Theorem grpraddf1o 19137

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 779	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐺 ∈ Grp)
5		simpr 490	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
6		simplr 781	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ 𝐵)
7	2, 3, 4, 5, 6	grpcld 19071	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑋) ∈ 𝐵)
8		simpll 779	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝐺 ∈ Grp)
9		simpr 490	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
10		eqid 2760	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
11		simplr 781	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝐵)
12	2, 10, 8, 11	grpinvcld 19112	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((invg‘𝐺)‘𝑋) ∈ 𝐵)
13	2, 3, 8, 9, 12	grpcld 19071	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑦 + ((invg‘𝐺)‘𝑋)) ∈ 𝐵)
14		eqcom 2767	. . 3 ⊢ (𝑥 = (𝑦 + ((invg‘𝐺)‘𝑋)) ↔ (𝑦 + ((invg‘𝐺)‘𝑋)) = 𝑥)
15		simpll 779	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Grp)
16	13	adantrl 729	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 + ((invg‘𝐺)‘𝑋)) ∈ 𝐵)
17		simprl 783	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
18		simplr 781	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
19	2, 3	grprcan 19097	. . . . 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 1399	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((𝑦 + ((invg‘𝐺)‘𝑋)) + 𝑋) = (𝑥 + 𝑋) ↔ (𝑦 + ((invg‘𝐺)‘𝑋)) = 𝑥))
21		simprr 785	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
22	12	adantrl 729	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((invg‘𝐺)‘𝑋) ∈ 𝐵)
23	2, 3, 15, 21, 22, 18	grpassd 19069	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑦 + ((invg‘𝐺)‘𝑋)) + 𝑋) = (𝑦 + (((invg‘𝐺)‘𝑋) + 𝑋)))
24		eqid 2760	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
25	2, 3, 24, 10	grplinv 19113	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (((invg‘𝐺)‘𝑋) + 𝑋) = (0g‘𝐺))
26	25	adantr 486	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((invg‘𝐺)‘𝑋) + 𝑋) = (0g‘𝐺))
27	26	oveq2d 7429	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 + (((invg‘𝐺)‘𝑋) + 𝑋)) = (𝑦 + (0g‘𝐺)))
28	2, 3, 24	grprid 19092	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → (𝑦 + (0g‘𝐺)) = 𝑦)
29	28	ad2ant2rl 762	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 + (0g‘𝐺)) = 𝑦)
30	23, 27, 29	3eqtrd 2799	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑦 + ((invg‘𝐺)‘𝑋)) + 𝑋) = 𝑦)
31	30	eqeq1d 2762	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((𝑦 + ((invg‘𝐺)‘𝑋)) + 𝑋) = (𝑥 + 𝑋) ↔ 𝑦 = (𝑥 + 𝑋)))
32	20, 31	bitr3d 284	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑦 + ((invg‘𝐺)‘𝑋)) = 𝑥 ↔ 𝑦 = (𝑥 + 𝑋)))
33	14, 32	bitrid 286	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = (𝑦 + ((invg‘𝐺)‘𝑋)) ↔ 𝑦 = (𝑥 + 𝑋)))
34	1, 7, 13, 33	f1o2d 7668	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 2145   ↦ cmpt 5186  –1-1-onto→wf1o 6532  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  +_gcplusg 17342  0_gc0g 17524  Grpcgrp 19057  inv_gcminusg 19058

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-0g 17526  df-mgm 18730  df-sgrp 18821  df-mnd 18837  df-grp 19060  df-minusg 19061

This theorem is used by: (None)