Theorem grpridd 5717

Description: Deduce right identity from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
grprinvlem.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
grprinvlem.o	⊢ (𝜑 → 𝑂 ∈ 𝐵)
grprinvlem.i	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = 𝑥)
grprinvlem.a	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
grprinvlem.n	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂)

Assertion

Ref	Expression
grpridd	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑂) = 𝑥)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝑂,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥, + ,𝑦,𝑧

Proof of Theorem grpridd

Dummy variables 𝑢 𝑛 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grprinvlem.n	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂)
2		oveq1 5539	. . . . . 6 ⊢ (𝑦 = 𝑛 → (𝑦 + 𝑥) = (𝑛 + 𝑥))
3	2	eqeq1d 2089	. . . . 5 ⊢ (𝑦 = 𝑛 → ((𝑦 + 𝑥) = 𝑂 ↔ (𝑛 + 𝑥) = 𝑂))
4	3	cbvrexv 2578	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∃𝑛 ∈ 𝐵 (𝑛 + 𝑥) = 𝑂)
5	1, 4	sylib 120	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑛 ∈ 𝐵 (𝑛 + 𝑥) = 𝑂)
6		grprinvlem.a	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
7	6	caovassg 5679	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
8	7	adantlr 460	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
9		simprl 497	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → 𝑥 ∈ 𝐵)
10		simprrl 505	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → 𝑛 ∈ 𝐵)
11	8, 9, 10, 9	caovassd 5680	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → ((𝑥 + 𝑛) + 𝑥) = (𝑥 + (𝑛 + 𝑥)))
12		grprinvlem.c	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
13		grprinvlem.o	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ 𝐵)
14		grprinvlem.i	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = 𝑥)
15		simprrr 506	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑛 + 𝑥) = 𝑂)
16	12, 13, 14, 6, 1, 9, 10, 15	grprinvd 5716	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑥 + 𝑛) = 𝑂)
17	16	oveq1d 5547	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → ((𝑥 + 𝑛) + 𝑥) = (𝑂 + 𝑥))
18	15	oveq2d 5548	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑥 + (𝑛 + 𝑥)) = (𝑥 + 𝑂))
19	11, 17, 18	3eqtr3d 2121	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
20	19	anassrs 392	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂)) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
21	5, 20	rexlimddv 2481	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
22	21, 14	eqtr3d 2115	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑂) = 𝑥)

Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ∃wrex 2349  (class class class)co 5532

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by: (None)