 | 11, 7, 33 | syl2anc 693 | . . . . 5 ⊢ (𝜑 → (𝑋(.r‘𝑅) 1 ) = 𝑋) |
35 | iftrue 4092 | . . . . . 6 ⊢ (𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 𝑋) | |
36 | 35 | eqcomd 2628 | . . . . 5 ⊢ (𝑦 = 𝐾 → 𝑋 = if(𝑦 = 𝐾, 𝑋, 0 )) |
37 | 34, 36 | sylan9eq 2676 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐾) → (𝑋(.r‘𝑅) 1 ) = if(𝑦 = 𝐾, 𝑋, 0 )) |
38 | 3, 5, 8 | ringrz 18588 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋(.r‘𝑅) 0 ) = 0 ) |
39 | 11, 7, 38 | syl2anc 693 | . . . . 5 ⊢ (𝜑 → (𝑋(.r‘𝑅) 0 ) = 0 ) |
40 | iffalse 4095 | . . . . . 6 ⊢ (¬ 𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 0 ) | |
41 | 40 | eqcomd 2628 | . . . . 5 ⊢ (¬ 𝑦 = 𝐾 → 0 = if(𝑦 = 𝐾, 𝑋, 0 )) |
42 | 39, 41 | sylan9eq 2676 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑦 = 𝐾) → (𝑋(.r‘𝑅) 0 ) = if(𝑦 = 𝐾, 𝑋, 0 )) |
43 | 30, 32, 37, 42 | ifbothda 4123 | . . 3 ⊢ (𝜑 → (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )) = if(𝑦 = 𝐾, 𝑋, 0 )) |
44 | 43 | mpteq2dv 4745 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 ))) |
45 | 14, 28, 44 | 3eqtrd 2660 | 1 ⊢ (𝜑 → (𝑋 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 ifcif 4086 {csn 4177 ↦ cmpt 4729 × cxp 5112 ◡ccnv 5113 “ cima 5117 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 ↑𝑚 cmap 7857 Fincfn 7955 ℕcn 11020 ℕ0cn0 11292 Basecbs 15857 .rcmulr 15942 ·𝑠 cvsca 15945 0gc0g 16100 1rcur 18501 Ringcrg 18547 mPoly cmpl 19353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-tset 15960 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-mgp 18490 df-ur 18502 df-ring 18549 df-psr 19356 df-mpl 19358 |
This theorem is referenced by: mplascl 19496 mplmon2cl 19500 mplmon2mul 19501 mplcoe4 19503 coe1tm 19643 |
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