Theorem grpidinv2 19022

Description: A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised 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
grpidinv2	⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → ((( 0 + 𝐴) = 𝐴 ∧ (𝐴 + 0 ) = 𝐴) ∧ ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))

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

Proof of Theorem grpidinv2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grplrinv.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		grplrinv.p	. . 3 ⊢ + = (+g‘𝐺)
3		grplrinv.i	. . 3 ⊢ 0 = (0g‘𝐺)
4	1, 2, 3	grplid 18992	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → ( 0 + 𝐴) = 𝐴)
5	1, 2, 3	grprid 18993	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → (𝐴 + 0 ) = 𝐴)
6	1, 2, 3	grplrinv 19021	. . 3 ⊢ (𝐺 ∈ Grp → ∀𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ))
7		oveq2 7400	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (𝑦 + 𝑧) = (𝑦 + 𝐴))
8	7	eqeq1d 2763	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝑦 + 𝑧) = 0 ↔ (𝑦 + 𝐴) = 0 ))
9		oveq1 7399	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (𝑧 + 𝑦) = (𝐴 + 𝑦))
10	9	eqeq1d 2763	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝑧 + 𝑦) = 0 ↔ (𝐴 + 𝑦) = 0 ))
11	8, 10	anbi12d 641	. . . . 5 ⊢ (𝑧 = 𝐴 → (((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) ↔ ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
12	11	rexbidv 3185	. . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) ↔ ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
13	12	rspcv 3577	. . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) → ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
14	6, 13	mpan9 514	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 ))
15	4, 5, 14	jca31 522	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → ((( 0 + 𝐴) = 𝐴 ∧ (𝐴 + 0 ) = 𝐴) ∧ ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  +_gcplusg 17269  0_gc0g 17451  Grpcgrp 18958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fv 6525  df-riota 7349  df-ov 7395  df-0g 17453  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-grp 18961  df-minusg 18962

This theorem is referenced by:  grpidinv  19023