Theorem grplmulf1o 13450

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 13384	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑋 + 𝑥) ∈ 𝐵)
5	4	3expa 1206	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑋 + 𝑥) ∈ 𝐵)
6		eqid 2206	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
7	2, 6	grpinvcl 13424	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((invg‘𝐺)‘𝑋) ∈ 𝐵)
8	2, 3	grpcl 13384	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ ((invg‘𝐺)‘𝑋) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((invg‘𝐺)‘𝑋) + 𝑦) ∈ 𝐵)
9	8	3expa 1206	. . 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 2208	. . 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 529	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
15		simplr 528	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
16	2, 3	grplcan 13438	. . . . 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 1252	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 + (((invg‘𝐺)‘𝑋) + 𝑦)) = (𝑋 + 𝑥) ↔ (((invg‘𝐺)‘𝑋) + 𝑦) = 𝑥))
18		eqid 2206	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
19	2, 3, 18, 6	grprinv 13427	. . . . . . . 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 5966	. . . . . 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 531	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
24	2, 3	grpass 13385	. . . . . . 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 1252	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑋 + ((invg‘𝐺)‘𝑋)) + 𝑦) = (𝑋 + (((invg‘𝐺)‘𝑋) + 𝑦)))
26	2, 3, 18	grplid 13407	. . . . . . 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 2247	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑋 + (((invg‘𝐺)‘𝑋) + 𝑦)) = 𝑦)
29	28	eqeq1d 2215	. . . 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 6158	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝐹:𝐵–1-1-onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2177   ↦ cmpt 4109  –1-1-onto→wf1o 5275  ‘cfv 5276  (class class class)co 5951  Basecbs 12876  +_gcplusg 12953  0_gc0g 13132  Grpcgrp 13376  inv_gcminusg 13377

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-cnex 8023  ax-resscn 8024  ax-1re 8026  ax-addrcl 8029

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-riota 5906  df-ov 5954  df-inn 9044  df-2 9102  df-ndx 12879  df-slot 12880  df-base 12882  df-plusg 12966  df-0g 13134  df-mgm 13232  df-sgrp 13278  df-mnd 13293  df-grp 13379  df-minusg 13380

This theorem is referenced by: (None)