Theorem grplmulf1o 18942

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 18871	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑋 + 𝑥) ∈ 𝐵)
5	4	3expa 1115	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑋 + 𝑥) ∈ 𝐵)
6		eqid 2726	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
7	2, 6	grpinvcl 18917	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((invg‘𝐺)‘𝑋) ∈ 𝐵)
8	2, 3	grpcl 18871	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ ((invg‘𝐺)‘𝑋) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
9	8	3expa 1115	. . 3 ⊢ (((𝐺 ∈ Grp ∧ ((invg‘𝐺)‘𝑋) ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
10	7, 9	syldanl 601	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
11		eqcom 2733	. . 3 ⊢ (𝑥 = (((invg‘𝐺)‘𝑋) + 𝑦) ↔ (((invg‘𝐺)‘𝑋) + 𝑦) = 𝑥)
12		simpll 764	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Grp)
13	10	adantrl 713	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
14		simprl 768	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
15		simplr 766	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
16	2, 3	grplcan 18930	. . . . 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 1369	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 + (((invg‘𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ (((invg‘𝐺)‘𝑋) + 𝑦) = 𝑥))
18		eqid 2726	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
19	2, 3, 18, 6	grprinv 18920	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + ((invg‘𝐺)‘𝑋)) = (0g‘𝐺))
20	19	adantr 480	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑋 + ((invg‘𝐺)‘𝑋)) = (0g‘𝐺))
21	20	oveq1d 7420	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 + ((invg‘𝐺)‘𝑋)) + 𝑦) = ((0g‘𝐺) + 𝑦))
22	7	adantr 480	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((invg‘𝐺)‘𝑋) ∈ 𝐵)
23		simprr 770	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
24	2, 3	grpass 18872	. . . . . . 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 1369	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 + ((invg‘𝐺)‘𝑋)) + 𝑦) = (𝑋 + (((invg‘𝐺)‘𝑋) + 𝑦)))
26	2, 3, 18	grplid 18897	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → ((0g‘𝐺) + 𝑦) = 𝑦)
27	26	ad2ant2rl 746	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((0g‘𝐺) + 𝑦) = 𝑦)
28	21, 25, 27	3eqtr3d 2774	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑋 + (((invg‘𝐺)‘𝑋) + 𝑦)) = 𝑦)
29	28	eqeq1d 2728	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 + (((invg‘𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ 𝑦 = (𝑋 + 𝑥)))
30	17, 29	bitr3d 281	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑋) + 𝑦) = 𝑥 ↔ 𝑦 = (𝑋 + 𝑥)))
31	11, 30	bitrid 283	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = (((invg‘𝐺)‘𝑋) + 𝑦) ↔ 𝑦 = (𝑋 + 𝑥)))
32	1, 5, 10, 31	f1o2d 7657	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝐹:𝐵–1-1-onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ↦ cmpt 5224  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7405  Basecbs 17153  +_gcplusg 17206  0_gc0g 17394  Grpcgrp 18863  inv_gcminusg 18864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-minusg 18867

This theorem is referenced by:  sylow1lem2  19519  sylow2blem1  19540