| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | f1ghm0to0.a | . . . . . 6
⊢ 𝐴 = (Base‘𝑅) | 
| 2 |  | f1ghm0to0.b | . . . . . 6
⊢ 𝐵 = (Base‘𝑆) | 
| 3 |  | f1ghm0to0.n | . . . . . 6
⊢ 𝑁 = (0g‘𝑅) | 
| 4 |  | f1ghm0to0.0 | . . . . . 6
⊢  0 =
(0g‘𝑆) | 
| 5 | 1, 2, 3, 4 | f1ghm0to0 19264 | . . . . 5
⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = 𝑁)) | 
| 6 | 5 | 3expa 1118 | . . . 4
⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = 𝑁)) | 
| 7 | 6 | biimpd 229 | . . 3
⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) | 
| 8 | 7 | ralrimiva 3145 | . 2
⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) | 
| 9 | 1, 2 | ghmf 19239 | . . . 4
⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:𝐴⟶𝐵) | 
| 10 | 9 | adantr 480 | . . 3
⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) → 𝐹:𝐴⟶𝐵) | 
| 11 |  | eqid 2736 | . . . . . . . . . 10
⊢
(-g‘𝑅) = (-g‘𝑅) | 
| 12 |  | eqid 2736 | . . . . . . . . . 10
⊢
(-g‘𝑆) = (-g‘𝑆) | 
| 13 | 1, 11, 12 | ghmsub 19243 | . . . . . . . . 9
⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹‘(𝑦(-g‘𝑅)𝑧)) = ((𝐹‘𝑦)(-g‘𝑆)(𝐹‘𝑧))) | 
| 14 | 13 | 3expb 1120 | . . . . . . . 8
⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐹‘(𝑦(-g‘𝑅)𝑧)) = ((𝐹‘𝑦)(-g‘𝑆)(𝐹‘𝑧))) | 
| 15 | 14 | adantlr 715 | . . . . . . 7
⊢ (((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 2738 | . . . . . 6
⊢ (((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 6914 | . . . . . . . 8
⊢ (𝑥 = (𝑦(-g‘𝑅)𝑧) → ((𝐹‘𝑥) = 0 ↔ (𝐹‘(𝑦(-g‘𝑅)𝑧)) = 0 )) | 
| 18 |  | eqeq1 2740 | . . . . . . . 8
⊢ (𝑥 = (𝑦(-g‘𝑅)𝑧) → (𝑥 = 𝑁 ↔ (𝑦(-g‘𝑅)𝑧) = 𝑁)) | 
| 19 | 17, 18 | imbi12d 344 | . . . . . . 7
⊢ (𝑥 = (𝑦(-g‘𝑅)𝑧) → (((𝐹‘𝑥) = 0 → 𝑥 = 𝑁) ↔ ((𝐹‘(𝑦(-g‘𝑅)𝑧)) = 0 → (𝑦(-g‘𝑅)𝑧) = 𝑁))) | 
| 20 |  | simplr 768 | . . . . . . 7
⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) | 
| 21 |  | ghmgrp1 19237 | . . . . . . . . 9
⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp) | 
| 22 | 21 | adantr 480 | . . . . . . . 8
⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) → 𝑅 ∈ Grp) | 
| 23 | 1, 11 | grpsubcl 19039 | . . . . . . . . 9
⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑦(-g‘𝑅)𝑧) ∈ 𝐴) | 
| 24 | 23 | 3expb 1120 | . . . . . . . 8
⊢ ((𝑅 ∈ Grp ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦(-g‘𝑅)𝑧) ∈ 𝐴) | 
| 25 | 22, 24 | sylan 580 | . . . . . . 7
⊢ (((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 3622 | . . . . . 6
⊢ (((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 260 | . . . . 5
⊢ (((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 19238 | . . . . . . 7
⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑆 ∈ Grp) | 
| 29 | 28 | ad2antrr 726 | . . . . . 6
⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑆 ∈ Grp) | 
| 30 | 9 | ad2antrr 726 | . . . . . . 7
⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐹:𝐴⟶𝐵) | 
| 31 |  | simprl 770 | . . . . . . 7
⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑦 ∈ 𝐴) | 
| 32 | 30, 31 | ffvelcdmd 7104 | . . . . . 6
⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐹‘𝑦) ∈ 𝐵) | 
| 33 |  | simprr 772 | . . . . . . 7
⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ 𝐴) | 
| 34 | 30, 33 | ffvelcdmd 7104 | . . . . . 6
⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐹‘𝑧) ∈ 𝐵) | 
| 35 | 2, 4, 12 | grpsubeq0 19045 | . . . . . 6
⊢ ((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 1372 | . . . . 5
⊢ (((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 726 | . . . . . 6
⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑅 ∈ Grp) | 
| 38 | 1, 3, 11 | grpsubeq0 19045 | . . . . . 6
⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑦(-g‘𝑅)𝑧) = 𝑁 ↔ 𝑦 = 𝑧)) | 
| 39 | 37, 31, 33, 38 | syl3anc 1372 | . . . . 5
⊢ (((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 293 | . . . 4
⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)) | 
| 41 | 40 | ralrimivva 3201 | . . 3
⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)) | 
| 42 |  | dff13 7276 | . . 3
⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))) | 
| 43 | 10, 41, 42 | sylanbrc 583 | . 2
⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)) → 𝐹:𝐴–1-1→𝐵) | 
| 44 | 8, 43 | impbida 800 | 1
⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁))) |