Theorem grplmulf1o 19110

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 19039	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑋 + 𝑥) ∈ 𝐵)
5	4	3expa 1136	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑋 + 𝑥) ∈ 𝐵)
6		eqid 2766	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
7	2, 6	grpinvcl 19085	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((invg‘𝐺)‘𝑋) ∈ 𝐵)
8	2, 3	grpcl 19039	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ ((invg‘𝐺)‘𝑋) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
9	8	3expa 1136	. . 3 ⊢ (((𝐺 ∈ Grp ∧ ((invg‘𝐺)‘𝑋) ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
10	7, 9	syldanl 614	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
11		eqcom 2773	. . 3 ⊢ (𝑥 = (((invg‘𝐺)‘𝑋) + 𝑦) ↔ (((invg‘𝐺)‘𝑋) + 𝑦) = 𝑥)
12		simpll 779	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Grp)
13	10	adantrl 729	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
14		simprl 783	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
15		simplr 781	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
16	2, 3	grplcan 19098	. . . . 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 1399	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 + (((invg‘𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ (((invg‘𝐺)‘𝑋) + 𝑦) = 𝑥))
18		eqid 2766	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
19	2, 3, 18, 6	grprinv 19088	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + ((invg‘𝐺)‘𝑋)) = (0g‘𝐺))
20	19	adantr 486	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑋 + ((invg‘𝐺)‘𝑋)) = (0g‘𝐺))
21	20	oveq1d 7438	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 + ((invg‘𝐺)‘𝑋)) + 𝑦) = ((0g‘𝐺) + 𝑦))
22	7	adantr 486	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((invg‘𝐺)‘𝑋) ∈ 𝐵)
23		simprr 785	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
24	2, 3, 12, 15, 22, 23	grpassd 19043	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 + ((invg‘𝐺)‘𝑋)) + 𝑦) = (𝑋 + (((invg‘𝐺)‘𝑋) + 𝑦)))
25	2, 3, 18	grplid 19065	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → ((0g‘𝐺) + 𝑦) = 𝑦)
26	25	ad2ant2rl 762	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((0g‘𝐺) + 𝑦) = 𝑦)
27	21, 24, 26	3eqtr3d 2809	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑋 + (((invg‘𝐺)‘𝑋) + 𝑦)) = 𝑦)
28	27	eqeq1d 2768	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 + (((invg‘𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ 𝑦 = (𝑋 + 𝑥)))
29	17, 28	bitr3d 284	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑋) + 𝑦) = 𝑥 ↔ 𝑦 = (𝑋 + 𝑥)))
30	11, 29	bitrid 286	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = (((invg‘𝐺)‘𝑋) + 𝑦) ↔ 𝑦 = (𝑋 + 𝑥)))
31	1, 5, 10, 30	f1o2d 7677	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝐹:𝐵–1-1-onto→𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ↦ cmpt 5197  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  +_gcplusg 17335  0_gc0g 17517  Grpcgrp 19031  inv_gcminusg 19032

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-0g 17519  df-mgm 18723  df-sgrp 18806  df-mnd 18822  df-grp 19034  df-minusg 19035

This theorem is used by:  sylow1lem2  19700  sylow2blem1  19721