Metamath Proof Explorer

Theorem ghmf1

Description: Two ways of saying a group homomorphism is 1-1 into its codomain. (Contributed by Paul Chapman, 3-Mar-2008) (Revised by Mario Carneiro, 13-Jan-2015) (Proof shortened by AV, 4-Apr-2025)

Ref Expression
Hypotheses f1ghm0to0.a ⊢ 𝐴 = ( Base ‘ 𝑅 )
f1ghm0to0.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
f1ghm0to0.n ⊢ 𝑁 = ( 0g ‘ 𝑅 )
f1ghm0to0.0 ⊢ 0 = ( 0g ‘ 𝑆 )
Assertion ghmf1 ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) )

Proof

Step Hyp Ref Expression
1 f1ghm0to0.a ⊢ 𝐴 = ( Base ‘ 𝑅 )
2 f1ghm0to0.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
3 f1ghm0to0.n ⊢ 𝑁 = ( 0g ‘ 𝑅 )
4 f1ghm0to0.0 ⊢ 0 = ( 0g ‘ 𝑆 )
5 1 2 3 4 f1ghm0to0 ⊢ ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑥 ) = 0 ↔ 𝑥 = 𝑁 ) )
6 5 3expa ⊢ ( ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑥 ) = 0 ↔ 𝑥 = 𝑁 ) )
7 6 biimpd ⊢ ( ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) )
8 7 ralrimiva ⊢ ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) )
9 1 2 ghmf ⊢ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) → 𝐹 : 𝐴 ⟶ 𝐵 )
10 9 adantr ⊢ ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) → 𝐹 : 𝐴 ⟶ 𝐵 )
11 eqid ⊢ ( -g ‘ 𝑅 ) = ( -g ‘ 𝑅 )
12 eqid ⊢ ( -g ‘ 𝑆 ) = ( -g ‘ 𝑆 )
13 1 11 12 ghmsub ⊢ ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) → ( 𝐹 ‘ ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) ) = ( ( 𝐹 ‘ 𝑦 ) ( -g ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) )
14 13 3expb ⊢ ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( 𝐹 ‘ ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) ) = ( ( 𝐹 ‘ 𝑦 ) ( -g ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) )
15 14 adantlr ⊢ ( ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( 𝐹 ‘ ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) ) = ( ( 𝐹 ‘ 𝑦 ) ( -g ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) )
16 15 eqeq1d ⊢ ( ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( ( 𝐹 ‘ ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) ) = 0 ↔ ( ( 𝐹 ‘ 𝑦 ) ( -g ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) = 0 ) )
17 fveqeq2 ⊢ ( 𝑥 = ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) → ( ( 𝐹 ‘ 𝑥 ) = 0 ↔ ( 𝐹 ‘ ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) ) = 0 ) )
18 eqeq1 ⊢ ( 𝑥 = ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) → ( 𝑥 = 𝑁 ↔ ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) = 𝑁 ) )
19 17 18 imbi12d ⊢ ( 𝑥 = ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) → ( ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ↔ ( ( 𝐹 ‘ ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) ) = 0 → ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) = 𝑁 ) ) )
20 simplr ⊢ ( ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) )
21 ghmgrp1 ⊢ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) → 𝑅 ∈ Grp )
22 21 adantr ⊢ ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) → 𝑅 ∈ Grp )
23 1 11 grpsubcl ⊢ ( ( 𝑅 ∈ Grp ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) → ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) ∈ 𝐴 )
24 23 3expb ⊢ ( ( 𝑅 ∈ Grp ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) ∈ 𝐴 )
25 22 24 sylan ⊢ ( ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) ∈ 𝐴 )
26 19 20 25 rspcdva ⊢ ( ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( ( 𝐹 ‘ ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) ) = 0 → ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) = 𝑁 ) )
27 16 26 sylbird ⊢ ( ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( ( ( 𝐹 ‘ 𝑦 ) ( -g ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) = 0 → ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) = 𝑁 ) )
28 ghmgrp2 ⊢ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) → 𝑆 ∈ Grp )
29 28 ad2antrr ⊢ ( ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → 𝑆 ∈ Grp )
30 9 ad2antrr ⊢ ( ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → 𝐹 : 𝐴 ⟶ 𝐵 )
31 simprl ⊢ ( ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → 𝑦 ∈ 𝐴 )
32 30 31 ffvelcdmd ⊢ ( ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝐵 )
33 simprr ⊢ ( ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → 𝑧 ∈ 𝐴 )
34 30 33 ffvelcdmd ⊢ ( ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝐵 )
35 2 4 12 grpsubeq0 ⊢ ( ( 𝑆 ∈ Grp ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝐵 ∧ ( 𝐹 ‘ 𝑧 ) ∈ 𝐵 ) → ( ( ( 𝐹 ‘ 𝑦 ) ( -g ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) = 0 ↔ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ) )
36 29 32 34 35 syl3anc ⊢ ( ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( ( ( 𝐹 ‘ 𝑦 ) ( -g ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) = 0 ↔ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ) )
37 21 ad2antrr ⊢ ( ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → 𝑅 ∈ Grp )
38 1 3 11 grpsubeq0 ⊢ ( ( 𝑅 ∈ Grp ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) = 𝑁 ↔ 𝑦 = 𝑧 ) )
39 37 31 33 38 syl3anc ⊢ ( ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( ( 𝑦 ( -g ‘ 𝑅 ) 𝑧 ) = 𝑁 ↔ 𝑦 = 𝑧 ) )
40 27 36 39 3imtr3d ⊢ ( ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) → 𝑦 = 𝑧 ) )
41 40 ralrimivva ⊢ ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) → ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) → 𝑦 = 𝑧 ) )
42 dff13 ⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) )
43 10 41 42 sylanbrc ⊢ ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) → 𝐹 : 𝐴 –1-1→ 𝐵 )
44 8 43 impbida ⊢ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = 0 → 𝑥 = 𝑁 ) ) )