Theorem grpidlcan 19102

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 19065	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋)
5	4	3adant3 1150	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋)
6	5	eqeq2d 2777	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑋) = ( 0 + 𝑋) ↔ (𝑍 + 𝑋) = 𝑋))
7		simp1 1154	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → 𝐺 ∈ Grp)
8		simp3 1156	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ 𝐵)
9	1, 3	grpidcl 19063	. . . 4 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
10	9	3ad2ant1 1151	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → 0 ∈ 𝐵)
11		simp2 1155	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → 𝑋 ∈ 𝐵)
12	1, 2	grprcan 19071	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑍 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑍 + 𝑋) = ( 0 + 𝑋) ↔ 𝑍 = 0 ))
13	7, 8, 10, 11, 12	syl13anc 1399	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑋) = ( 0 + 𝑋) ↔ 𝑍 = 0 ))
14	6, 13	bitr3d 284	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑋) = 𝑋 ↔ 𝑍 = 0 ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  +_gcplusg 17335  0_gc0g 17517  Grpcgrp 19031

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-riota 7380  df-ov 7426  df-0g 17519  df-mgm 18723  df-sgrp 18806  df-mnd 18822  df-grp 19034

This theorem is used by:  grpidssd  19113