Theorem grplrinv 18966

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 2737	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	grpinvcl 18957	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
4		oveq1 7368	. . . . . 6 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → (𝑦 + 𝑥) = (((invg‘𝐺)‘𝑥) + 𝑥))
5	4	eqeq1d 2739	. . . . 5 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → ((𝑦 + 𝑥) = 0 ↔ (((invg‘𝐺)‘𝑥) + 𝑥) = 0 ))
6		oveq2 7369	. . . . . 6 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → (𝑥 + 𝑦) = (𝑥 + ((invg‘𝐺)‘𝑥)))
7	6	eqeq1d 2739	. . . . 5 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → ((𝑥 + 𝑦) = 0 ↔ (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 ))
8	5, 7	anbi12d 633	. . . 4 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → (((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ) ↔ ((((invg‘𝐺)‘𝑥) + 𝑥) = 0 ∧ (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 )))
9	8	adantl 481	. . 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 18959	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (((invg‘𝐺)‘𝑥) + 𝑥) = 0 )
13	1, 10, 11, 2	grprinv 18960	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 )
14	12, 13	jca 511	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((((invg‘𝐺)‘𝑥) + 𝑥) = 0 ∧ (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 ))
15	3, 9, 14	rspcedvd 3567	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ))
16	15	ralrimiva 3130	1 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  ‘cfv 6493  (class class class)co 7361  Basecbs 17173  +_gcplusg 17214  0_gc0g 17396  Grpcgrp 18903  inv_gcminusg 18904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-riota 7318  df-ov 7364  df-0g 17398  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-grp 18906  df-minusg 18907

This theorem is referenced by:  grpidinv2  18967