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Theorem psrlinv 14888

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	. . . 4 ⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
2		fnmap 6891	. . . . 5 ⊢ ↑_𝑚 Fn (V × V)
3		nn0ex 9507	. . . . 5 ⊢ ℕ₀ ∈ V
4		psrgrp.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
5	4	elexd 2829	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
6		fnovex 6085	. . . . 5 ⊢ (( ↑_𝑚 Fn (V × V) ∧ ℕ₀ ∈ V ∧ 𝐼 ∈ V) → (ℕ₀ ↑_𝑚 𝐼) ∈ V)
7	2, 3, 5, 6	mp3an12i 1378	. . . 4 ⊢ (𝜑 → (ℕ₀ ↑_𝑚 𝐼) ∈ V)
8	1, 7	rabexd 4259	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
9		psrgrp.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
10		psrgrp.s	. . . . . 6 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
11		eqid 2234	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
12		psrnegcl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
13		psrnegcl.z	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
14	10, 11, 1, 12, 13	psrelbas 14879	. . . . 5 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
15	14	ffvelcdmda 5814	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑋‘𝑥) ∈ (Base‘𝑅))
16		psrnegcl.i	. . . . 5 ⊢ 𝑁 = (inv_g‘𝑅)
17	11, 16	grpinvcl 13782	. . . 4 ⊢ ((𝑅 ∈ Grp ∧ (𝑋‘𝑥) ∈ (Base‘𝑅)) → (𝑁‘(𝑋‘𝑥)) ∈ (Base‘𝑅))
18	9, 15, 17	syl2an2r 599	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑁‘(𝑋‘𝑥)) ∈ (Base‘𝑅))
19	14	feqmptd 5732	. . . 4 ⊢ (𝜑 → 𝑋 = (𝑥 ∈ 𝐷 ↦ (𝑋‘𝑥)))
20	11, 16, 9	grpinvf1o 13804	. . . . . 6 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
21		f1of 5616	. . . . . 6 ⊢ (𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅) → 𝑁:(Base‘𝑅)⟶(Base‘𝑅))
22	20, 21	syl 14	. . . . 5 ⊢ (𝜑 → 𝑁:(Base‘𝑅)⟶(Base‘𝑅))
23	22	feqmptd 5732	. . . 4 ⊢ (𝜑 → 𝑁 = (𝑦 ∈ (Base‘𝑅) ↦ (𝑁‘𝑦)))
24		fveq2 5672	. . . 4 ⊢ (𝑦 = (𝑋‘𝑥) → (𝑁‘𝑦) = (𝑁‘(𝑋‘𝑥)))
25	15, 19, 23, 24	fmptco 5845	. . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐷 ↦ (𝑁‘(𝑋‘𝑥))))
26	8, 18, 15, 25, 19	offval2 6284	. 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) ∘_𝑓 (+_g‘𝑅)𝑋) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))))
27		eqid 2234	. . 3 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
28		psrlinv.p	. . 3 ⊢ + = (+_g‘𝑆)
29	10, 4, 9, 1, 16, 12, 13	psrnegcl 14887	. . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵)
30	10, 12, 27, 28, 29, 13	psradd 14883	. 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = ((𝑁 ∘ 𝑋) ∘_𝑓 (+_g‘𝑅)𝑋))
31		fconstmpt 4799	. . 3 ⊢ (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ 0 )
32		psrlinv.o	. . . . . 6 ⊢ 0 = (0_g‘𝑅)
33	11, 27, 32, 16	grplinv 13784	. . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (𝑋‘𝑥) ∈ (Base‘𝑅)) → ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥)) = 0 )
34	9, 15, 33	syl2an2r 599	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥)) = 0 )
35	34	mpteq2dva 4202	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))) = (𝑥 ∈ 𝐷 ↦ 0 ))
36	31, 35	eqtr4id 2286	. 2 ⊢ (𝜑 → (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))))
37	26, 30, 36	3eqtr4d 2277	1 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = (𝐷 × { 0 }))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1398 ∈ wcel 2205 {crab 2526 Vcvv 2815 {csn 3691 ↦ cmpt 4173 × cxp 4749 ^◡ccnv 4750 “ cima 4754 ∘ ccom 4755 Fn wfn 5349 ⟶wf 5350 –1-1-onto→wf1o 5353 ‘cfv 5354 (class class class)co 6052 ∘_𝑓 cof 6266 ↑_𝑚 cmap 6884 Fincfn 6977 ℕcn 9242 ℕ₀cn0 9501 Basecbs 13233 +_gcplusg 13311 0_gc0g 13490 Grpcgrp 13734 inv_gcminusg 13735 mPwSer cmps 14858

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4227 ax-sep 4230 ax-pow 4289 ax-pr 4324 ax-un 4556 ax-setind 4661 ax-cnex 8223 ax-resscn 8224 ax-1cn 8225 ax-1re 8226 ax-icn 8227 ax-addcl 8228 ax-addrcl 8229 ax-mulcl 8230 ax-addcom 8232 ax-addass 8234 ax-distr 8236 ax-i2m1 8237 ax-0lt1 8238 ax-0id 8240 ax-rnegex 8241 ax-cnre 8243 ax-pre-ltirr 8244 ax-pre-ltwlin 8245 ax-pre-lttrn 8246 ax-pre-apti 8247 ax-pre-ltadd 8248

This theorem depends on definitions: df-bi 117 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3045 df-csb 3141 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-nul 3511 df-pw 3673 df-sn 3697 df-pr 3698 df-tp 3699 df-op 3700 df-uni 3917 df-int 3952 df-iun 3995 df-br 4112 df-opab 4174 df-mpt 4175 df-id 4416 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-rn 4762 df-res 4763 df-ima 4764 df-iota 5314 df-fun 5356 df-fn 5357 df-f 5358 df-f1 5359 df-fo 5360 df-f1o 5361 df-fv 5362 df-riota 6005 df-ov 6055 df-oprab 6056 df-mpo 6057 df-of 6268 df-1st 6336 df-2nd 6337 df-map 6886 df-ixp 6936 df-pnf 8315 df-mnf 8316 df-xr 8317 df-ltxr 8318 df-le 8319 df-sub 8451 df-neg 8452 df-inn 9243 df-2 9301 df-3 9302 df-4 9303 df-5 9304 df-6 9305 df-7 9306 df-8 9307 df-9 9308 df-n0 9502 df-z 9583 df-uz 9860 df-fz 10349 df-struct 13235 df-ndx 13236 df-slot 13237 df-base 13239 df-plusg 13324 df-mulr 13325 df-sca 13327 df-vsca 13328 df-tset 13330 df-rest 13475 df-topn 13476 df-0g 13492 df-topgen 13494 df-pt 13495 df-mgm 13590 df-sgrp 13636 df-mnd 13651 df-grp 13737 df-minusg 13738 df-psr 14860

This theorem is referenced by: psrneg 14891

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