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Theorem grplidd 13095
Description: The identity element of a group is a left identity. Deduction associated with grplid 13093. (Contributed by SN, 29-Jan-2025.)
Hypotheses
Ref Expression
grpbn0.b  |-  B  =  ( Base `  G
)
grplid.p  |-  .+  =  ( +g  `  G )
grplid.o  |-  .0.  =  ( 0g `  G )
grplidd.g  |-  ( ph  ->  G  e.  Grp )
grplidd.1  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
grplidd  |-  ( ph  ->  (  .0.  .+  X
)  =  X )

Proof of Theorem grplidd
StepHypRef Expression
1 grplidd.g . 2  |-  ( ph  ->  G  e.  Grp )
2 grplidd.1 . 2  |-  ( ph  ->  X  e.  B )
3 grpbn0.b . . 3  |-  B  =  ( Base `  G
)
4 grplid.p . . 3  |-  .+  =  ( +g  `  G )
5 grplid.o . . 3  |-  .0.  =  ( 0g `  G )
63, 4, 5grplid 13093 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  .+  X
)  =  X )
71, 2, 6syl2anc 411 1  |-  ( ph  ->  (  .0.  .+  X
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2164   ` cfv 5246  (class class class)co 5910   Basecbs 12608   +g cplusg 12685   0gc0g 12857   Grpcgrp 13062
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-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-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-cnex 7953  ax-resscn 7954  ax-1re 7956  ax-addrcl 7959
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  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-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-iota 5207  df-fun 5248  df-fn 5249  df-fv 5254  df-riota 5865  df-ov 5913  df-inn 8973  df-2 9031  df-ndx 12611  df-slot 12612  df-base 12614  df-plusg 12698  df-0g 12859  df-mgm 12929  df-sgrp 12975  df-mnd 12988  df-grp 13065
This theorem is referenced by:  conjnmz  13338
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