psrnegcl - Metamath Proof Explorer

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Theorem psrnegcl 21908

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 2734	. . . . . 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 18937	. . . . 5 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
5		f1of 6772	. . . . 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 21888	. . . 4 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
12		fco 6684	. . . 4 ⊢ ((𝑁:(Base‘𝑅)⟶(Base‘𝑅) ∧ 𝑋:𝐷⟶(Base‘𝑅)) → (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅))
13	6, 11, 12	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅))
14		fvex 6845	. . . 4 ⊢ (Base‘𝑅) ∈ V
15		ovex 7389	. . . . 5 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
16	8, 15	rabex2 5284	. . . 4 ⊢ 𝐷 ∈ V
17	14, 16	elmap 8807	. . 3 ⊢ ((𝑁 ∘ 𝑋) ∈ ((Base‘𝑅) ↑_m 𝐷) ↔ (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅))
18	13, 17	sylibr 234	. 2 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ ((Base‘𝑅) ↑_m 𝐷))
19		psrgrp.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
20	7, 1, 8, 9, 19	psrbas 21887	. 2 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑_m 𝐷))
21	18, 20	eleqtrrd 2837	1 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 {crab 3397 ^◡ccnv 5621 “ cima 5625 ∘ ccom 5626 ⟶wf 6486 –1-1-onto→wf1o 6489 ‘cfv 6490 (class class class)co 7356 ↑_m cmap 8761 Fincfn 8881 ℕcn 12143 ℕ₀cn0 12399 Basecbs 17134 Grpcgrp 18861 inv_gcminusg 18862 mPwSer cmps 21858

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8763 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-uz 12750 df-fz 13422 df-struct 17072 df-slot 17107 df-ndx 17119 df-base 17135 df-plusg 17188 df-mulr 17189 df-sca 17191 df-vsca 17192 df-tset 17194 df-0g 17359 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18864 df-minusg 18865 df-psr 21863

This theorem is referenced by: psrlinv 21909 psrneg 21912

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