Theorem psrgrp 14702

Description: The ring of power series is a group. (Contributed by Mario Carneiro, 29-Dec-2014.) (Proof shortened by SN, 7-Feb-2025.)

Hypotheses

Ref	Expression
psrgrp.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
psrgrp.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
psrgrp.r	⊢ (𝜑 → 𝑅 ∈ Grp)

Assertion

Ref	Expression
psrgrp	⊢ (𝜑 → 𝑆 ∈ Grp)

Proof of Theorem psrgrp

Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrgrp.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Grp)
2		eqid 2231	. . . 4 ⊢ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
3		fnmap 6824	. . . . 5 ⊢ ↑𝑚 Fn (V × V)
4		nn0ex 9408	. . . . 5 ⊢ ℕ0 ∈ V
5		psrgrp.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
6	5	elexd 2816	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
7		fnovex 6051	. . . . 5 ⊢ (( ↑𝑚 Fn (V × V) ∧ ℕ0 ∈ V ∧ 𝐼 ∈ V) → (ℕ0 ↑𝑚 𝐼) ∈ V)
8	3, 4, 6, 7	mp3an12i 1377	. . . 4 ⊢ (𝜑 → (ℕ0 ↑𝑚 𝐼) ∈ V)
9	2, 8	rabexd 4235	. . 3 ⊢ (𝜑 → {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V)
10		eqid 2231	. . . 4 ⊢ (𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) = (𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})
11	10	pwsgrp 13696	. . 3 ⊢ ((𝑅 ∈ Grp ∧ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V) → (𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∈ Grp)
12	1, 9, 11	syl2anc 411	. 2 ⊢ (𝜑 → (𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∈ Grp)
13		eqid 2231	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
14	10, 13	pwsbas 13377	. . . 4 ⊢ ((𝑅 ∈ Grp ∧ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V) → ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) = (Base‘(𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})))
15	1, 9, 14	syl2anc 411	. . 3 ⊢ (𝜑 → ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) = (Base‘(𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})))
16		psrgrp.s	. . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
17		eqid 2231	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
18	16, 13, 2, 17, 5, 1	psrbasg 14691	. . . 4 ⊢ (𝜑 → (Base‘𝑆) = ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))
19	18	eqcomd 2237	. . 3 ⊢ (𝜑 → ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) = (Base‘𝑆))
20		eqid 2231	. . . . 5 ⊢ (Base‘(𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) = (Base‘(𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))
21	1	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))) → 𝑅 ∈ Grp)
22	9	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))) → {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V)
23	15	eleq2d 2301	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ↔ 𝑥 ∈ (Base‘(𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))))
24	23	biimpa 296	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) → 𝑥 ∈ (Base‘(𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})))
25	24	adantrr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))) → 𝑥 ∈ (Base‘(𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})))
26	15	eleq2d 2301	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ↔ 𝑦 ∈ (Base‘(𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))))
27	26	biimpa 296	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) → 𝑦 ∈ (Base‘(𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})))
28	27	adantrl 478	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))) → 𝑦 ∈ (Base‘(𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})))
29		eqid 2231	. . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅)
30		eqid 2231	. . . . 5 ⊢ (+g‘(𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) = (+g‘(𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))
31	10, 20, 21, 22, 25, 28, 29, 30	pwsplusgval 13380	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))) → (𝑥(+g‘(𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))𝑦) = (𝑥 ∘𝑓 (+g‘𝑅)𝑦))
32		eqid 2231	. . . . 5 ⊢ (+g‘𝑆) = (+g‘𝑆)
33	18	eleq2d 2301	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑆) ↔ 𝑥 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})))
34	33	biimpar 297	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) → 𝑥 ∈ (Base‘𝑆))
35	34	adantrr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))) → 𝑥 ∈ (Base‘𝑆))
36	18	eleq2d 2301	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ (Base‘𝑆) ↔ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})))
37	36	biimpar 297	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) → 𝑦 ∈ (Base‘𝑆))
38	37	adantrl 478	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))) → 𝑦 ∈ (Base‘𝑆))
39	16, 17, 29, 32, 35, 38	psradd 14696	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))) → (𝑥(+g‘𝑆)𝑦) = (𝑥 ∘𝑓 (+g‘𝑅)𝑦))
40	31, 39	eqtr4d 2267	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))) → (𝑥(+g‘(𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))𝑦) = (𝑥(+g‘𝑆)𝑦))
41	15, 19, 40	grppropd 13602	. 2 ⊢ (𝜑 → ((𝑅 ↑s {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∈ Grp ↔ 𝑆 ∈ Grp))
42	12, 41	mpbid 147	1 ⊢ (𝜑 → 𝑆 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  {crab 2514  Vcvv 2802   × cxp 4723  ^◡ccnv 4724   “ cima 4728   Fn wfn 5321  ‘cfv 5326  (class class class)co 6018   ∘_𝑓 cof 6233   ↑_𝑚 cmap 6817  Fincfn 6909  ℕcn 9143  ℕ₀cn0 9402  Basecbs 13084  +_gcplusg 13162   ↑_s cpws 13351  Grpcgrp 13585   mPwSer cmps 14678

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-tp 3677  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-of 6235  df-1st 6303  df-2nd 6304  df-map 6819  df-ixp 6868  df-sup 7183  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-z 9480  df-dec 9612  df-uz 9756  df-fz 10244  df-struct 13086  df-ndx 13087  df-slot 13088  df-base 13090  df-plusg 13175  df-mulr 13176  df-sca 13178  df-vsca 13179  df-ip 13180  df-tset 13181  df-ple 13182  df-ds 13184  df-hom 13186  df-cco 13187  df-rest 13326  df-topn 13327  df-0g 13343  df-topgen 13345  df-pt 13346  df-prds 13352  df-pws 13375  df-mgm 13441  df-sgrp 13487  df-mnd 13502  df-grp 13588  df-minusg 13589  df-psr 14680

This theorem is referenced by:  psr0  14703  psrneg  14704  mplsubgfi  14718