Theorem psrgrpOLD 22000

Description: Obsolete proof of psrgrp 21999 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 2741	. 2 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2		eqidd 2741	. 2 ⊢ (𝜑 → (+g‘𝑆) = (+g‘𝑆))
3		psrgrp.s	. . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
4		eqid 2740	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
5		eqid 2740	. . 3 ⊢ (+g‘𝑆) = (+g‘𝑆)
6		psrgrp.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp)
7	6	grpmgmd 19001	. . . 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 21981	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
12		ovex 7481	. . . . . . 7 ⊢ (ℕ0 ↑m 𝐼) ∈ V
13	12	rabex 5357	. . . . . 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 2740	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
16		eqid 2740	. . . . . 6 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
17		simpr1 1194	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
18	3, 15, 16, 4, 17	psrelbas 21977	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
19		simpr2 1195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
20	3, 15, 16, 4, 19	psrelbas 21977	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
21		simpr3 1196	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
22	3, 15, 16, 4, 21	psrelbas 21977	. . . . 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 2740	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
25	15, 24	grpass 18982	. . . . . 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 579	. . . . 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 7752	. . . 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 21980	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑆)𝑦) = (𝑥 ∘f (+g‘𝑅)𝑦))
29	28	oveq1d 7463	. . . 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 21980	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g‘𝑆)𝑧) = (𝑦 ∘f (+g‘𝑅)𝑧))
31	30	oveq2d 7464	. . . 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 2790	. . 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 1184	. . . 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 21980	. . 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 21981	. . . 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 21980	. . 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 2790	. 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 2740	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
41	3, 39, 6, 16, 40, 4	psr0cl 21995	. 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 21996	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(0g‘𝑅)})(+g‘𝑆)𝑥) = 𝑥)
46		eqid 2740	. . 3 ⊢ (invg‘𝑅) = (invg‘𝑅)
47	3, 42, 43, 16, 46, 4, 44	psrnegcl 21997	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → ((invg‘𝑅) ∘ 𝑥) ∈ (Base‘𝑆))
48	3, 42, 43, 16, 46, 4, 44, 40, 5	psrlinv 21998	. 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 18998	1 ⊢ (𝜑 → 𝑆 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  {crab 3443  Vcvv 3488  {csn 4648   × cxp 5698  ^◡ccnv 5699   “ cima 5703   ∘ ccom 5704  ‘cfv 6573  (class class class)co 7448   ∘_f cof 7712   ↑_m cmap 8884  Fincfn 9003  ℕcn 12293  ℕ₀cn0 12553  Basecbs 17258  +_gcplusg 17311  0_gc0g 17499  Mgmcmgm 18676  Grpcgrp 18973  inv_gcminusg 18974   mPwSer cmps 21947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-struct 17194  df-slot 17229  df-ndx 17241  df-base 17259  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-tset 17330  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977  df-psr 21952

This theorem is referenced by: (None)