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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 2734 | . . 3 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
5 | eqid 2734 | . . 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 22070 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) ∈ (Base‘𝑃)) |
14 | 1, 2, 3, 4, 5, 6, 7, 13 | mplvsca 22052 | . 2 ⊢ (𝜑 → (𝑋 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = ((𝐷 × {𝑋}) ∘f (.r‘𝑅)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )))) |
15 | ovex 7463 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
16 | 6, 15 | rabex2 5346 | . . . 4 ⊢ 𝐷 ∈ V |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
18 | 7 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝑋 ∈ 𝐵) |
19 | 9 | fvexi 6920 | . . . . 5 ⊢ 1 ∈ V |
20 | 8 | fvexi 6920 | . . . . 5 ⊢ 0 ∈ V |
21 | 19, 20 | ifex 4580 | . . . 4 ⊢ if(𝑦 = 𝐾, 1 , 0 ) ∈ V |
22 | 21 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → if(𝑦 = 𝐾, 1 , 0 ) ∈ V) |
23 | fconstmpt 5750 | . . . 4 ⊢ (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋) | |
24 | 23 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋)) |
25 | eqidd 2735 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) | |
26 | 17, 18, 22, 24, 25 | offval2 7716 | . 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 7438 | . . . . 5 ⊢ ( 1 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(.r‘𝑅) 1 ) = (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) | |
28 | 27 | eqeq1d 2736 | . . . 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 7438 | . . . . 5 ⊢ ( 0 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(.r‘𝑅) 0 ) = (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) | |
30 | 29 | eqeq1d 2736 | . . . 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 20283 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋(.r‘𝑅) 1 ) = 𝑋) |
32 | 11, 7, 31 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑋(.r‘𝑅) 1 ) = 𝑋) |
33 | iftrue 4536 | . . . . . 6 ⊢ (𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 𝑋) | |
34 | 33 | eqcomd 2740 | . . . . 5 ⊢ (𝑦 = 𝐾 → 𝑋 = if(𝑦 = 𝐾, 𝑋, 0 )) |
35 | 32, 34 | sylan9eq 2794 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐾) → (𝑋(.r‘𝑅) 1 ) = if(𝑦 = 𝐾, 𝑋, 0 )) |
36 | 3, 5, 8 | ringrz 20307 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋(.r‘𝑅) 0 ) = 0 ) |
37 | 11, 7, 36 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑋(.r‘𝑅) 0 ) = 0 ) |
38 | iffalse 4539 | . . . . . 6 ⊢ (¬ 𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 0 ) | |
39 | 38 | eqcomd 2740 | . . . . 5 ⊢ (¬ 𝑦 = 𝐾 → 0 = if(𝑦 = 𝐾, 𝑋, 0 )) |
40 | 37, 39 | sylan9eq 2794 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑦 = 𝐾) → (𝑋(.r‘𝑅) 0 ) = if(𝑦 = 𝐾, 𝑋, 0 )) |
41 | 28, 30, 35, 40 | ifbothda 4568 | . . 3 ⊢ (𝜑 → (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )) = if(𝑦 = 𝐾, 𝑋, 0 )) |
42 | 41 | mpteq2dv 5249 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 ))) |
43 | 14, 26, 42 | 3eqtrd 2778 | 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 395 = wceq 1536 ∈ wcel 2105 {crab 3432 Vcvv 3477 ifcif 4530 {csn 4630 ↦ cmpt 5230 × cxp 5686 ◡ccnv 5687 “ cima 5691 ‘cfv 6562 (class class class)co 7430 ∘f cof 7694 ↑m cmap 8864 Fincfn 8983 ℕcn 12263 ℕ0cn0 12523 Basecbs 17244 .rcmulr 17298 ·𝑠 cvsca 17301 0gc0g 17485 1rcur 20198 Ringcrg 20250 mPoly cmpl 21943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-tset 17316 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-psr 21946 df-mpl 21948 |
This theorem is referenced by: mplascl 22105 mplmon2cl 22109 mplmon2mul 22110 mplcoe4 22112 coe1tm 22291 |
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