Theorem grpraddf1o 18988

Description: Right addition by a group element is a bijection on any group. (Contributed by SN, 28-Apr-2012.)

Hypotheses

Ref	Expression
grpraddf1o.b	⊢ 𝐵 = (Base‘𝐺)
grpraddf1o.p	⊢ + = (+g‘𝐺)
grpraddf1o.n	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 + 𝑋))

Assertion

Ref	Expression
grpraddf1o	⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝐹:𝐵–1-1-onto→𝐵)

Distinct variable groups:   𝑥,𝐵   𝑥,𝐺   𝑥, +   𝑥,𝑋

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem grpraddf1o

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpraddf1o.n	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 + 𝑋))
2		grpraddf1o.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
3		grpraddf1o.p	. . 3 ⊢ + = (+g‘𝐺)
4		simpll 772	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐺 ∈ Grp)
5		simpr 485	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
6		simplr 774	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ 𝐵)
7	2, 3, 4, 5, 6	grpcld 18921	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑋) ∈ 𝐵)
8		simpll 772	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝐺 ∈ Grp)
9		simpr 485	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
10		eqid 2740	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
11		simplr 774	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝐵)
12	2, 10, 8, 11	grpinvcld 18962	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((invg‘𝐺)‘𝑋) ∈ 𝐵)
13	2, 3, 8, 9, 12	grpcld 18921	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑦 + ((invg‘𝐺)‘𝑋)) ∈ 𝐵)
14		eqcom 2747	. . 3 ⊢ (𝑥 = (𝑦 + ((invg‘𝐺)‘𝑋)) ↔ (𝑦 + ((invg‘𝐺)‘𝑋)) = 𝑥)
15		simpll 772	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Grp)
16	13	adantrl 722	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 + ((invg‘𝐺)‘𝑋)) ∈ 𝐵)
17		simprl 776	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
18		simplr 774	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
19	2, 3	grprcan 18947	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ ((𝑦 + ((invg‘𝐺)‘𝑋)) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (((𝑦 + ((invg‘𝐺)‘𝑋)) + 𝑋) = (𝑥 + 𝑋) ↔ (𝑦 + ((invg‘𝐺)‘𝑋)) = 𝑥))
20	15, 16, 17, 18, 19	syl13anc 1380	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((𝑦 + ((invg‘𝐺)‘𝑋)) + 𝑋) = (𝑥 + 𝑋) ↔ (𝑦 + ((invg‘𝐺)‘𝑋)) = 𝑥))
21		simprr 778	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
22	12	adantrl 722	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((invg‘𝐺)‘𝑋) ∈ 𝐵)
23	2, 3, 15, 21, 22, 18	grpassd 18919	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑦 + ((invg‘𝐺)‘𝑋)) + 𝑋) = (𝑦 + (((invg‘𝐺)‘𝑋) + 𝑋)))
24		eqid 2740	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
25	2, 3, 24, 10	grplinv 18963	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (((invg‘𝐺)‘𝑋) + 𝑋) = (0g‘𝐺))
26	25	adantr 481	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((invg‘𝐺)‘𝑋) + 𝑋) = (0g‘𝐺))
27	26	oveq2d 7379	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 + (((invg‘𝐺)‘𝑋) + 𝑋)) = (𝑦 + (0g‘𝐺)))
28	2, 3, 24	grprid 18942	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → (𝑦 + (0g‘𝐺)) = 𝑦)
29	28	ad2ant2rl 755	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 + (0g‘𝐺)) = 𝑦)
30	23, 27, 29	3eqtrd 2779	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑦 + ((invg‘𝐺)‘𝑋)) + 𝑋) = 𝑦)
31	30	eqeq1d 2742	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((𝑦 + ((invg‘𝐺)‘𝑋)) + 𝑋) = (𝑥 + 𝑋) ↔ 𝑦 = (𝑥 + 𝑋)))
32	20, 31	bitr3d 282	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑦 + ((invg‘𝐺)‘𝑋)) = 𝑥 ↔ 𝑦 = (𝑥 + 𝑋)))
33	14, 32	bitrid 284	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = (𝑦 + ((invg‘𝐺)‘𝑋)) ↔ 𝑦 = (𝑥 + 𝑋)))
34	1, 7, 13, 33	f1o2d 7617	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝐹:𝐵–1-1-onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ↦ cmpt 5160  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  +_gcplusg 17218  0_gc0g 17400  Grpcgrp 18907  inv_gcminusg 18908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  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 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-0g 17402  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-grp 18910  df-minusg 18911

This theorem is referenced by: (None)