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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 2737 | . . 3 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
| 5 | eqid 2737 | . . 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 22053 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) ∈ (Base‘𝑃)) |
| 14 | 1, 2, 3, 4, 5, 6, 7, 13 | mplvsca 22035 | . 2 ⊢ (𝜑 → (𝑋 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = ((𝐷 × {𝑋}) ∘f (.r‘𝑅)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )))) |
| 15 | ovex 7464 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
| 16 | 6, 15 | rabex2 5341 | . . . 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 4576 | . . . 4 ⊢ if(𝑦 = 𝐾, 1 , 0 ) ∈ V |
| 22 | 21 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → if(𝑦 = 𝐾, 1 , 0 ) ∈ V) |
| 23 | fconstmpt 5747 | . . . 4 ⊢ (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋) | |
| 24 | 23 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋)) |
| 25 | eqidd 2738 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) | |
| 26 | 17, 18, 22, 24, 25 | offval2 7717 | . 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 7439 | . . . . 5 ⊢ ( 1 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(.r‘𝑅) 1 ) = (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) | |
| 28 | 27 | eqeq1d 2739 | . . . 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 7439 | . . . . 5 ⊢ ( 0 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(.r‘𝑅) 0 ) = (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) | |
| 30 | 29 | eqeq1d 2739 | . . . 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 20267 | . . . . . 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 4531 | . . . . . 6 ⊢ (𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 𝑋) | |
| 34 | 33 | eqcomd 2743 | . . . . 5 ⊢ (𝑦 = 𝐾 → 𝑋 = if(𝑦 = 𝐾, 𝑋, 0 )) |
| 35 | 32, 34 | sylan9eq 2797 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐾) → (𝑋(.r‘𝑅) 1 ) = if(𝑦 = 𝐾, 𝑋, 0 )) |
| 36 | 3, 5, 8 | ringrz 20291 | . . . . . 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 4534 | . . . . . 6 ⊢ (¬ 𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 0 ) | |
| 39 | 38 | eqcomd 2743 | . . . . 5 ⊢ (¬ 𝑦 = 𝐾 → 0 = if(𝑦 = 𝐾, 𝑋, 0 )) |
| 40 | 37, 39 | sylan9eq 2797 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑦 = 𝐾) → (𝑋(.r‘𝑅) 0 ) = if(𝑦 = 𝐾, 𝑋, 0 )) |
| 41 | 28, 30, 35, 40 | ifbothda 4564 | . . 3 ⊢ (𝜑 → (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )) = if(𝑦 = 𝐾, 𝑋, 0 )) |
| 42 | 41 | mpteq2dv 5244 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ (𝑋(.r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 ))) |
| 43 | 14, 26, 42 | 3eqtrd 2781 | 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 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ifcif 4525 {csn 4626 ↦ cmpt 5225 × cxp 5683 ◡ccnv 5684 “ cima 5688 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 ↑m cmap 8866 Fincfn 8985 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 .rcmulr 17298 ·𝑠 cvsca 17301 0gc0g 17484 1rcur 20178 Ringcrg 20230 mPoly cmpl 21926 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-tset 17316 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-psr 21929 df-mpl 21931 |
| This theorem is referenced by: mplascl 22088 mplmon2cl 22092 mplmon2mul 22093 mplcoe4 22095 coe1tm 22276 |
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