Theorem grpridd 12698

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 5876	. . . . . 6 ⊢ (𝑦 = 𝑛 → (𝑦 + 𝑥) = (𝑛 + 𝑥))
3	2	eqeq1d 2186	. . . . 5 ⊢ (𝑦 = 𝑛 → ((𝑦 + 𝑥) = 𝑂 ↔ (𝑛 + 𝑥) = 𝑂))
4	3	cbvrexvw 2708	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∃𝑛 ∈ 𝐵 (𝑛 + 𝑥) = 𝑂)
5	1, 4	sylib 122	. . 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 6027	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
8	7	adantlr 477	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
9		simprl 529	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → 𝑥 ∈ 𝐵)
10		simprrl 539	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → 𝑛 ∈ 𝐵)
11	8, 9, 10, 9	caovassd 6028	. . . . 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 540	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑛 + 𝑥) = 𝑂)
16	12, 13, 14, 6, 1, 9, 10, 15	grprinvd 12697	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑥 + 𝑛) = 𝑂)
17	16	oveq1d 5884	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → ((𝑥 + 𝑛) + 𝑥) = (𝑂 + 𝑥))
18	15	oveq2d 5885	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑥 + (𝑛 + 𝑥)) = (𝑥 + 𝑂))
19	11, 17, 18	3eqtr3d 2218	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
20	19	anassrs 400	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂)) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
21	5, 20	rexlimddv 2599	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
22	21, 14	eqtr3d 2212	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑂) = 𝑥)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ∃wrex 2456  (class class class)co 5869

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-iota 5174  df-fv 5220  df-ov 5872

This theorem is referenced by:  isgrpde  12788