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Theorem psrnegcl 19596

Description: The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)

**Hypotheses**
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 = (inv_g‘𝑅)
psrnegcl.b	⊢ 𝐵 = (Base‘𝑆)
psrnegcl.z	⊢ (𝜑 → 𝑋 ∈ 𝐵)

**Assertion**
Ref	Expression
psrnegcl	⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵)

Distinct variable group: 𝑓,𝐼 Allowed substitution hints: 𝜑(𝑓) 𝐵(𝑓) 𝐷(𝑓) 𝑅(𝑓) 𝑆(𝑓) 𝑁(𝑓) 𝑉(𝑓) 𝑋(𝑓)

**Proof of Theorem psrnegcl**
Step	Hyp	Ref	Expression
1		eqid 2758	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		psrnegcl.i	. . . . . 6 ⊢ 𝑁 = (inv_g‘𝑅)
3		psrgrp.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
4	1, 2, 3	grpinvf1o 17684	. . . . 5 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
5		f1of 6296	. . . . 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 19579	. . . 4 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
12		fco 6217	. . . 4 ⊢ ((𝑁:(Base‘𝑅)⟶(Base‘𝑅) ∧ 𝑋:𝐷⟶(Base‘𝑅)) → (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅))
13	6, 11, 12	syl2anc 696	. . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅))
14		fvex 6360	. . . 4 ⊢ (Base‘𝑅) ∈ V
15		ovex 6839	. . . . 5 ⊢ (ℕ₀ ↑_𝑚 𝐼) ∈ V
16	8, 15	rabex2 4964	. . . 4 ⊢ 𝐷 ∈ V
17	14, 16	elmap 8050	. . 3 ⊢ ((𝑁 ∘ 𝑋) ∈ ((Base‘𝑅) ↑_𝑚 𝐷) ↔ (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅))
18	13, 17	sylibr 224	. 2 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ ((Base‘𝑅) ↑_𝑚 𝐷))
19		psrgrp.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
20	7, 1, 8, 9, 19	psrbas 19578	. 2 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑_𝑚 𝐷))
21	18, 20	eleqtrrd 2840	1 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1630 ∈ wcel 2137 {crab 3052 ^◡ccnv 5263 “ cima 5267 ∘ ccom 5268 ⟶wf 6043 –1-1-onto→wf1o 6046 ‘cfv 6047 (class class class)co 6811 ↑_𝑚 cmap 8021 Fincfn 8119 ℕcn 11210 ℕ₀cn0 11482 Basecbs 16057 Grpcgrp 17621 inv_gcminusg 17622 mPwSer cmps 19551

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-8 2139 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-rep 4921 ax-sep 4931 ax-nul 4939 ax-pow 4990 ax-pr 5053 ax-un 7112 ax-cnex 10182 ax-resscn 10183 ax-1cn 10184 ax-icn 10185 ax-addcl 10186 ax-addrcl 10187 ax-mulcl 10188 ax-mulrcl 10189 ax-mulcom 10190 ax-addass 10191 ax-mulass 10192 ax-distr 10193 ax-i2m1 10194 ax-1ne0 10195 ax-1rid 10196 ax-rnegex 10197 ax-rrecex 10198 ax-cnre 10199 ax-pre-lttri 10200 ax-pre-lttrn 10201 ax-pre-ltadd 10202 ax-pre-mulgt0 10203

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-mo 2610 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-ne 2931 df-nel 3034 df-ral 3053 df-rex 3054 df-reu 3055 df-rmo 3056 df-rab 3057 df-v 3340 df-sbc 3575 df-csb 3673 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-pss 3729 df-nul 4057 df-if 4229 df-pw 4302 df-sn 4320 df-pr 4322 df-tp 4324 df-op 4326 df-uni 4587 df-int 4626 df-iun 4672 df-br 4803 df-opab 4863 df-mpt 4880 df-tr 4903 df-id 5172 df-eprel 5177 df-po 5185 df-so 5186 df-fr 5223 df-we 5225 df-xp 5270 df-rel 5271 df-cnv 5272 df-co 5273 df-dm 5274 df-rn 5275 df-res 5276 df-ima 5277 df-pred 5839 df-ord 5885 df-on 5886 df-lim 5887 df-suc 5888 df-iota 6010 df-fun 6049 df-fn 6050 df-f 6051 df-f1 6052 df-fo 6053 df-f1o 6054 df-fv 6055 df-riota 6772 df-ov 6814 df-oprab 6815 df-mpt2 6816 df-of 7060 df-om 7229 df-1st 7331 df-2nd 7332 df-supp 7462 df-wrecs 7574 df-recs 7635 df-rdg 7673 df-1o 7727 df-oadd 7731 df-er 7909 df-map 8023 df-en 8120 df-dom 8121 df-sdom 8122 df-fin 8123 df-fsupp 8439 df-pnf 10266 df-mnf 10267 df-xr 10268 df-ltxr 10269 df-le 10270 df-sub 10458 df-neg 10459 df-nn 11211 df-2 11269 df-3 11270 df-4 11271 df-5 11272 df-6 11273 df-7 11274 df-8 11275 df-9 11276 df-n0 11483 df-z 11568 df-uz 11878 df-fz 12518 df-struct 16059 df-ndx 16060 df-slot 16061 df-base 16063 df-plusg 16154 df-mulr 16155 df-sca 16157 df-vsca 16158 df-tset 16160 df-0g 16302 df-mgm 17441 df-sgrp 17483 df-mnd 17494 df-grp 17624 df-minusg 17625 df-psr 19556

This theorem is referenced by: psrlinv 19597 psrgrp 19598 psrneg 19600

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