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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 2821 | . . 3 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
5 | eqid 2821 | . . 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 20243 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) ∈ (Base‘𝑃)) |
14 | 1, 2, 3, 4, 5, 6, 7, 13 | mplvsca 20226 | . 2 ⊢ (𝜑 → (𝑋 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = ((𝐷 × {𝑋}) ∘f (.r‘𝑅)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )))) |
15 | ovex 7188 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
16 | 6, 15 | rabex2 5236 | . . . 4 ⊢ 𝐷 ∈ V |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
18 | 7 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝑋 ∈ 𝐵) |
19 | 9 | fvexi 6683 | . . . . 5 ⊢ 1 ∈ V |
20 | 8 | fvexi 6683 | . . . . 5 ⊢ 0 ∈ V |
21 | 19, 20 | ifex 4514 | . . . 4 ⊢ if(𝑦 = 𝐾, 1 , 0 ) ∈ V |
22 | 21 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → if(𝑦 = 𝐾, 1 , 0 ) ∈ V) |
23 | fconstmpt 5613 | . . . 4 ⊢ (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋) | |
24 | 23 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋)) |
25 | eqidd 2822 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) | |
26 | 17, 18, 22, 24, 25 | offval2 7425 | . 2 ⊢ (𝜑 → ((𝐷 × {𝑋}) ∘f (.r‘𝑅)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )))) |
27 | oveq2 7163 | . . . . 5 ⊢ ( 1 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(.r‘𝑅) 1 ) = (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) | |
28 | 27 | eqeq1d 2823 | . . . 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 ))) |
29 | oveq2 7163 | . . . . 5 ⊢ ( 0 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(.r‘𝑅) 0 ) = (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) | |
30 | 29 | eqeq1d 2823 | . . . 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 ))) |
31 | 3, 5, 9 | ringridm 19321 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋(.r‘𝑅) 1 ) = 𝑋) |
32 | 11, 7, 31 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝑋(.r‘𝑅) 1 ) = 𝑋) |
33 | iftrue 4472 | . . . . . 6 ⊢ (𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 𝑋) | |
34 | 33 | eqcomd 2827 | . . . . 5 ⊢ (𝑦 = 𝐾 → 𝑋 = if(𝑦 = 𝐾, 𝑋, 0 )) |
35 | 32, 34 | sylan9eq 2876 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐾) → (𝑋(.r‘𝑅) 1 ) = if(𝑦 = 𝐾, 𝑋, 0 )) |
36 | 3, 5, 8 | ringrz 19337 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋(.r‘𝑅) 0 ) = 0 ) |
37 | 11, 7, 36 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝑋(.r‘𝑅) 0 ) = 0 ) |
38 | iffalse 4475 | . . . . . 6 ⊢ (¬ 𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 0 ) | |
39 | 38 | eqcomd 2827 | . . . . 5 ⊢ (¬ 𝑦 = 𝐾 → 0 = if(𝑦 = 𝐾, 𝑋, 0 )) |
40 | 37, 39 | sylan9eq 2876 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑦 = 𝐾) → (𝑋(.r‘𝑅) 0 ) = if(𝑦 = 𝐾, 𝑋, 0 )) |
41 | 28, 30, 35, 40 | ifbothda 4503 | . . 3 ⊢ (𝜑 → (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )) = if(𝑦 = 𝐾, 𝑋, 0 )) |
42 | 41 | mpteq2dv 5161 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 ))) |
43 | 14, 26, 42 | 3eqtrd 2860 | 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 398 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3494 ifcif 4466 {csn 4566 ↦ cmpt 5145 × cxp 5552 ◡ccnv 5553 “ cima 5557 ‘cfv 6354 (class class class)co 7155 ∘f cof 7406 ↑m cmap 8405 Fincfn 8508 ℕcn 11637 ℕ0cn0 11896 Basecbs 16482 .rcmulr 16565 ·𝑠 cvsca 16568 0gc0g 16712 1rcur 19250 Ringcrg 19296 mPoly cmpl 20132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-tset 16583 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-mgp 19239 df-ur 19251 df-ring 19298 df-psr 20135 df-mpl 20137 |
This theorem is referenced by: mplascl 20275 mplmon2cl 20279 mplmon2mul 20280 mplcoe4 20282 coe1tm 20440 |
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