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Mirrors > Home > MPE Home > Th. List > mplmon | Structured version Visualization version GIF version |
Description: A monomial is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.) |
Ref | Expression |
---|---|
mplmon.s | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplmon.b | ⊢ 𝐵 = (Base‘𝑃) |
mplmon.z | ⊢ 0 = (0g‘𝑅) |
mplmon.o | ⊢ 1 = (1r‘𝑅) |
mplmon.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
mplmon.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
mplmon.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mplmon.x | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
Ref | Expression |
---|---|
mplmon | ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mplmon.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | eqid 2736 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | mplmon.o | . . . . . . . . 9 ⊢ 1 = (1r‘𝑅) | |
4 | 2, 3 | ringidcl 19902 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
5 | mplmon.z | . . . . . . . . 9 ⊢ 0 = (0g‘𝑅) | |
6 | 2, 5 | ring0cl 19903 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅)) |
7 | 4, 6 | ifcld 4519 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → if(𝑦 = 𝑋, 1 , 0 ) ∈ (Base‘𝑅)) |
8 | 1, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → if(𝑦 = 𝑋, 1 , 0 ) ∈ (Base‘𝑅)) |
9 | 8 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → if(𝑦 = 𝑋, 1 , 0 ) ∈ (Base‘𝑅)) |
10 | 9 | fmpttd 7045 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
11 | fvex 6838 | . . . . 5 ⊢ (Base‘𝑅) ∈ V | |
12 | mplmon.d | . . . . . 6 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
13 | ovex 7370 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
14 | 12, 13 | rabex2 5278 | . . . . 5 ⊢ 𝐷 ∈ V |
15 | 11, 14 | elmap 8730 | . . . 4 ⊢ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ ((Base‘𝑅) ↑m 𝐷) ↔ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
16 | 10, 15 | sylibr 233 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ ((Base‘𝑅) ↑m 𝐷)) |
17 | eqid 2736 | . . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
18 | eqid 2736 | . . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
19 | mplmon.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
20 | 17, 2, 12, 18, 19 | psrbas 21253 | . . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑m 𝐷)) |
21 | 16, 20 | eleqtrrd 2840 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅))) |
22 | 14 | mptex 7155 | . . . . 5 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ V |
23 | funmpt 6522 | . . . . 5 ⊢ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) | |
24 | 5 | fvexi 6839 | . . . . 5 ⊢ 0 ∈ V |
25 | 22, 23, 24 | 3pm3.2i 1338 | . . . 4 ⊢ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ V ∧ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∧ 0 ∈ V) |
26 | 25 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ V ∧ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∧ 0 ∈ V)) |
27 | snfi 8909 | . . . 4 ⊢ {𝑋} ∈ Fin | |
28 | 27 | a1i 11 | . . 3 ⊢ (𝜑 → {𝑋} ∈ Fin) |
29 | eldifsni 4737 | . . . . . . 7 ⊢ (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦 ≠ 𝑋) | |
30 | 29 | adantl 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦 ≠ 𝑋) |
31 | 30 | neneqd 2945 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋) |
32 | 31 | iffalsed 4484 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 ) |
33 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ V) |
34 | 32, 33 | suppss2 8086 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋}) |
35 | suppssfifsupp 9241 | . . 3 ⊢ ((((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ V ∧ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∧ 0 ∈ V) ∧ ({𝑋} ∈ Fin ∧ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋})) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) finSupp 0 ) | |
36 | 26, 28, 34, 35 | syl12anc 834 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) finSupp 0 ) |
37 | mplmon.s | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
38 | mplmon.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
39 | 37, 17, 18, 5, 38 | mplelbas 21305 | . 2 ⊢ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵 ↔ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) finSupp 0 )) |
40 | 21, 36, 39 | sylanbrc 583 | 1 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 {crab 3403 Vcvv 3441 ∖ cdif 3895 ⊆ wss 3898 ifcif 4473 {csn 4573 class class class wbr 5092 ↦ cmpt 5175 ◡ccnv 5619 “ cima 5623 Fun wfun 6473 ⟶wf 6475 ‘cfv 6479 (class class class)co 7337 supp csupp 8047 ↑m cmap 8686 Fincfn 8804 finSupp cfsupp 9226 ℕcn 12074 ℕ0cn0 12334 Basecbs 17009 0gc0g 17247 1rcur 19832 Ringcrg 19878 mPwSer cmps 21213 mPoly cmpl 21215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-of 7595 df-om 7781 df-1st 7899 df-2nd 7900 df-supp 8048 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-er 8569 df-map 8688 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-fsupp 9227 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-3 12138 df-4 12139 df-5 12140 df-6 12141 df-7 12142 df-8 12143 df-9 12144 df-n0 12335 df-z 12421 df-uz 12684 df-fz 13341 df-struct 16945 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-mulr 17073 df-sca 17075 df-vsca 17076 df-tset 17078 df-0g 17249 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-grp 18676 df-mgp 19816 df-ur 19833 df-ring 19880 df-psr 21218 df-mpl 21220 |
This theorem is referenced by: mplmonmul 21343 mplcoe1 21344 mplbas2 21349 mplmon2 21375 mplmon2cl 21382 mplmon2mul 21383 |
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