Theorem psrgrpOLD 21917

Description: Obsolete version of psrgrp 21916 as of 7-Feb-2025. (Contributed by Mario Carneiro, 29-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
psrgrpOLD	⊢ (𝜑 → 𝑆 ∈ Grp)

Proof of Theorem psrgrpOLD

Dummy variables 𝑥 𝑠 𝑟 𝑡 𝑦 𝑧 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2736	. 2 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2		eqidd 2736	. 2 ⊢ (𝜑 → (+g‘𝑆) = (+g‘𝑆))
3		psrgrp.s	. . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
4		eqid 2735	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
5		eqid 2735	. . 3 ⊢ (+g‘𝑆) = (+g‘𝑆)
6		psrgrp.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp)
7	6	grpmgmd 18944	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Mgm)
8	7	3ad2ant1 1133	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Mgm)
9		simp2 1137	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
10		simp3 1138	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
11	3, 4, 5, 8, 9, 10	psraddcl 21898	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
12		ovex 7438	. . . . . . 7 ⊢ (ℕ0 ↑m 𝐼) ∈ V
13	12	rabex 5309	. . . . . 6 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V
14	13	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V)
15		eqid 2735	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
16		eqid 2735	. . . . . 6 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
17		simpr1 1195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
18	3, 15, 16, 4, 17	psrelbas 21894	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
19		simpr2 1196	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
20	3, 15, 16, 4, 19	psrelbas 21894	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
21		simpr3 1197	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
22	3, 15, 16, 4, 21	psrelbas 21894	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
23	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Grp)
24		eqid 2735	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
25	15, 24	grpass 18925	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(+g‘𝑅)𝑠)(+g‘𝑅)𝑡) = (𝑟(+g‘𝑅)(𝑠(+g‘𝑅)𝑡)))
26	23, 25	sylan 580	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(+g‘𝑅)𝑠)(+g‘𝑅)𝑡) = (𝑟(+g‘𝑅)(𝑠(+g‘𝑅)𝑡)))
27	14, 18, 20, 22, 26	caofass 7711	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥 ∘f (+g‘𝑅)𝑦) ∘f (+g‘𝑅)𝑧) = (𝑥 ∘f (+g‘𝑅)(𝑦 ∘f (+g‘𝑅)𝑧)))
28	3, 4, 24, 5, 17, 19	psradd 21897	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑆)𝑦) = (𝑥 ∘f (+g‘𝑅)𝑦))
29	28	oveq1d 7420	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g‘𝑆)𝑦) ∘f (+g‘𝑅)𝑧) = ((𝑥 ∘f (+g‘𝑅)𝑦) ∘f (+g‘𝑅)𝑧))
30	3, 4, 24, 5, 19, 21	psradd 21897	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g‘𝑆)𝑧) = (𝑦 ∘f (+g‘𝑅)𝑧))
31	30	oveq2d 7421	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥 ∘f (+g‘𝑅)(𝑦(+g‘𝑆)𝑧)) = (𝑥 ∘f (+g‘𝑅)(𝑦 ∘f (+g‘𝑅)𝑧)))
32	27, 29, 31	3eqtr4d 2780	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g‘𝑆)𝑦) ∘f (+g‘𝑅)𝑧) = (𝑥 ∘f (+g‘𝑅)(𝑦(+g‘𝑆)𝑧)))
33	11	3adant3r3 1185	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
34	3, 4, 24, 5, 33, 21	psradd 21897	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g‘𝑆)𝑦)(+g‘𝑆)𝑧) = ((𝑥(+g‘𝑆)𝑦) ∘f (+g‘𝑅)𝑧))
35	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Mgm)
36	3, 4, 5, 35, 19, 21	psraddcl 21898	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g‘𝑆)𝑧) ∈ (Base‘𝑆))
37	3, 4, 24, 5, 17, 36	psradd 21897	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑆)(𝑦(+g‘𝑆)𝑧)) = (𝑥 ∘f (+g‘𝑅)(𝑦(+g‘𝑆)𝑧)))
38	32, 34, 37	3eqtr4d 2780	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g‘𝑆)𝑦)(+g‘𝑆)𝑧) = (𝑥(+g‘𝑆)(𝑦(+g‘𝑆)𝑧)))
39		psrgrp.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
40		eqid 2735	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
41	3, 39, 6, 16, 40, 4	psr0cl 21912	. 2 ⊢ (𝜑 → ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(0g‘𝑅)}) ∈ (Base‘𝑆))
42	39	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝐼 ∈ 𝑉)
43	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Grp)
44		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
45	3, 42, 43, 16, 40, 4, 5, 44	psr0lid 21913	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(0g‘𝑅)})(+g‘𝑆)𝑥) = 𝑥)
46		eqid 2735	. . 3 ⊢ (invg‘𝑅) = (invg‘𝑅)
47	3, 42, 43, 16, 46, 4, 44	psrnegcl 21914	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → ((invg‘𝑅) ∘ 𝑥) ∈ (Base‘𝑆))
48	3, 42, 43, 16, 46, 4, 44, 40, 5	psrlinv 21915	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (((invg‘𝑅) ∘ 𝑥)(+g‘𝑆)𝑥) = ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(0g‘𝑅)}))
49	1, 2, 11, 38, 41, 45, 47, 48	isgrpd 18941	1 ⊢ (𝜑 → 𝑆 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  {crab 3415  Vcvv 3459  {csn 4601   × cxp 5652  ^◡ccnv 5653   “ cima 5657   ∘ ccom 5658  ‘cfv 6531  (class class class)co 7405   ∘_f cof 7669   ↑_m cmap 8840  Fincfn 8959  ℕcn 12240  ℕ₀cn0 12501  Basecbs 17228  +_gcplusg 17271  0_gc0g 17453  Mgmcmgm 18616  Grpcgrp 18916  inv_gcminusg 18917   mPwSer cmps 21864

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7671  df-om 7862  df-1st 7988  df-2nd 7989  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fsupp 9374  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-uz 12853  df-fz 13525  df-struct 17166  df-slot 17201  df-ndx 17213  df-base 17229  df-plusg 17284  df-mulr 17285  df-sca 17287  df-vsca 17288  df-tset 17290  df-0g 17455  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-grp 18919  df-minusg 18920  df-psr 21869

This theorem is referenced by: (None)