Theorem grplmulf1o 13871

Description: Left multiplication by a group element is a bijection on any group. (Contributed by Mario Carneiro, 17-Jan-2015.)

Hypotheses

Ref	Expression
grplmulf1o.b	⊢ 𝐵 = (Base‘𝐺)
grplmulf1o.p	⊢ + = (+g‘𝐺)
grplmulf1o.n	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑋 + 𝑥))

Assertion

Ref	Expression
grplmulf1o	⊢ ((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 grplmulf1o

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grplmulf1o.n	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑋 + 𝑥))
2		grplmulf1o.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3		grplmulf1o.p	. . . 4 ⊢ + = (+g‘𝐺)
4	2, 3	grpcl 13805	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑋 + 𝑥) ∈ 𝐵)
5	4	3expa 1230	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑋 + 𝑥) ∈ 𝐵)
6		eqid 2234	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
7	2, 6	grpinvcl 13845	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((invg‘𝐺)‘𝑋) ∈ 𝐵)
8	2, 3	grpcl 13805	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ ((invg‘𝐺)‘𝑋) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
9	8	3expa 1230	. . 3 ⊢ (((𝐺 ∈ Grp ∧ ((invg‘𝐺)‘𝑋) ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
10	7, 9	syldanl 449	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
11		eqcom 2236	. . 3 ⊢ (𝑥 = (((invg‘𝐺)‘𝑋) + 𝑦) ↔ (((invg‘𝐺)‘𝑋) + 𝑦) = 𝑥)
12		simpll 527	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Grp)
13	10	adantrl 478	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
14		simprl 531	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
15		simplr 529	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
16	2, 3	grplcan 13859	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ ((((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 + (((invg‘𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ (((invg‘𝐺)‘𝑋) + 𝑦) = 𝑥))
17	12, 13, 14, 15, 16	syl13anc 1276	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 + (((invg‘𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ (((invg‘𝐺)‘𝑋) + 𝑦) = 𝑥))
18		eqid 2234	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
19	2, 3, 18, 6	grprinv 13848	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + ((invg‘𝐺)‘𝑋)) = (0g‘𝐺))
20	19	adantr 276	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑋 + ((invg‘𝐺)‘𝑋)) = (0g‘𝐺))
21	20	oveq1d 6073	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 + ((invg‘𝐺)‘𝑋)) + 𝑦) = ((0g‘𝐺) + 𝑦))
22	7	adantr 276	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((invg‘𝐺)‘𝑋) ∈ 𝐵)
23		simprr 533	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
24	2, 3	grpass 13806	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑋) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 + ((invg‘𝐺)‘𝑋)) + 𝑦) = (𝑋 + (((invg‘𝐺)‘𝑋) + 𝑦)))
25	12, 15, 22, 23, 24	syl13anc 1276	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 + ((invg‘𝐺)‘𝑋)) + 𝑦) = (𝑋 + (((invg‘𝐺)‘𝑋) + 𝑦)))
26	2, 3, 18	grplid 13828	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → ((0g‘𝐺) + 𝑦) = 𝑦)
27	26	ad2ant2rl 511	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((0g‘𝐺) + 𝑦) = 𝑦)
28	21, 25, 27	3eqtr3d 2275	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑋 + (((invg‘𝐺)‘𝑋) + 𝑦)) = 𝑦)
29	28	eqeq1d 2243	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 + (((invg‘𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ 𝑦 = (𝑋 + 𝑥)))
30	17, 29	bitr3d 190	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑋) + 𝑦) = 𝑥 ↔ 𝑦 = (𝑋 + 𝑥)))
31	11, 30	bitrid 192	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = (((invg‘𝐺)‘𝑋) + 𝑦) ↔ 𝑦 = (𝑋 + 𝑥)))
32	1, 5, 10, 31	f1o2d 6268	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝐹:𝐵–1-1-onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205   ↦ cmpt 4176  –1-1-onto→wf1o 5356  ‘cfv 5357  (class class class)co 6058  Basecbs 13296  +_gcplusg 13374  0_gc0g 13553  Grpcgrp 13797  inv_gcminusg 13798

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-mgm 13653  df-sgrp 13699  df-mnd 13714  df-grp 13800  df-minusg 13801

This theorem is referenced by: (None)