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Theorem psrlinv 14691

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 6819	. . . . 5 ⊢ ↑_𝑚 Fn (V × V)
3		nn0ex 9401	. . . . 5 ⊢ ℕ₀ ∈ V
4		psrgrp.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
5	4	elexd 2814	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
6		fnovex 6046	. . . . 5 ⊢ (( ↑_𝑚 Fn (V × V) ∧ ℕ₀ ∈ V ∧ 𝐼 ∈ V) → (ℕ₀ ↑_𝑚 𝐼) ∈ V)
7	2, 3, 5, 6	mp3an12i 1375	. . . 4 ⊢ (𝜑 → (ℕ₀ ↑_𝑚 𝐼) ∈ V)
8	1, 7	rabexd 4233	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
9		psrgrp.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
10		psrgrp.s	. . . . . 6 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
11		eqid 2229	. . . . . 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 14682	. . . . 5 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
15	14	ffvelcdmda 5778	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑋‘𝑥) ∈ (Base‘𝑅))
16		psrnegcl.i	. . . . 5 ⊢ 𝑁 = (inv_g‘𝑅)
17	11, 16	grpinvcl 13624	. . . 4 ⊢ ((𝑅 ∈ Grp ∧ (𝑋‘𝑥) ∈ (Base‘𝑅)) → (𝑁‘(𝑋‘𝑥)) ∈ (Base‘𝑅))
18	9, 15, 17	syl2an2r 597	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑁‘(𝑋‘𝑥)) ∈ (Base‘𝑅))
19	14	feqmptd 5695	. . . 4 ⊢ (𝜑 → 𝑋 = (𝑥 ∈ 𝐷 ↦ (𝑋‘𝑥)))
20	11, 16, 9	grpinvf1o 13646	. . . . . 6 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
21		f1of 5580	. . . . . 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 5695	. . . 4 ⊢ (𝜑 → 𝑁 = (𝑦 ∈ (Base‘𝑅) ↦ (𝑁‘𝑦)))
24		fveq2 5635	. . . 4 ⊢ (𝑦 = (𝑋‘𝑥) → (𝑁‘𝑦) = (𝑁‘(𝑋‘𝑥)))
25	15, 19, 23, 24	fmptco 5809	. . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐷 ↦ (𝑁‘(𝑋‘𝑥))))
26	8, 18, 15, 25, 19	offval2 6246	. 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) ∘_𝑓 (+_g‘𝑅)𝑋) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))))
27		eqid 2229	. . 3 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
28		psrlinv.p	. . 3 ⊢ + = (+_g‘𝑆)
29	10, 4, 9, 1, 16, 12, 13	psrnegcl 14690	. . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵)
30	10, 12, 27, 28, 29, 13	psradd 14686	. 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = ((𝑁 ∘ 𝑋) ∘_𝑓 (+_g‘𝑅)𝑋))
31		fconstmpt 4771	. . 3 ⊢ (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ 0 )
32		psrlinv.o	. . . . . 6 ⊢ 0 = (0_g‘𝑅)
33	11, 27, 32, 16	grplinv 13626	. . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (𝑋‘𝑥) ∈ (Base‘𝑅)) → ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥)) = 0 )
34	9, 15, 33	syl2an2r 597	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥)) = 0 )
35	34	mpteq2dva 4177	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))) = (𝑥 ∈ 𝐷 ↦ 0 ))
36	31, 35	eqtr4id 2281	. 2 ⊢ (𝜑 → (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))))
37	26, 30, 36	3eqtr4d 2272	1 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = (𝐷 × { 0 }))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1395 ∈ wcel 2200 {crab 2512 Vcvv 2800 {csn 3667 ↦ cmpt 4148 × cxp 4721 ^◡ccnv 4722 “ cima 4726 ∘ ccom 4727 Fn wfn 5319 ⟶wf 5320 –1-1-onto→wf1o 5323 ‘cfv 5324 (class class class)co 6013 ∘_𝑓 cof 6228 ↑_𝑚 cmap 6812 Fincfn 6904 ℕcn 9136 ℕ₀cn0 9395 Basecbs 13075 +_gcplusg 13153 0_gc0g 13332 Grpcgrp 13576 inv_gcminusg 13577 mPwSer cmps 14668

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-addcom 8125 ax-addass 8127 ax-distr 8129 ax-i2m1 8130 ax-0lt1 8131 ax-0id 8133 ax-rnegex 8134 ax-cnre 8136 ax-pre-ltirr 8137 ax-pre-ltwlin 8138 ax-pre-lttrn 8139 ax-pre-apti 8140 ax-pre-ltadd 8141

This theorem depends on definitions: df-bi 117 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-pw 3652 df-sn 3673 df-pr 3674 df-tp 3675 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-id 4388 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-of 6230 df-1st 6298 df-2nd 6299 df-map 6814 df-ixp 6863 df-pnf 8209 df-mnf 8210 df-xr 8211 df-ltxr 8212 df-le 8213 df-sub 8345 df-neg 8346 df-inn 9137 df-2 9195 df-3 9196 df-4 9197 df-5 9198 df-6 9199 df-7 9200 df-8 9201 df-9 9202 df-n0 9396 df-z 9473 df-uz 9749 df-fz 10237 df-struct 13077 df-ndx 13078 df-slot 13079 df-base 13081 df-plusg 13166 df-mulr 13167 df-sca 13169 df-vsca 13170 df-tset 13172 df-rest 13317 df-topn 13318 df-0g 13334 df-topgen 13336 df-pt 13337 df-mgm 13432 df-sgrp 13478 df-mnd 13493 df-grp 13579 df-minusg 13580 df-psr 14670

This theorem is referenced by: psrneg 14694

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