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Theorem psrlinv 21871

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 7423	. . . . 5 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
3	1, 2	rabex2 5299	. . . 4 ⊢ 𝐷 ∈ V
4	3	a1i 11	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
5		fvexd 6876	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑁‘(𝑋‘𝑥)) ∈ V)
6		psrgrp.s	. . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
7		eqid 2730	. . . . 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 21850	. . . 4 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
11	10	ffvelcdmda 7059	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑋‘𝑥) ∈ (Base‘𝑅))
12	10	feqmptd 6932	. . . 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 18948	. . . . . 6 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
16		f1of 6803	. . . . . 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 6932	. . . 4 ⊢ (𝜑 → 𝑁 = (𝑦 ∈ (Base‘𝑅) ↦ (𝑁‘𝑦)))
19		fveq2 6861	. . . 4 ⊢ (𝑦 = (𝑋‘𝑥) → (𝑁‘𝑦) = (𝑁‘(𝑋‘𝑥)))
20	11, 12, 18, 19	fmptco 7104	. . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐷 ↦ (𝑁‘(𝑋‘𝑥))))
21	4, 5, 11, 20, 12	offval2 7676	. 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) ∘_f (+_g‘𝑅)𝑋) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))))
22		eqid 2730	. . 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 21870	. . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵)
26	6, 8, 22, 23, 25, 9	psradd 21853	. 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = ((𝑁 ∘ 𝑋) ∘_f (+_g‘𝑅)𝑋))
27		fconstmpt 5703	. . 3 ⊢ (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ 0 )
28		psrlinv.o	. . . . . 6 ⊢ 0 = (0_g‘𝑅)
29	7, 22, 28, 13	grplinv 18928	. . . . 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 5203	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))) = (𝑥 ∈ 𝐷 ↦ 0 ))
32	27, 31	eqtr4id 2784	. 2 ⊢ (𝜑 → (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))))
33	21, 26, 32	3eqtr4d 2775	1 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = (𝐷 × { 0 }))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {crab 3408 Vcvv 3450 {csn 4592 ↦ cmpt 5191 × cxp 5639 ^◡ccnv 5640 “ cima 5644 ∘ ccom 5645 ⟶wf 6510 –1-1-onto→wf1o 6513 ‘cfv 6514 (class class class)co 7390 ∘_f cof 7654 ↑_m cmap 8802 Fincfn 8921 ℕcn 12193 ℕ₀cn0 12449 Basecbs 17186 +_gcplusg 17227 0_gc0g 17409 Grpcgrp 18872 inv_gcminusg 18873 mPwSer cmps 21820

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-struct 17124 df-slot 17159 df-ndx 17171 df-base 17187 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-tset 17246 df-0g 17411 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-grp 18875 df-minusg 18876 df-psr 21825

This theorem is referenced by: psrgrpOLD 21873 psrneg 21875

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