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Theorem psrlinv 21992

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)
psrlinv.o	⊢ 0 = (0_g‘𝑅)
psrlinv.p	⊢ + = (+_g‘𝑆)

**Assertion**
Ref	Expression
psrlinv	⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = (𝐷 × { 0 }))

Distinct variable group: 𝑓,𝐼 Allowed substitution hints: 𝜑(𝑓) 𝐵(𝑓) 𝐷(𝑓) + (𝑓) 𝑅(𝑓) 𝑆(𝑓) 𝑁(𝑓) 𝑉(𝑓) 𝑋(𝑓) 0 (𝑓)

Proof of Theorem psrlinv Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrnegcl.d	. . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
2		ovex 7463	. . . . 5 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
3	1, 2	rabex2 5346	. . . 4 ⊢ 𝐷 ∈ V
4	3	a1i 11	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
5		fvexd 6921	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑁‘(𝑋‘𝑥)) ∈ V)
6		psrgrp.s	. . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
7		eqid 2734	. . . . 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 21971	. . . 4 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
11	10	ffvelcdmda 7103	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑋‘𝑥) ∈ (Base‘𝑅))
12	10	feqmptd 6976	. . . 4 ⊢ (𝜑 → 𝑋 = (𝑥 ∈ 𝐷 ↦ (𝑋‘𝑥)))
13		psrnegcl.i	. . . . . . 7 ⊢ 𝑁 = (inv_g‘𝑅)
14		psrgrp.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Grp)
15	7, 13, 14	grpinvf1o 19039	. . . . . 6 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
16		f1of 6848	. . . . . 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 6976	. . . 4 ⊢ (𝜑 → 𝑁 = (𝑦 ∈ (Base‘𝑅) ↦ (𝑁‘𝑦)))
19		fveq2 6906	. . . 4 ⊢ (𝑦 = (𝑋‘𝑥) → (𝑁‘𝑦) = (𝑁‘(𝑋‘𝑥)))
20	11, 12, 18, 19	fmptco 7148	. . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐷 ↦ (𝑁‘(𝑋‘𝑥))))
21	4, 5, 11, 20, 12	offval2 7716	. 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) ∘_f (+_g‘𝑅)𝑋) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))))
22		eqid 2734	. . 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 21991	. . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵)
26	6, 8, 22, 23, 25, 9	psradd 21974	. 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = ((𝑁 ∘ 𝑋) ∘_f (+_g‘𝑅)𝑋))
27		fconstmpt 5750	. . 3 ⊢ (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ 0 )
28		psrlinv.o	. . . . . 6 ⊢ 0 = (0_g‘𝑅)
29	7, 22, 28, 13	grplinv 19019	. . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (𝑋‘𝑥) ∈ (Base‘𝑅)) → ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥)) = 0 )
30	14, 11, 29	syl2an2r 685	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥)) = 0 )
31	30	mpteq2dva 5247	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))) = (𝑥 ∈ 𝐷 ↦ 0 ))
32	27, 31	eqtr4id 2793	. 2 ⊢ (𝜑 → (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))))
33	21, 26, 32	3eqtr4d 2784	1 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = (𝐷 × { 0 }))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {crab 3432 Vcvv 3477 {csn 4630 ↦ cmpt 5230 × cxp 5686 ^◡ccnv 5687 “ cima 5691 ∘ ccom 5692 ⟶wf 6558 –1-1-onto→wf1o 6561 ‘cfv 6562 (class class class)co 7430 ∘_f cof 7694 ↑_m cmap 8864 Fincfn 8983 ℕcn 12263 ℕ₀cn0 12523 Basecbs 17244 +_gcplusg 17297 0_gc0g 17485 Grpcgrp 18963 inv_gcminusg 18964 mPwSer cmps 21941

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-tset 17316 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-psr 21946

This theorem is referenced by: psrgrpOLD 21994 psrneg 21996

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