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Theorem psrgrpOLD 21977
Description: Obsolete version of psrgrp 21976 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 (𝜑𝐼𝑉)
psrgrp.r (𝜑𝑅 ∈ Grp)
Assertion
Ref Expression
psrgrpOLD (𝜑𝑆 ∈ Grp)

Proof of Theorem psrgrpOLD
Dummy variables 𝑥 𝑠 𝑟 𝑡 𝑦 𝑧 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2738 . 2 (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2 eqidd 2738 . 2 (𝜑 → (+g𝑆) = (+g𝑆))
3 psrgrp.s . . 3 𝑆 = (𝐼 mPwSer 𝑅)
4 eqid 2737 . . 3 (Base‘𝑆) = (Base‘𝑆)
5 eqid 2737 . . 3 (+g𝑆) = (+g𝑆)
6 psrgrp.r . . . . 5 (𝜑𝑅 ∈ Grp)
76grpmgmd 18979 . . . 4 (𝜑𝑅 ∈ Mgm)
873ad2ant1 1134 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Mgm)
9 simp2 1138 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
10 simp3 1139 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
113, 4, 5, 8, 9, 10psraddcl 21958 . 2 ((𝜑𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
12 ovex 7464 . . . . . . 7 (ℕ0m 𝐼) ∈ V
1312rabex 5339 . . . . . 6 {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V
1413a1i 11 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V)
15 eqid 2737 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
16 eqid 2737 . . . . . 6 {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
17 simpr1 1195 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
183, 15, 16, 4, 17psrelbas 21954 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
19 simpr2 1196 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
203, 15, 16, 4, 19psrelbas 21954 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
21 simpr3 1197 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
223, 15, 16, 4, 21psrelbas 21954 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
236adantr 480 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Grp)
24 eqid 2737 . . . . . . 7 (+g𝑅) = (+g𝑅)
2515, 24grpass 18960 . . . . . 6 ((𝑅 ∈ Grp ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(+g𝑅)𝑠)(+g𝑅)𝑡) = (𝑟(+g𝑅)(𝑠(+g𝑅)𝑡)))
2623, 25sylan 580 . . . . 5 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(+g𝑅)𝑠)(+g𝑅)𝑡) = (𝑟(+g𝑅)(𝑠(+g𝑅)𝑡)))
2714, 18, 20, 22, 26caofass 7737 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥f (+g𝑅)𝑦) ∘f (+g𝑅)𝑧) = (𝑥f (+g𝑅)(𝑦f (+g𝑅)𝑧)))
283, 4, 24, 5, 17, 19psradd 21957 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)𝑦) = (𝑥f (+g𝑅)𝑦))
2928oveq1d 7446 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g𝑆)𝑦) ∘f (+g𝑅)𝑧) = ((𝑥f (+g𝑅)𝑦) ∘f (+g𝑅)𝑧))
303, 4, 24, 5, 19, 21psradd 21957 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g𝑆)𝑧) = (𝑦f (+g𝑅)𝑧))
3130oveq2d 7447 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥f (+g𝑅)(𝑦(+g𝑆)𝑧)) = (𝑥f (+g𝑅)(𝑦f (+g𝑅)𝑧)))
3227, 29, 313eqtr4d 2787 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g𝑆)𝑦) ∘f (+g𝑅)𝑧) = (𝑥f (+g𝑅)(𝑦(+g𝑆)𝑧)))
33113adant3r3 1185 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
343, 4, 24, 5, 33, 21psradd 21957 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g𝑆)𝑦)(+g𝑆)𝑧) = ((𝑥(+g𝑆)𝑦) ∘f (+g𝑅)𝑧))
357adantr 480 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Mgm)
363, 4, 5, 35, 19, 21psraddcl 21958 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g𝑆)𝑧) ∈ (Base‘𝑆))
373, 4, 24, 5, 17, 36psradd 21957 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)(𝑦(+g𝑆)𝑧)) = (𝑥f (+g𝑅)(𝑦(+g𝑆)𝑧)))
3832, 34, 373eqtr4d 2787 . 2 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g𝑆)𝑦)(+g𝑆)𝑧) = (𝑥(+g𝑆)(𝑦(+g𝑆)𝑧)))
39 psrgrp.i . . 3 (𝜑𝐼𝑉)
40 eqid 2737 . . 3 (0g𝑅) = (0g𝑅)
413, 39, 6, 16, 40, 4psr0cl 21972 . 2 (𝜑 → ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)}) ∈ (Base‘𝑆))
4239adantr 480 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝐼𝑉)
436adantr 480 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Grp)
44 simpr 484 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
453, 42, 43, 16, 40, 4, 5, 44psr0lid 21973 . 2 ((𝜑𝑥 ∈ (Base‘𝑆)) → (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)})(+g𝑆)𝑥) = 𝑥)
46 eqid 2737 . . 3 (invg𝑅) = (invg𝑅)
473, 42, 43, 16, 46, 4, 44psrnegcl 21974 . 2 ((𝜑𝑥 ∈ (Base‘𝑆)) → ((invg𝑅) ∘ 𝑥) ∈ (Base‘𝑆))
483, 42, 43, 16, 46, 4, 44, 40, 5psrlinv 21975 . 2 ((𝜑𝑥 ∈ (Base‘𝑆)) → (((invg𝑅) ∘ 𝑥)(+g𝑆)𝑥) = ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(0g𝑅)}))
491, 2, 11, 38, 41, 45, 47, 48isgrpd 18976 1 (𝜑𝑆 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  {csn 4626   × cxp 5683  ccnv 5684  cima 5688  ccom 5689  cfv 6561  (class class class)co 7431  f cof 7695  m cmap 8866  Fincfn 8985  cn 12266  0cn0 12526  Basecbs 17247  +gcplusg 17297  0gc0g 17484  Mgmcmgm 18651  Grpcgrp 18951  invgcminusg 18952   mPwSer cmps 21924
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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-tset 17316  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-psr 21929
This theorem is referenced by: (None)
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