| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | ghmgrp2 19237 | . . . . 5
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp) | 
| 2 |  | ghmgrp1 19236 | . . . . 5
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp) | 
| 3 | 1, 2 | jca 511 | . . . 4
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑇 ∈ Grp ∧ 𝑆 ∈ Grp)) | 
| 4 | 3 | adantr 480 | . . 3
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑇 ∈ Grp ∧ 𝑆 ∈ Grp)) | 
| 5 |  | f1ocnv 6860 | . . . . . 6
⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋) | 
| 6 | 5 | adantl 481 | . . . . 5
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹:𝑌–1-1-onto→𝑋) | 
| 7 |  | f1of 6848 | . . . . 5
⊢ (◡𝐹:𝑌–1-1-onto→𝑋 → ◡𝐹:𝑌⟶𝑋) | 
| 8 | 6, 7 | syl 17 | . . . 4
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹:𝑌⟶𝑋) | 
| 9 |  | simpll 767 | . . . . . . . 8
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | 
| 10 | 8 | adantr 480 | . . . . . . . . 9
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ◡𝐹:𝑌⟶𝑋) | 
| 11 |  | simprl 771 | . . . . . . . . 9
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ 𝑌) | 
| 12 | 10, 11 | ffvelcdmd 7105 | . . . . . . . 8
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (◡𝐹‘𝑥) ∈ 𝑋) | 
| 13 |  | simprr 773 | . . . . . . . . 9
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑦 ∈ 𝑌) | 
| 14 | 10, 13 | ffvelcdmd 7105 | . . . . . . . 8
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (◡𝐹‘𝑦) ∈ 𝑋) | 
| 15 |  | ghmf1o.x | . . . . . . . . 9
⊢ 𝑋 = (Base‘𝑆) | 
| 16 |  | eqid 2737 | . . . . . . . . 9
⊢
(+g‘𝑆) = (+g‘𝑆) | 
| 17 |  | eqid 2737 | . . . . . . . . 9
⊢
(+g‘𝑇) = (+g‘𝑇) | 
| 18 | 15, 16, 17 | ghmlin 19239 | . . . . . . . 8
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (◡𝐹‘𝑥) ∈ 𝑋 ∧ (◡𝐹‘𝑦) ∈ 𝑋) → (𝐹‘((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦))) = ((𝐹‘(◡𝐹‘𝑥))(+g‘𝑇)(𝐹‘(◡𝐹‘𝑦)))) | 
| 19 | 9, 12, 14, 18 | syl3anc 1373 | . . . . . . 7
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝐹‘((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦))) = ((𝐹‘(◡𝐹‘𝑥))(+g‘𝑇)(𝐹‘(◡𝐹‘𝑦)))) | 
| 20 |  | simplr 769 | . . . . . . . . 9
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝐹:𝑋–1-1-onto→𝑌) | 
| 21 |  | f1ocnvfv2 7297 | . . . . . . . . 9
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥) | 
| 22 | 20, 11, 21 | syl2anc 584 | . . . . . . . 8
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥) | 
| 23 |  | f1ocnvfv2 7297 | . . . . . . . . 9
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑦 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑦)) = 𝑦) | 
| 24 | 20, 13, 23 | syl2anc 584 | . . . . . . . 8
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑦)) = 𝑦) | 
| 25 | 22, 24 | oveq12d 7449 | . . . . . . 7
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((𝐹‘(◡𝐹‘𝑥))(+g‘𝑇)(𝐹‘(◡𝐹‘𝑦))) = (𝑥(+g‘𝑇)𝑦)) | 
| 26 | 19, 25 | eqtrd 2777 | . . . . . 6
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝐹‘((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦))) = (𝑥(+g‘𝑇)𝑦)) | 
| 27 | 9, 2 | syl 17 | . . . . . . . 8
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑆 ∈ Grp) | 
| 28 | 15, 16 | grpcl 18959 | . . . . . . . 8
⊢ ((𝑆 ∈ Grp ∧ (◡𝐹‘𝑥) ∈ 𝑋 ∧ (◡𝐹‘𝑦) ∈ 𝑋) → ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦)) ∈ 𝑋) | 
| 29 | 27, 12, 14, 28 | syl3anc 1373 | . . . . . . 7
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦)) ∈ 𝑋) | 
| 30 |  | f1ocnvfv 7298 | . . . . . . 7
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦)) ∈ 𝑋) → ((𝐹‘((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦))) = (𝑥(+g‘𝑇)𝑦) → (◡𝐹‘(𝑥(+g‘𝑇)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦)))) | 
| 31 | 20, 29, 30 | syl2anc 584 | . . . . . 6
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((𝐹‘((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦))) = (𝑥(+g‘𝑇)𝑦) → (◡𝐹‘(𝑥(+g‘𝑇)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦)))) | 
| 32 | 26, 31 | mpd 15 | . . . . 5
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (◡𝐹‘(𝑥(+g‘𝑇)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦))) | 
| 33 | 32 | ralrimivva 3202 | . . . 4
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (◡𝐹‘(𝑥(+g‘𝑇)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦))) | 
| 34 | 8, 33 | jca 511 | . . 3
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (◡𝐹:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (◡𝐹‘(𝑥(+g‘𝑇)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦)))) | 
| 35 |  | ghmf1o.y | . . . 4
⊢ 𝑌 = (Base‘𝑇) | 
| 36 | 35, 15, 17, 16 | isghm 19233 | . . 3
⊢ (◡𝐹 ∈ (𝑇 GrpHom 𝑆) ↔ ((𝑇 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (◡𝐹:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (◡𝐹‘(𝑥(+g‘𝑇)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑆)(◡𝐹‘𝑦))))) | 
| 37 | 4, 34, 36 | sylanbrc 583 | . 2
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) | 
| 38 | 15, 35 | ghmf 19238 | . . . . 5
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋⟶𝑌) | 
| 39 | 38 | adantr 480 | . . . 4
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑋⟶𝑌) | 
| 40 | 39 | ffnd 6737 | . . 3
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹 Fn 𝑋) | 
| 41 | 35, 15 | ghmf 19238 | . . . . 5
⊢ (◡𝐹 ∈ (𝑇 GrpHom 𝑆) → ◡𝐹:𝑌⟶𝑋) | 
| 42 | 41 | adantl 481 | . . . 4
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → ◡𝐹:𝑌⟶𝑋) | 
| 43 | 42 | ffnd 6737 | . . 3
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → ◡𝐹 Fn 𝑌) | 
| 44 |  | dff1o4 6856 | . . 3
⊢ (𝐹:𝑋–1-1-onto→𝑌 ↔ (𝐹 Fn 𝑋 ∧ ◡𝐹 Fn 𝑌)) | 
| 45 | 40, 43, 44 | sylanbrc 583 | . 2
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑋–1-1-onto→𝑌) | 
| 46 | 37, 45 | impbida 801 | 1
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 GrpHom 𝑆))) |