Theorem grplrinv 13701

Description: In a group, every member has a left and right inverse. (Contributed by AV, 1-Sep-2021.)

Hypotheses

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

Assertion

Ref	Expression
grplrinv	⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ))

Distinct variable groups:   𝑦,𝐵   𝑥,𝐺,𝑦   𝑦, +   𝑦, 0

Allowed substitution hints:   𝐵(𝑥)   + (𝑥)   0 (𝑥)

Proof of Theorem grplrinv

Step	Hyp	Ref	Expression
1		grplrinv.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2		eqid 2231	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	grpinvcl 13692	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
4		oveq1 6035	. . . . . 6 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → (𝑦 + 𝑥) = (((invg‘𝐺)‘𝑥) + 𝑥))
5	4	eqeq1d 2240	. . . . 5 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → ((𝑦 + 𝑥) = 0 ↔ (((invg‘𝐺)‘𝑥) + 𝑥) = 0 ))
6		oveq2 6036	. . . . . 6 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → (𝑥 + 𝑦) = (𝑥 + ((invg‘𝐺)‘𝑥)))
7	6	eqeq1d 2240	. . . . 5 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → ((𝑥 + 𝑦) = 0 ↔ (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 ))
8	5, 7	anbi12d 473	. . . 4 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → (((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ) ↔ ((((invg‘𝐺)‘𝑥) + 𝑥) = 0 ∧ (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 )))
9	8	adantl 277	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 = ((invg‘𝐺)‘𝑥)) → (((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ) ↔ ((((invg‘𝐺)‘𝑥) + 𝑥) = 0 ∧ (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 )))
10		grplrinv.p	. . . . 5 ⊢ + = (+g‘𝐺)
11		grplrinv.i	. . . . 5 ⊢ 0 = (0g‘𝐺)
12	1, 10, 11, 2	grplinv 13694	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (((invg‘𝐺)‘𝑥) + 𝑥) = 0 )
13	1, 10, 11, 2	grprinv 13695	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 )
14	12, 13	jca 306	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((((invg‘𝐺)‘𝑥) + 𝑥) = 0 ∧ (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 ))
15	3, 9, 14	rspcedvd 2917	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ))
16	15	ralrimiva 2606	1 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512  ‘cfv 5333  (class class class)co 6028  Basecbs 13143  +_gcplusg 13221  0_gc0g 13400  Grpcgrp 13644  inv_gcminusg 13645

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-inn 9187  df-2 9245  df-ndx 13146  df-slot 13147  df-base 13149  df-plusg 13234  df-0g 13402  df-mgm 13500  df-sgrp 13546  df-mnd 13561  df-grp 13647  df-minusg 13648

This theorem is referenced by:  grpidinv2  13702