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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 2823 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | mplmon.o | . . . . . . . . 9 ⊢ 1 = (1r‘𝑅) | |
4 | 2, 3 | ringidcl 19320 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
5 | mplmon.z | . . . . . . . . 9 ⊢ 0 = (0g‘𝑅) | |
6 | 2, 5 | ring0cl 19321 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅)) |
7 | 4, 6 | ifcld 4514 | . . . . . . 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 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → if(𝑦 = 𝑋, 1 , 0 ) ∈ (Base‘𝑅)) |
10 | 9 | fmpttd 6881 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅)) |
11 | fvex 6685 | . . . . 5 ⊢ (Base‘𝑅) ∈ V | |
12 | mplmon.d | . . . . . 6 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
13 | ovex 7191 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
14 | 12, 13 | rabex2 5239 | . . . . 5 ⊢ 𝐷 ∈ V |
15 | 11, 14 | elmap 8437 | . . . 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 236 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ ((Base‘𝑅) ↑m 𝐷)) |
17 | eqid 2823 | . . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
18 | eqid 2823 | . . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
19 | mplmon.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
20 | 17, 2, 12, 18, 19 | psrbas 20160 | . . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑m 𝐷)) |
21 | 16, 20 | eleqtrrd 2918 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅))) |
22 | 14 | mptex 6988 | . . . . 5 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ V |
23 | funmpt 6395 | . . . . 5 ⊢ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) | |
24 | 5 | fvexi 6686 | . . . . 5 ⊢ 0 ∈ V |
25 | 22, 23, 24 | 3pm3.2i 1335 | . . . 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 8596 | . . . 4 ⊢ {𝑋} ∈ Fin | |
28 | 27 | a1i 11 | . . 3 ⊢ (𝜑 → {𝑋} ∈ Fin) |
29 | eldifsni 4724 | . . . . . . 7 ⊢ (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦 ≠ 𝑋) | |
30 | 29 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦 ≠ 𝑋) |
31 | 30 | neneqd 3023 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋) |
32 | 31 | iffalsed 4480 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 ) |
33 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ V) |
34 | 32, 33 | suppss2 7866 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋}) |
35 | suppssfifsupp 8850 | . . 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 20212 | . 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 585 | 1 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 {crab 3144 Vcvv 3496 ∖ cdif 3935 ⊆ wss 3938 ifcif 4469 {csn 4569 class class class wbr 5068 ↦ cmpt 5148 ◡ccnv 5556 “ cima 5560 Fun wfun 6351 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 supp csupp 7832 ↑m cmap 8408 Fincfn 8511 finSupp cfsupp 8835 ℕcn 11640 ℕ0cn0 11900 Basecbs 16485 0gc0g 16715 1rcur 19253 Ringcrg 19299 mPwSer cmps 20133 mPoly cmpl 20135 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-tset 16586 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-mgp 19242 df-ur 19254 df-ring 19301 df-psr 20138 df-mpl 20140 |
This theorem is referenced by: mplmonmul 20247 mplcoe1 20248 mplbas2 20253 mplmon2 20275 mplmon2cl 20282 mplmon2mul 20283 |
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