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Theorem psrlinv 14312

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 6723	. . . . 5 ⊢ ↑_𝑚 Fn (V × V)
3		nn0ex 9272	. . . . 5 ⊢ ℕ₀ ∈ V
4		psrgrp.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
5	4	elexd 2776	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
6		fnovex 5958	. . . . 5 ⊢ (( ↑_𝑚 Fn (V × V) ∧ ℕ₀ ∈ V ∧ 𝐼 ∈ V) → (ℕ₀ ↑_𝑚 𝐼) ∈ V)
7	2, 3, 5, 6	mp3an12i 1352	. . . 4 ⊢ (𝜑 → (ℕ₀ ↑_𝑚 𝐼) ∈ V)
8	1, 7	rabexd 4179	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
9		psrgrp.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
10		psrgrp.s	. . . . . 6 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
11		eqid 2196	. . . . . 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 14304	. . . . 5 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
15	14	ffvelcdmda 5700	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑋‘𝑥) ∈ (Base‘𝑅))
16		psrnegcl.i	. . . . 5 ⊢ 𝑁 = (inv_g‘𝑅)
17	11, 16	grpinvcl 13250	. . . 4 ⊢ ((𝑅 ∈ Grp ∧ (𝑋‘𝑥) ∈ (Base‘𝑅)) → (𝑁‘(𝑋‘𝑥)) ∈ (Base‘𝑅))
18	9, 15, 17	syl2an2r 595	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑁‘(𝑋‘𝑥)) ∈ (Base‘𝑅))
19	14	feqmptd 5617	. . . 4 ⊢ (𝜑 → 𝑋 = (𝑥 ∈ 𝐷 ↦ (𝑋‘𝑥)))
20	11, 16, 9	grpinvf1o 13272	. . . . . 6 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
21		f1of 5507	. . . . . 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 5617	. . . 4 ⊢ (𝜑 → 𝑁 = (𝑦 ∈ (Base‘𝑅) ↦ (𝑁‘𝑦)))
24		fveq2 5561	. . . 4 ⊢ (𝑦 = (𝑋‘𝑥) → (𝑁‘𝑦) = (𝑁‘(𝑋‘𝑥)))
25	15, 19, 23, 24	fmptco 5731	. . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐷 ↦ (𝑁‘(𝑋‘𝑥))))
26	8, 18, 15, 25, 19	offval2 6155	. 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) ∘_𝑓 (+_g‘𝑅)𝑋) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))))
27		eqid 2196	. . 3 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
28		psrlinv.p	. . 3 ⊢ + = (+_g‘𝑆)
29	10, 4, 9, 1, 16, 12, 13	psrnegcl 14311	. . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵)
30	10, 12, 27, 28, 29, 13	psradd 14307	. 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = ((𝑁 ∘ 𝑋) ∘_𝑓 (+_g‘𝑅)𝑋))
31		fconstmpt 4711	. . 3 ⊢ (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ 0 )
32		psrlinv.o	. . . . . 6 ⊢ 0 = (0_g‘𝑅)
33	11, 27, 32, 16	grplinv 13252	. . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (𝑋‘𝑥) ∈ (Base‘𝑅)) → ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥)) = 0 )
34	9, 15, 33	syl2an2r 595	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥)) = 0 )
35	34	mpteq2dva 4124	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))) = (𝑥 ∈ 𝐷 ↦ 0 ))
36	31, 35	eqtr4id 2248	. 2 ⊢ (𝜑 → (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))))
37	26, 30, 36	3eqtr4d 2239	1 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = (𝐷 × { 0 }))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2167 {crab 2479 Vcvv 2763 {csn 3623 ↦ cmpt 4095 × cxp 4662 ^◡ccnv 4663 “ cima 4667 ∘ ccom 4668 Fn wfn 5254 ⟶wf 5255 –1-1-onto→wf1o 5258 ‘cfv 5259 (class class class)co 5925 ∘_𝑓 cof 6137 ↑_𝑚 cmap 6716 Fincfn 6808 ℕcn 9007 ℕ₀cn0 9266 Basecbs 12703 +_gcplusg 12780 0_gc0g 12958 Grpcgrp 13202 inv_gcminusg 13203 mPwSer cmps 14293

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-addcom 7996 ax-addass 7998 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-0id 8004 ax-rnegex 8005 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-pw 3608 df-sn 3629 df-pr 3630 df-tp 3631 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-id 4329 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-of 6139 df-1st 6207 df-2nd 6208 df-map 6718 df-ixp 6767 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-5 9069 df-6 9070 df-7 9071 df-8 9072 df-9 9073 df-n0 9267 df-z 9344 df-uz 9619 df-fz 10101 df-struct 12705 df-ndx 12706 df-slot 12707 df-base 12709 df-plusg 12793 df-mulr 12794 df-sca 12796 df-vsca 12797 df-tset 12799 df-rest 12943 df-topn 12944 df-0g 12960 df-topgen 12962 df-pt 12963 df-mgm 13058 df-sgrp 13104 df-mnd 13119 df-grp 13205 df-minusg 13206 df-psr 14294

This theorem is referenced by: psrneg 14315

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