Theorem psrgrp 14766

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 6867	. . . . 5 ⊢ ↑𝑚 Fn (V × V)
4		nn0ex 9451	. . . . 5 ⊢ ℕ0 ∈ V
5		psrgrp.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
6	5	elexd 2817	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
7		fnovex 6061	. . . . 5 ⊢ (( ↑𝑚 Fn (V × V) ∧ ℕ0 ∈ V ∧ 𝐼 ∈ V) → (ℕ0 ↑𝑚 𝐼) ∈ V)
8	3, 4, 6, 7	mp3an12i 1378	. . . 4 ⊢ (𝜑 → (ℕ0 ↑𝑚 𝐼) ∈ V)
9	2, 8	rabexd 4240	. . 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 13755	. . 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 13436	. . . 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 14755	. . . 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 13439	. . . 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 14760	. . . 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 13661	. 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 1398   ∈ wcel 2202  {crab 2515  Vcvv 2803   × cxp 4729  ^◡ccnv 4730   “ cima 4734   Fn wfn 5328  ‘cfv 5333  (class class class)co 6028   ∘_𝑓 cof 6242   ↑_𝑚 cmap 6860  Fincfn 6952  ℕcn 9186  ℕ₀cn0 9445  Basecbs 13143  +_gcplusg 13221   ↑_s cpws 13410  Grpcgrp 13644   mPwSer cmps 14737

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-of 6244  df-1st 6312  df-2nd 6313  df-map 6862  df-ixp 6911  df-sup 7226  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-9 9252  df-n0 9446  df-z 9523  df-dec 9655  df-uz 9799  df-fz 10287  df-struct 13145  df-ndx 13146  df-slot 13147  df-base 13149  df-plusg 13234  df-mulr 13235  df-sca 13237  df-vsca 13238  df-ip 13239  df-tset 13240  df-ple 13241  df-ds 13243  df-hom 13245  df-cco 13246  df-rest 13385  df-topn 13386  df-0g 13402  df-topgen 13404  df-pt 13405  df-prds 13411  df-pws 13434  df-mgm 13500  df-sgrp 13546  df-mnd 13561  df-grp 13647  df-minusg 13648  df-psr 14739

This theorem is referenced by:  psr0  14767  psrneg  14768  mplsubgfi  14782