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Theorem psrlinv 19218

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 6577	. . . . 5 ⊢ (ℕ₀ ↑_𝑚 𝐼) ∈ V
3	1, 2	rabex2 4742	. . . 4 ⊢ 𝐷 ∈ V
4	3	a1i 11	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
5		fvex 6113	. . . 4 ⊢ (𝑁‘(𝑋‘𝑥)) ∈ V
6	5	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑁‘(𝑋‘𝑥)) ∈ V)
7		psrgrp.s	. . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
8		eqid 2610	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
9		psrnegcl.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑆)
10		psrnegcl.z	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11	7, 8, 1, 9, 10	psrelbas 19200	. . . 4 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
12	11	ffvelrnda 6267	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑋‘𝑥) ∈ (Base‘𝑅))
13	11	feqmptd 6159	. . . 4 ⊢ (𝜑 → 𝑋 = (𝑥 ∈ 𝐷 ↦ (𝑋‘𝑥)))
14		psrnegcl.i	. . . . . . 7 ⊢ 𝑁 = (inv_g‘𝑅)
15		psrgrp.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Grp)
16	8, 14, 15	grpinvf1o 17308	. . . . . 6 ⊢ (𝜑 → 𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅))
17		f1of 6050	. . . . . 6 ⊢ (𝑁:(Base‘𝑅)–1-1-onto→(Base‘𝑅) → 𝑁:(Base‘𝑅)⟶(Base‘𝑅))
18	16, 17	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁:(Base‘𝑅)⟶(Base‘𝑅))
19	18	feqmptd 6159	. . . 4 ⊢ (𝜑 → 𝑁 = (𝑦 ∈ (Base‘𝑅) ↦ (𝑁‘𝑦)))
20		fveq2 6103	. . . 4 ⊢ (𝑦 = (𝑋‘𝑥) → (𝑁‘𝑦) = (𝑁‘(𝑋‘𝑥)))
21	12, 13, 19, 20	fmptco 6303	. . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) = (𝑥 ∈ 𝐷 ↦ (𝑁‘(𝑋‘𝑥))))
22	4, 6, 12, 21, 13	offval2 6812	. 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) ∘_𝑓 (+_g‘𝑅)𝑋) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))))
23		eqid 2610	. . 3 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
24		psrlinv.p	. . 3 ⊢ + = (+_g‘𝑆)
25		psrgrp.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
26	7, 25, 15, 1, 14, 9, 10	psrnegcl 19217	. . 3 ⊢ (𝜑 → (𝑁 ∘ 𝑋) ∈ 𝐵)
27	7, 9, 23, 24, 26, 10	psradd 19203	. 2 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = ((𝑁 ∘ 𝑋) ∘_𝑓 (+_g‘𝑅)𝑋))
28		psrlinv.o	. . . . . 6 ⊢ 0 = (0_g‘𝑅)
29	8, 23, 28, 14	grplinv 17291	. . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (𝑋‘𝑥) ∈ (Base‘𝑅)) → ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥)) = 0 )
30	15, 12, 29	syl2an2r 872	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥)) = 0 )
31	30	mpteq2dva 4672	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))) = (𝑥 ∈ 𝐷 ↦ 0 ))
32		fconstmpt 5085	. . 3 ⊢ (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ 0 )
33	31, 32	syl6reqr 2663	. 2 ⊢ (𝜑 → (𝐷 × { 0 }) = (𝑥 ∈ 𝐷 ↦ ((𝑁‘(𝑋‘𝑥))(+_g‘𝑅)(𝑋‘𝑥))))
34	22, 27, 33	3eqtr4d 2654	1 ⊢ (𝜑 → ((𝑁 ∘ 𝑋) + 𝑋) = (𝐷 × { 0 }))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 {csn 4125 ↦ cmpt 4643 × cxp 5036 ^◡ccnv 5037 “ cima 5041 ∘ ccom 5042 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ∘_𝑓 cof 6793 ↑_𝑚 cmap 7744 Fincfn 7841 ℕcn 10897 ℕ₀cn0 11169 Basecbs 15695 +_gcplusg 15768 0_gc0g 15923 Grpcgrp 17245 inv_gcminusg 17246 mPwSer cmps 19172

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-tset 15787 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-psr 19177

This theorem is referenced by: psrgrp 19219 psrneg 19221

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