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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 2798 | . . 3 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
5 | eqid 2798 | . . 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 20703 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) ∈ (Base‘𝑃)) |
14 | 1, 2, 3, 4, 5, 6, 7, 13 | mplvsca 20686 | . 2 ⊢ (𝜑 → (𝑋 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = ((𝐷 × {𝑋}) ∘f (.r‘𝑅)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )))) |
15 | ovex 7168 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
16 | 6, 15 | rabex2 5201 | . . . 4 ⊢ 𝐷 ∈ V |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
18 | 7 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝑋 ∈ 𝐵) |
19 | 9 | fvexi 6659 | . . . . 5 ⊢ 1 ∈ V |
20 | 8 | fvexi 6659 | . . . . 5 ⊢ 0 ∈ V |
21 | 19, 20 | ifex 4473 | . . . 4 ⊢ if(𝑦 = 𝐾, 1 , 0 ) ∈ V |
22 | 21 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → if(𝑦 = 𝐾, 1 , 0 ) ∈ V) |
23 | fconstmpt 5578 | . . . 4 ⊢ (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋) | |
24 | 23 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋)) |
25 | eqidd 2799 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) | |
26 | 17, 18, 22, 24, 25 | offval2 7406 | . 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 7143 | . . . . 5 ⊢ ( 1 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(.r‘𝑅) 1 ) = (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) | |
28 | 27 | eqeq1d 2800 | . . . 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 7143 | . . . . 5 ⊢ ( 0 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(.r‘𝑅) 0 ) = (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) | |
30 | 29 | eqeq1d 2800 | . . . 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 19318 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋(.r‘𝑅) 1 ) = 𝑋) |
32 | 11, 7, 31 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → (𝑋(.r‘𝑅) 1 ) = 𝑋) |
33 | iftrue 4431 | . . . . . 6 ⊢ (𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 𝑋) | |
34 | 33 | eqcomd 2804 | . . . . 5 ⊢ (𝑦 = 𝐾 → 𝑋 = if(𝑦 = 𝐾, 𝑋, 0 )) |
35 | 32, 34 | sylan9eq 2853 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐾) → (𝑋(.r‘𝑅) 1 ) = if(𝑦 = 𝐾, 𝑋, 0 )) |
36 | 3, 5, 8 | ringrz 19334 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋(.r‘𝑅) 0 ) = 0 ) |
37 | 11, 7, 36 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → (𝑋(.r‘𝑅) 0 ) = 0 ) |
38 | iffalse 4434 | . . . . . 6 ⊢ (¬ 𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 0 ) | |
39 | 38 | eqcomd 2804 | . . . . 5 ⊢ (¬ 𝑦 = 𝐾 → 0 = if(𝑦 = 𝐾, 𝑋, 0 )) |
40 | 37, 39 | sylan9eq 2853 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑦 = 𝐾) → (𝑋(.r‘𝑅) 0 ) = if(𝑦 = 𝐾, 𝑋, 0 )) |
41 | 28, 30, 35, 40 | ifbothda 4462 | . . 3 ⊢ (𝜑 → (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )) = if(𝑦 = 𝐾, 𝑋, 0 )) |
42 | 41 | mpteq2dv 5126 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 ))) |
43 | 14, 26, 42 | 3eqtrd 2837 | 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 399 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ifcif 4425 {csn 4525 ↦ cmpt 5110 × cxp 5517 ◡ccnv 5518 “ cima 5522 ‘cfv 6324 (class class class)co 7135 ∘f cof 7387 ↑m cmap 8389 Fincfn 8492 ℕcn 11625 ℕ0cn0 11885 Basecbs 16475 .rcmulr 16558 ·𝑠 cvsca 16561 0gc0g 16705 1rcur 19244 Ringcrg 19290 mPoly cmpl 20591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 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-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-tset 16576 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-mgp 19233 df-ur 19245 df-ring 19292 df-psr 20594 df-mpl 20596 |
This theorem is referenced by: mplascl 20735 mplmon2cl 20739 mplmon2mul 20740 mplcoe4 20742 coe1tm 20902 |
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