psrnegcl - Metamath Proof Explorer

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Theorem psrnegcl 21245

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 2736	. . . . . 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 18718	. . . . 5 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
5		f1of 6753	. . . . 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 21228	. . . 4 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
12		fco 6661	. . . 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 6824	. . . 4 ⊢ (Base‘𝑅) ∈ V
15		ovex 7349	. . . . 5 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
16	8, 15	rabex2 5272	. . . 4 ⊢ 𝐷 ∈ V
17	14, 16	elmap 8708	. . 3 ⊢ ((𝑁 ∘ 𝑋) ∈ ((Base‘𝑅) ↑_m 𝐷) ↔ (𝑁 ∘ 𝑋):𝐷⟶(Base‘𝑅))
18	13, 17	sylibr 233	. 2 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ ((Base‘𝑅) ↑_m 𝐷))
19		psrgrp.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
20	7, 1, 8, 9, 19	psrbas 21227	. 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 1540 ∈ wcel 2105 {crab 3403 ^◡ccnv 5606 “ cima 5610 ∘ ccom 5611 ⟶wf 6461 –1-1-onto→wf1o 6464 ‘cfv 6465 (class class class)co 7316 ↑_m cmap 8664 Fincfn 8782 ℕcn 12052 ℕ₀cn0 12312 Basecbs 16986 Grpcgrp 18650 inv_gcminusg 18651 mPwSer cmps 21187

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5223 ax-sep 5237 ax-nul 5244 ax-pow 5302 ax-pr 5366 ax-un 7629 ax-cnex 11006 ax-resscn 11007 ax-1cn 11008 ax-icn 11009 ax-addcl 11010 ax-addrcl 11011 ax-mulcl 11012 ax-mulrcl 11013 ax-mulcom 11014 ax-addass 11015 ax-mulass 11016 ax-distr 11017 ax-i2m1 11018 ax-1ne0 11019 ax-1rid 11020 ax-rnegex 11021 ax-rrecex 11022 ax-cnre 11023 ax-pre-lttri 11024 ax-pre-lttrn 11025 ax-pre-ltadd 11026 ax-pre-mulgt0 11027

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3442 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4471 df-pw 4546 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4850 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5170 df-tr 5204 df-id 5506 df-eprel 5512 df-po 5520 df-so 5521 df-fr 5562 df-we 5564 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-rn 5618 df-res 5619 df-ima 5620 df-pred 6224 df-ord 6291 df-on 6292 df-lim 6293 df-suc 6294 df-iota 6417 df-fun 6467 df-fn 6468 df-f 6469 df-f1 6470 df-fo 6471 df-f1o 6472 df-fv 6473 df-riota 7273 df-ov 7319 df-oprab 7320 df-mpo 7321 df-of 7574 df-om 7759 df-1st 7877 df-2nd 7878 df-supp 8026 df-frecs 8145 df-wrecs 8176 df-recs 8250 df-rdg 8289 df-1o 8345 df-er 8547 df-map 8666 df-en 8783 df-dom 8784 df-sdom 8785 df-fin 8786 df-fsupp 9205 df-pnf 11090 df-mnf 11091 df-xr 11092 df-ltxr 11093 df-le 11094 df-sub 11286 df-neg 11287 df-nn 12053 df-2 12115 df-3 12116 df-4 12117 df-5 12118 df-6 12119 df-7 12120 df-8 12121 df-9 12122 df-n0 12313 df-z 12399 df-uz 12662 df-fz 13319 df-struct 16922 df-slot 16957 df-ndx 16969 df-base 16987 df-plusg 17049 df-mulr 17050 df-sca 17052 df-vsca 17053 df-tset 17055 df-0g 17226 df-mgm 18400 df-sgrp 18449 df-mnd 18460 df-grp 18653 df-minusg 18654 df-psr 21192

This theorem is referenced by: psrlinv 21246 psrgrp 21247 psrneg 21249

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