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Theorem ablressid 13408
Description: A commutative group restricted to its base set is a commutative group. It will usually be the original group exactly, of course, but to show that needs additional conditions such as those in strressid 12692. (Contributed by Jim Kingdon, 5-May-2025.)
Hypothesis
Ref Expression
ablressid.b 𝐵 = (Base‘𝐺)
Assertion
Ref Expression
ablressid (𝐺 ∈ Abel → (𝐺s 𝐵) ∈ Abel)

Proof of Theorem ablressid
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2194 . . 3 (𝐺 ∈ Abel → (𝐺s 𝐵) = (𝐺s 𝐵))
2 ablressid.b . . . 4 𝐵 = (Base‘𝐺)
32a1i 9 . . 3 (𝐺 ∈ Abel → 𝐵 = (Base‘𝐺))
4 id 19 . . 3 (𝐺 ∈ Abel → 𝐺 ∈ Abel)
5 ssidd 3201 . . 3 (𝐺 ∈ Abel → 𝐵𝐵)
61, 3, 4, 5ressbas2d 12689 . 2 (𝐺 ∈ Abel → 𝐵 = (Base‘(𝐺s 𝐵)))
7 eqidd 2194 . . 3 (𝐺 ∈ Abel → (+g𝐺) = (+g𝐺))
8 basfn 12679 . . . . 5 Base Fn V
9 elex 2771 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ V)
10 funfvex 5572 . . . . . 6 ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
1110funfni 5355 . . . . 5 ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
128, 9, 11sylancr 414 . . . 4 (𝐺 ∈ Abel → (Base‘𝐺) ∈ V)
132, 12eqeltrid 2280 . . 3 (𝐺 ∈ Abel → 𝐵 ∈ V)
141, 7, 13, 9ressplusgd 12749 . 2 (𝐺 ∈ Abel → (+g𝐺) = (+g‘(𝐺s 𝐵)))
15 ablgrp 13362 . . 3 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
162grpressid 13136 . . 3 (𝐺 ∈ Grp → (𝐺s 𝐵) ∈ Grp)
1715, 16syl 14 . 2 (𝐺 ∈ Abel → (𝐺s 𝐵) ∈ Grp)
18 eqid 2193 . . 3 (+g𝐺) = (+g𝐺)
192, 18ablcom 13376 . 2 ((𝐺 ∈ Abel ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
206, 14, 17, 19isabld 13372 1 (𝐺 ∈ Abel → (𝐺s 𝐵) ∈ Abel)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2164  Vcvv 2760   Fn wfn 5250  cfv 5255  (class class class)co 5919  Basecbs 12621  s cress 12622  +gcplusg 12698  Grpcgrp 13075  Abelcabl 13358
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-ltadd 7990
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079  df-cmn 13359  df-abl 13360
This theorem is referenced by:  rngressid  13453
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