Theorem grpidlcan 13025

Description: If left adding an element of a group to an arbitrary element of the group results in this element, the added element is the identity element and vice versa. (Contributed by AV, 15-Mar-2019.)

Hypotheses

Ref	Expression
grpidrcan.b	⊢ 𝐵 = (Base‘𝐺)
grpidrcan.p	⊢ + = (+g‘𝐺)
grpidrcan.o	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
grpidlcan	⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑋) = 𝑋 ↔ 𝑍 = 0 ))

Proof of Theorem grpidlcan

Step	Hyp	Ref	Expression
1		grpidrcan.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
2		grpidrcan.p	. . . . 5 ⊢ + = (+g‘𝐺)
3		grpidrcan.o	. . . . 5 ⊢ 0 = (0g‘𝐺)
4	1, 2, 3	grplid 12990	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋)
5	4	3adant3 1019	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋)
6	5	eqeq2d 2201	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑋) = ( 0 + 𝑋) ↔ (𝑍 + 𝑋) = 𝑋))
7		simp1 999	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → 𝐺 ∈ Grp)
8		simp3 1001	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ 𝐵)
9	1, 3	grpidcl 12988	. . . 4 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
10	9	3ad2ant1 1020	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → 0 ∈ 𝐵)
11		simp2 1000	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → 𝑋 ∈ 𝐵)
12	1, 2	grprcan 12996	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑍 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑍 + 𝑋) = ( 0 + 𝑋) ↔ 𝑍 = 0 ))
13	7, 8, 10, 11, 12	syl13anc 1251	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑋) = ( 0 + 𝑋) ↔ 𝑍 = 0 ))
14	6, 13	bitr3d 190	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑋) = 𝑋 ↔ 𝑍 = 0 ))

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2160  ‘cfv 5235  (class class class)co 5897  Basecbs 12515  +_gcplusg 12592  0_gc0g 12764  Grpcgrp 12960

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-cnex 7933  ax-resscn 7934  ax-1re 7936  ax-addrcl 7939

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-riota 5852  df-ov 5900  df-inn 8951  df-2 9009  df-ndx 12518  df-slot 12519  df-base 12521  df-plusg 12605  df-0g 12766  df-mgm 12835  df-sgrp 12880  df-mnd 12893  df-grp 12963

This theorem is referenced by:  grpidssd  13035