Proof of Theorem rhmf1o
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | rhmrcl2 20477 | . . . . 5
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring) | 
| 2 |  | rhmrcl1 20476 | . . . . 5
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring) | 
| 3 | 1, 2 | jca 511 | . . . 4
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝑆 ∈ Ring ∧ 𝑅 ∈ Ring)) | 
| 4 | 3 | adantr 480 | . . 3
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (𝑆 ∈ Ring ∧ 𝑅 ∈ Ring)) | 
| 5 |  | simpr 484 | . . . . 5
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹:𝐵–1-1-onto→𝐶) | 
| 6 |  | rhmghm 20484 | . . . . . . 7
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | 
| 7 | 6 | adantr 480 | . . . . . 6
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | 
| 8 |  | rhmf1o.b | . . . . . . . 8
⊢ 𝐵 = (Base‘𝑅) | 
| 9 |  | rhmf1o.c | . . . . . . . 8
⊢ 𝐶 = (Base‘𝑆) | 
| 10 | 8, 9 | ghmf1o 19266 | . . . . . . 7
⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 GrpHom 𝑅))) | 
| 11 | 10 | bicomd 223 | . . . . . 6
⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (◡𝐹 ∈ (𝑆 GrpHom 𝑅) ↔ 𝐹:𝐵–1-1-onto→𝐶)) | 
| 12 | 7, 11 | syl 17 | . . . . 5
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹 ∈ (𝑆 GrpHom 𝑅) ↔ 𝐹:𝐵–1-1-onto→𝐶)) | 
| 13 | 5, 12 | mpbird 257 | . . . 4
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ (𝑆 GrpHom 𝑅)) | 
| 14 |  | eqidd 2738 | . . . . . . 7
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 = 𝐹) | 
| 15 |  | eqid 2737 | . . . . . . . . 9
⊢
(mulGrp‘𝑅) =
(mulGrp‘𝑅) | 
| 16 | 15, 8 | mgpbas 20142 | . . . . . . . 8
⊢ 𝐵 =
(Base‘(mulGrp‘𝑅)) | 
| 17 | 16 | a1i 11 | . . . . . . 7
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐵 = (Base‘(mulGrp‘𝑅))) | 
| 18 |  | eqid 2737 | . . . . . . . . 9
⊢
(mulGrp‘𝑆) =
(mulGrp‘𝑆) | 
| 19 | 18, 9 | mgpbas 20142 | . . . . . . . 8
⊢ 𝐶 =
(Base‘(mulGrp‘𝑆)) | 
| 20 | 19 | a1i 11 | . . . . . . 7
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐶 = (Base‘(mulGrp‘𝑆))) | 
| 21 | 14, 17, 20 | f1oeq123d 6842 | . . . . . 6
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ 𝐹:(Base‘(mulGrp‘𝑅))–1-1-onto→(Base‘(mulGrp‘𝑆)))) | 
| 22 | 21 | biimpa 476 | . . . . 5
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹:(Base‘(mulGrp‘𝑅))–1-1-onto→(Base‘(mulGrp‘𝑆))) | 
| 23 | 15, 18 | rhmmhm 20479 | . . . . . . 7
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))) | 
| 24 | 23 | adantr 480 | . . . . . 6
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))) | 
| 25 |  | eqid 2737 | . . . . . . . 8
⊢
(Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅)) | 
| 26 |  | eqid 2737 | . . . . . . . 8
⊢
(Base‘(mulGrp‘𝑆)) = (Base‘(mulGrp‘𝑆)) | 
| 27 | 25, 26 | mhmf1o 18809 | . . . . . . 7
⊢ (𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)) → (𝐹:(Base‘(mulGrp‘𝑅))–1-1-onto→(Base‘(mulGrp‘𝑆)) ↔ ◡𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑅)))) | 
| 28 | 27 | bicomd 223 | . . . . . 6
⊢ (𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)) → (◡𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑅)) ↔ 𝐹:(Base‘(mulGrp‘𝑅))–1-1-onto→(Base‘(mulGrp‘𝑆)))) | 
| 29 | 24, 28 | syl 17 | . . . . 5
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑅)) ↔ 𝐹:(Base‘(mulGrp‘𝑅))–1-1-onto→(Base‘(mulGrp‘𝑆)))) | 
| 30 | 22, 29 | mpbird 257 | . . . 4
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑅))) | 
| 31 | 13, 30 | jca 511 | . . 3
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹 ∈ (𝑆 GrpHom 𝑅) ∧ ◡𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑅)))) | 
| 32 | 18, 15 | isrhm 20478 | . . 3
⊢ (◡𝐹 ∈ (𝑆 RingHom 𝑅) ↔ ((𝑆 ∈ Ring ∧ 𝑅 ∈ Ring) ∧ (◡𝐹 ∈ (𝑆 GrpHom 𝑅) ∧ ◡𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑅))))) | 
| 33 | 4, 31, 32 | sylanbrc 583 | . 2
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ (𝑆 RingHom 𝑅)) | 
| 34 | 8, 9 | rhmf 20485 | . . . . 5
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:𝐵⟶𝐶) | 
| 35 | 34 | adantr 480 | . . . 4
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅)) → 𝐹:𝐵⟶𝐶) | 
| 36 | 35 | ffnd 6737 | . . 3
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅)) → 𝐹 Fn 𝐵) | 
| 37 | 9, 8 | rhmf 20485 | . . . . 5
⊢ (◡𝐹 ∈ (𝑆 RingHom 𝑅) → ◡𝐹:𝐶⟶𝐵) | 
| 38 | 37 | adantl 481 | . . . 4
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅)) → ◡𝐹:𝐶⟶𝐵) | 
| 39 | 38 | ffnd 6737 | . . 3
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅)) → ◡𝐹 Fn 𝐶) | 
| 40 |  | dff1o4 6856 | . . 3
⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹 Fn 𝐵 ∧ ◡𝐹 Fn 𝐶)) | 
| 41 | 36, 39, 40 | sylanbrc 583 | . 2
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅)) → 𝐹:𝐵–1-1-onto→𝐶) | 
| 42 | 33, 41 | impbida 801 | 1
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 RingHom 𝑅))) |