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Mirrors > Home > MPE Home > Th. List > psrnegcl | 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) |
Ref | Expression |
---|---|
psrnegcl | ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | psrnegcl.i | . . . . . 6 ⊢ 𝑁 = (invg‘𝑅) | |
3 | psrgrp.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
4 | 1, 2, 3 | grpinvf1o 18163 | . . . . 5 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅)) |
5 | f1of 6609 | . . . . 5 ⊢ (𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅) → 𝑁:(Base‘𝑅)⟶(Base‘𝑅)) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁:(Base‘𝑅)⟶(Base‘𝑅)) |
7 | psrgrp.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
8 | psrnegcl.d | . . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
9 | psrnegcl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑆) | |
10 | psrnegcl.z | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | 7, 1, 8, 9, 10 | psrelbas 20153 | . . . 4 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅)) |
12 | fco 6525 | . . . 4 ⊢ ((𝑁:(Base‘𝑅)⟶(Base‘𝑅) ∧ 𝑋:𝐷⟶(Base‘𝑅)) → (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅)) | |
13 | 6, 11, 12 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅)) |
14 | fvex 6677 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
15 | ovex 7183 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
16 | 8, 15 | rabex2 5229 | . . . 4 ⊢ 𝐷 ∈ V |
17 | 14, 16 | elmap 8429 | . . 3 ⊢ ((𝑁 ∘ 𝑋) ∈ ((Base‘𝑅) ↑m 𝐷) ↔ (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅)) |
18 | 13, 17 | sylibr 236 | . 2 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ ((Base‘𝑅) ↑m 𝐷)) |
19 | psrgrp.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
20 | 7, 1, 8, 9, 19 | psrbas 20152 | . 2 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑m 𝐷)) |
21 | 18, 20 | eleqtrrd 2916 | 1 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {crab 3142 ◡ccnv 5548 “ cima 5552 ∘ ccom 5553 ⟶wf 6345 –1-1-onto→wf1o 6348 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 Fincfn 8503 ℕcn 11632 ℕ0cn0 11891 Basecbs 16477 Grpcgrp 18097 invgcminusg 18098 mPwSer cmps 20125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-tset 16578 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-psr 20130 |
This theorem is referenced by: psrlinv 20171 psrgrp 20172 psrneg 20174 |
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