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Mirrors > Home > MPE Home > Th. List > psrlinv | Structured version Visualization version GIF version |
Description: The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
psrgrp.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
psrgrp.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
psrgrp.r | ⊢ (𝜑 → 𝑅 ∈ Grp) |
psrnegcl.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
psrnegcl.i | ⊢ 𝑁 = (invg‘𝑅) |
psrnegcl.b | ⊢ 𝐵 = (Base‘𝑆) |
psrnegcl.z | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
psrlinv.o | ⊢ 0 = (0g‘𝑅) |
psrlinv.p | ⊢ + = (+g‘𝑆) |
Ref | Expression |
---|---|
psrlinv | ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = (𝐷 × { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrnegcl.d | . . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
2 | ovex 6678 | . . . . 5 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V | |
3 | 1, 2 | rabex2 4815 | . . . 4 ⊢ 𝐷 ∈ V |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
5 | fvexd 6203 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑁‘(𝑋‘𝑥)) ∈ V) | |
6 | psrgrp.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
7 | eqid 2622 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
8 | psrnegcl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑆) | |
9 | psrnegcl.z | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
10 | 6, 7, 1, 8, 9 | psrelbas 19379 | . . . 4 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅)) |
11 | 10 | ffvelrnda 6359 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑋‘𝑥) ∈ (Base‘𝑅)) |
12 | 10 | feqmptd 6249 | . . . 4 ⊢ (𝜑 → 𝑋 = (𝑥 ∈ 𝐷 ↦ (𝑋‘𝑥))) |
13 | psrnegcl.i | . . . . . . 7 ⊢ 𝑁 = (invg‘𝑅) | |
14 | psrgrp.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
15 | 7, 13, 14 | grpinvf1o 17485 | . . . . . 6 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅)) |
16 | f1of 6137 | . . . . . 6 ⊢ (𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅) → 𝑁:(Base‘𝑅)⟶(Base‘𝑅)) | |
17 | 15, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁:(Base‘𝑅)⟶(Base‘𝑅)) |
18 | 17 | feqmptd 6249 | . . . 4 ⊢ (𝜑 → 𝑁 = (𝑦 ∈ (Base‘𝑅) ↦ (𝑁‘𝑦))) |
19 | fveq2 6191 | . . . 4 ⊢ (𝑦 = (𝑋‘𝑥) → (𝑁‘𝑦) = (𝑁‘(𝑋‘𝑥))) | |
20 | 11, 12, 18, 19 | fmptco 6396 | . . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐷 ↦ (𝑁‘(𝑋‘𝑥)))) |
21 | 4, 5, 11, 20, 12 | offval2 6914 | . 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) ∘𝑓 (+g‘𝑅)𝑋) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+g‘𝑅)(𝑋‘𝑥)))) |
22 | eqid 2622 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
23 | psrlinv.p | . . 3 ⊢ + = (+g‘𝑆) | |
24 | psrgrp.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
25 | 6, 24, 14, 1, 13, 8, 9 | psrnegcl 19396 | . . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵) |
26 | 6, 8, 22, 23, 25, 9 | psradd 19382 | . 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = ((𝑁 ∘ 𝑋) ∘𝑓 (+g‘𝑅)𝑋)) |
27 | psrlinv.o | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
28 | 7, 22, 27, 13 | grplinv 17468 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (𝑋‘𝑥) ∈ (Base‘𝑅)) → ((𝑁‘(𝑋‘𝑥))(+g‘𝑅)(𝑋‘𝑥)) = 0 ) |
29 | 14, 11, 28 | syl2an2r 876 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑁‘(𝑋‘𝑥))(+g‘𝑅)(𝑋‘𝑥)) = 0 ) |
30 | 29 | mpteq2dva 4744 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+g‘𝑅)(𝑋‘𝑥))) = (𝑥 ∈ 𝐷 ↦ 0 )) |
31 | fconstmpt 5163 | . . 3 ⊢ (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ 0 ) | |
32 | 30, 31 | syl6reqr 2675 | . 2 ⊢ (𝜑 → (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+g‘𝑅)(𝑋‘𝑥)))) |
33 | 21, 26, 32 | 3eqtr4d 2666 | 1 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = (𝐷 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 {csn 4177 ↦ cmpt 4729 × cxp 5112 ◡ccnv 5113 “ cima 5117 ∘ ccom 5118 ⟶wf 5884 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 ↑𝑚 cmap 7857 Fincfn 7955 ℕcn 11020 ℕ0cn0 11292 Basecbs 15857 +gcplusg 15941 0gc0g 16100 Grpcgrp 17422 invgcminusg 17423 mPwSer cmps 19351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-tset 15960 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-psr 19356 |
This theorem is referenced by: psrgrp 19398 psrneg 19400 |
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