Theorem grpidlcan 13442

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 13407	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋)
5	4	3adant3 1020	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋)
6	5	eqeq2d 2218	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑋) = ( 0 + 𝑋) ↔ (𝑍 + 𝑋) = 𝑋))
7		simp1 1000	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → 𝐺 ∈ Grp)
8		simp3 1002	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ 𝐵)
9	1, 3	grpidcl 13405	. . . 4 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
10	9	3ad2ant1 1021	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → 0 ∈ 𝐵)
11		simp2 1001	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → 𝑋 ∈ 𝐵)
12	1, 2	grprcan 13413	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑍 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑍 + 𝑋) = ( 0 + 𝑋) ↔ 𝑍 = 0 ))
13	7, 8, 10, 11, 12	syl13anc 1252	. 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 981   = wceq 1373   ∈ wcel 2177  ‘cfv 5276  (class class class)co 5951  Basecbs 12876  +_gcplusg 12953  0_gc0g 13132  Grpcgrp 13376

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-cnex 8023  ax-resscn 8024  ax-1re 8026  ax-addrcl 8029

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-iota 5237  df-fun 5278  df-fn 5279  df-fv 5284  df-riota 5906  df-ov 5954  df-inn 9044  df-2 9102  df-ndx 12879  df-slot 12880  df-base 12882  df-plusg 12966  df-0g 13134  df-mgm 13232  df-sgrp 13278  df-mnd 13293  df-grp 13379

This theorem is referenced by:  grpidssd  13452