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Theorem psrlinv 14727

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 6829	. . . . 5 ⊢ ↑_𝑚 Fn (V × V)
3		nn0ex 9413	. . . . 5 ⊢ ℕ₀ ∈ V
4		psrgrp.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
5	4	elexd 2815	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
6		fnovex 6056	. . . . 5 ⊢ (( ↑_𝑚 Fn (V × V) ∧ ℕ₀ ∈ V ∧ 𝐼 ∈ V) → (ℕ₀ ↑_𝑚 𝐼) ∈ V)
7	2, 3, 5, 6	mp3an12i 1377	. . . 4 ⊢ (𝜑 → (ℕ₀ ↑_𝑚 𝐼) ∈ V)
8	1, 7	rabexd 4236	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
9		psrgrp.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
10		psrgrp.s	. . . . . 6 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
11		eqid 2230	. . . . . 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 14718	. . . . 5 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
15	14	ffvelcdmda 5785	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑋‘𝑥) ∈ (Base‘𝑅))
16		psrnegcl.i	. . . . 5 ⊢ 𝑁 = (inv_g‘𝑅)
17	11, 16	grpinvcl 13654	. . . 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 5702	. . . 4 ⊢ (𝜑 → 𝑋 = (𝑥 ∈ 𝐷 ↦ (𝑋‘𝑥)))
20	11, 16, 9	grpinvf1o 13676	. . . . . 6 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
21		f1of 5586	. . . . . 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 5702	. . . 4 ⊢ (𝜑 → 𝑁 = (𝑦 ∈ (Base‘𝑅) ↦ (𝑁‘𝑦)))
24		fveq2 5642	. . . 4 ⊢ (𝑦 = (𝑋‘𝑥) → (𝑁‘𝑦) = (𝑁‘(𝑋‘𝑥)))
25	15, 19, 23, 24	fmptco 5816	. . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐷 ↦ (𝑁‘(𝑋‘𝑥))))
26	8, 18, 15, 25, 19	offval2 6256	. 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) ∘_𝑓 (+_g‘𝑅)𝑋) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))))
27		eqid 2230	. . 3 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
28		psrlinv.p	. . 3 ⊢ + = (+_g‘𝑆)
29	10, 4, 9, 1, 16, 12, 13	psrnegcl 14726	. . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵)
30	10, 12, 27, 28, 29, 13	psradd 14722	. 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = ((𝑁 ∘ 𝑋) ∘_𝑓 (+_g‘𝑅)𝑋))
31		fconstmpt 4775	. . 3 ⊢ (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ 0 )
32		psrlinv.o	. . . . . 6 ⊢ 0 = (0_g‘𝑅)
33	11, 27, 32, 16	grplinv 13656	. . . . 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 4180	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))) = (𝑥 ∈ 𝐷 ↦ 0 ))
36	31, 35	eqtr4id 2282	. 2 ⊢ (𝜑 → (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))))
37	26, 30, 36	3eqtr4d 2273	1 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = (𝐷 × { 0 }))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1397 ∈ wcel 2201 {crab 2513 Vcvv 2801 {csn 3670 ↦ cmpt 4151 × cxp 4725 ^◡ccnv 4726 “ cima 4730 ∘ ccom 4731 Fn wfn 5323 ⟶wf 5324 –1-1-onto→wf1o 5327 ‘cfv 5328 (class class class)co 6023 ∘_𝑓 cof 6238 ↑_𝑚 cmap 6822 Fincfn 6914 ℕcn 9148 ℕ₀cn0 9407 Basecbs 13105 +_gcplusg 13183 0_gc0g 13362 Grpcgrp 13606 inv_gcminusg 13607 mPwSer cmps 14699

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2203 ax-14 2204 ax-ext 2212 ax-coll 4205 ax-sep 4208 ax-pow 4266 ax-pr 4301 ax-un 4532 ax-setind 4637 ax-cnex 8128 ax-resscn 8129 ax-1cn 8130 ax-1re 8131 ax-icn 8132 ax-addcl 8133 ax-addrcl 8134 ax-mulcl 8135 ax-addcom 8137 ax-addass 8139 ax-distr 8141 ax-i2m1 8142 ax-0lt1 8143 ax-0id 8145 ax-rnegex 8146 ax-cnre 8148 ax-pre-ltirr 8149 ax-pre-ltwlin 8150 ax-pre-lttrn 8151 ax-pre-apti 8152 ax-pre-ltadd 8153

This theorem depends on definitions: df-bi 117 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1810 df-eu 2081 df-mo 2082 df-clab 2217 df-cleq 2223 df-clel 2226 df-nfc 2362 df-ne 2402 df-nel 2497 df-ral 2514 df-rex 2515 df-reu 2516 df-rmo 2517 df-rab 2518 df-v 2803 df-sbc 3031 df-csb 3127 df-dif 3201 df-un 3203 df-in 3205 df-ss 3212 df-nul 3494 df-pw 3655 df-sn 3676 df-pr 3677 df-tp 3678 df-op 3679 df-uni 3895 df-int 3930 df-iun 3973 df-br 4090 df-opab 4152 df-mpt 4153 df-id 4392 df-xp 4733 df-rel 4734 df-cnv 4735 df-co 4736 df-dm 4737 df-rn 4738 df-res 4739 df-ima 4740 df-iota 5288 df-fun 5330 df-fn 5331 df-f 5332 df-f1 5333 df-fo 5334 df-f1o 5335 df-fv 5336 df-riota 5976 df-ov 6026 df-oprab 6027 df-mpo 6028 df-of 6240 df-1st 6308 df-2nd 6309 df-map 6824 df-ixp 6873 df-pnf 8221 df-mnf 8222 df-xr 8223 df-ltxr 8224 df-le 8225 df-sub 8357 df-neg 8358 df-inn 9149 df-2 9207 df-3 9208 df-4 9209 df-5 9210 df-6 9211 df-7 9212 df-8 9213 df-9 9214 df-n0 9408 df-z 9485 df-uz 9761 df-fz 10249 df-struct 13107 df-ndx 13108 df-slot 13109 df-base 13111 df-plusg 13196 df-mulr 13197 df-sca 13199 df-vsca 13200 df-tset 13202 df-rest 13347 df-topn 13348 df-0g 13364 df-topgen 13366 df-pt 13367 df-mgm 13462 df-sgrp 13508 df-mnd 13523 df-grp 13609 df-minusg 13610 df-psr 14701

This theorem is referenced by: psrneg 14730

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