Theorem grpraddf1o 19054

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 766	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐺 ∈ Grp)
5		simpr 484	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
6		simplr 768	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ 𝐵)
7	2, 3, 4, 5, 6	grpcld 18987	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑋) ∈ 𝐵)
8		simpll 766	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝐺 ∈ Grp)
9		simpr 484	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
10		eqid 2740	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
11		simplr 768	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝐵)
12	2, 10, 8, 11	grpinvcld 19028	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((invg‘𝐺)‘𝑋) ∈ 𝐵)
13	2, 3, 8, 9, 12	grpcld 18987	. 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 766	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Grp)
16	13	adantrl 715	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 + ((invg‘𝐺)‘𝑋)) ∈ 𝐵)
17		simprl 770	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
18		simplr 768	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
19	2, 3	grprcan 19013	. . . . 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 1372	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((𝑦 + ((invg‘𝐺)‘𝑋)) + 𝑋) = (𝑥 + 𝑋) ↔ (𝑦 + ((invg‘𝐺)‘𝑋)) = 𝑥))
21		simprr 772	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
22	12	adantrl 715	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((invg‘𝐺)‘𝑋) ∈ 𝐵)
23	2, 3, 15, 21, 22, 18	grpassd 18985	. . . . . 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 19029	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (((invg‘𝐺)‘𝑋) + 𝑋) = (0g‘𝐺))
26	25	adantr 480	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((invg‘𝐺)‘𝑋) + 𝑋) = (0g‘𝐺))
27	26	oveq2d 7464	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 + (((invg‘𝐺)‘𝑋) + 𝑋)) = (𝑦 + (0g‘𝐺)))
28	2, 3, 24	grprid 19008	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → (𝑦 + (0g‘𝐺)) = 𝑦)
29	28	ad2ant2rl 748	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 + (0g‘𝐺)) = 𝑦)
30	23, 27, 29	3eqtrd 2784	. . . . 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 281	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑦 + ((invg‘𝐺)‘𝑋)) = 𝑥 ↔ 𝑦 = (𝑥 + 𝑋)))
33	14, 32	bitrid 283	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = (𝑦 + ((invg‘𝐺)‘𝑋)) ↔ 𝑦 = (𝑥 + 𝑋)))
34	1, 7, 13, 33	f1o2d 7704	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝐹:𝐵–1-1-onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ↦ cmpt 5249  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  0_gc0g 17499  Grpcgrp 18973  inv_gcminusg 18974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977

This theorem is referenced by: (None)