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Theorem psrlinv 21880

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 7386	. . . . 5 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
3	1, 2	rabex2 5283	. . . 4 ⊢ 𝐷 ∈ V
4	3	a1i 11	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
5		fvexd 6841	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑁‘(𝑋‘𝑥)) ∈ V)
6		psrgrp.s	. . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
7		eqid 2729	. . . . 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 21859	. . . 4 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
11	10	ffvelcdmda 7022	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑋‘𝑥) ∈ (Base‘𝑅))
12	10	feqmptd 6895	. . . 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 18906	. . . . . 6 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
16		f1of 6768	. . . . . 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 6895	. . . 4 ⊢ (𝜑 → 𝑁 = (𝑦 ∈ (Base‘𝑅) ↦ (𝑁‘𝑦)))
19		fveq2 6826	. . . 4 ⊢ (𝑦 = (𝑋‘𝑥) → (𝑁‘𝑦) = (𝑁‘(𝑋‘𝑥)))
20	11, 12, 18, 19	fmptco 7067	. . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐷 ↦ (𝑁‘(𝑋‘𝑥))))
21	4, 5, 11, 20, 12	offval2 7637	. 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) ∘_f (+_g‘𝑅)𝑋) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))))
22		eqid 2729	. . 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 21879	. . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵)
26	6, 8, 22, 23, 25, 9	psradd 21862	. 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = ((𝑁 ∘ 𝑋) ∘_f (+_g‘𝑅)𝑋))
27		fconstmpt 5685	. . 3 ⊢ (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ 0 )
28		psrlinv.o	. . . . . 6 ⊢ 0 = (0_g‘𝑅)
29	7, 22, 28, 13	grplinv 18886	. . . . 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 5188	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))) = (𝑥 ∈ 𝐷 ↦ 0 ))
32	27, 31	eqtr4id 2783	. 2 ⊢ (𝜑 → (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))))
33	21, 26, 32	3eqtr4d 2774	1 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = (𝐷 × { 0 }))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {crab 3396 Vcvv 3438 {csn 4579 ↦ cmpt 5176 × cxp 5621 ^◡ccnv 5622 “ cima 5626 ∘ ccom 5627 ⟶wf 6482 –1-1-onto→wf1o 6485 ‘cfv 6486 (class class class)co 7353 ∘_f cof 7615 ↑_m cmap 8760 Fincfn 8879 ℕcn 12146 ℕ₀cn0 12402 Basecbs 17138 +_gcplusg 17179 0_gc0g 17361 Grpcgrp 18830 inv_gcminusg 18831 mPwSer cmps 21829

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 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 8632 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-uz 12754 df-fz 13429 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17139 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-tset 17198 df-0g 17363 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-grp 18833 df-minusg 18834 df-psr 21834

This theorem is referenced by: psrgrpOLD 21882 psrneg 21884

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