Proof of Theorem rnghmf1o
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | rnghmrcl 20438 | . . . . 5
⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝑅 ∈ Rng ∧ 𝑆 ∈ Rng)) | 
| 2 | 1 | ancomd 461 | . . . 4
⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝑆 ∈ Rng ∧ 𝑅 ∈ Rng)) | 
| 3 | 2 | adantr 480 | . . 3
⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (𝑆 ∈ Rng ∧ 𝑅 ∈ Rng)) | 
| 4 |  | simpr 484 | . . . . 5
⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹:𝐵–1-1-onto→𝐶) | 
| 5 |  | rnghmghm 20447 | . . . . . . 7
⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | 
| 6 | 5 | adantr 480 | . . . . . 6
⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | 
| 7 |  | rnghmf1o.b | . . . . . . . 8
⊢ 𝐵 = (Base‘𝑅) | 
| 8 |  | rnghmf1o.c | . . . . . . . 8
⊢ 𝐶 = (Base‘𝑆) | 
| 9 | 7, 8 | ghmf1o 19266 | . . . . . . 7
⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 GrpHom 𝑅))) | 
| 10 | 9 | bicomd 223 | . . . . . 6
⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (◡𝐹 ∈ (𝑆 GrpHom 𝑅) ↔ 𝐹:𝐵–1-1-onto→𝐶)) | 
| 11 | 6, 10 | syl 17 | . . . . 5
⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹 ∈ (𝑆 GrpHom 𝑅) ↔ 𝐹:𝐵–1-1-onto→𝐶)) | 
| 12 | 4, 11 | mpbird 257 | . . . 4
⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ (𝑆 GrpHom 𝑅)) | 
| 13 |  | eqidd 2738 | . . . . . . 7
⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 = 𝐹) | 
| 14 |  | eqid 2737 | . . . . . . . . 9
⊢
(mulGrp‘𝑅) =
(mulGrp‘𝑅) | 
| 15 | 14, 7 | mgpbas 20142 | . . . . . . . 8
⊢ 𝐵 =
(Base‘(mulGrp‘𝑅)) | 
| 16 | 15 | a1i 11 | . . . . . . 7
⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐵 = (Base‘(mulGrp‘𝑅))) | 
| 17 |  | eqid 2737 | . . . . . . . . 9
⊢
(mulGrp‘𝑆) =
(mulGrp‘𝑆) | 
| 18 | 17, 8 | mgpbas 20142 | . . . . . . . 8
⊢ 𝐶 =
(Base‘(mulGrp‘𝑆)) | 
| 19 | 18 | a1i 11 | . . . . . . 7
⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐶 = (Base‘(mulGrp‘𝑆))) | 
| 20 | 13, 16, 19 | f1oeq123d 6842 | . . . . . 6
⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ 𝐹:(Base‘(mulGrp‘𝑅))–1-1-onto→(Base‘(mulGrp‘𝑆)))) | 
| 21 | 20 | biimpa 476 | . . . . 5
⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹:(Base‘(mulGrp‘𝑅))–1-1-onto→(Base‘(mulGrp‘𝑆))) | 
| 22 | 14, 17 | rnghmmgmhm 20443 | . . . . . . 7
⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 ∈ ((mulGrp‘𝑅) MgmHom (mulGrp‘𝑆))) | 
| 23 | 22 | adantr 480 | . . . . . 6
⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹 ∈ ((mulGrp‘𝑅) MgmHom (mulGrp‘𝑆))) | 
| 24 |  | eqid 2737 | . . . . . . . 8
⊢
(Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅)) | 
| 25 |  | eqid 2737 | . . . . . . . 8
⊢
(Base‘(mulGrp‘𝑆)) = (Base‘(mulGrp‘𝑆)) | 
| 26 | 24, 25 | mgmhmf1o 18713 | . . . . . . 7
⊢ (𝐹 ∈ ((mulGrp‘𝑅) MgmHom (mulGrp‘𝑆)) → (𝐹:(Base‘(mulGrp‘𝑅))–1-1-onto→(Base‘(mulGrp‘𝑆)) ↔ ◡𝐹 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑅)))) | 
| 27 | 26 | bicomd 223 | . . . . . 6
⊢ (𝐹 ∈ ((mulGrp‘𝑅) MgmHom (mulGrp‘𝑆)) → (◡𝐹 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑅)) ↔ 𝐹:(Base‘(mulGrp‘𝑅))–1-1-onto→(Base‘(mulGrp‘𝑆)))) | 
| 28 | 23, 27 | syl 17 | . . . . 5
⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑅)) ↔ 𝐹:(Base‘(mulGrp‘𝑅))–1-1-onto→(Base‘(mulGrp‘𝑆)))) | 
| 29 | 21, 28 | mpbird 257 | . . . 4
⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑅))) | 
| 30 | 12, 29 | jca 511 | . . 3
⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹 ∈ (𝑆 GrpHom 𝑅) ∧ ◡𝐹 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑅)))) | 
| 31 | 17, 14 | isrnghmmul 20442 | . . 3
⊢ (◡𝐹 ∈ (𝑆 RngHom 𝑅) ↔ ((𝑆 ∈ Rng ∧ 𝑅 ∈ Rng) ∧ (◡𝐹 ∈ (𝑆 GrpHom 𝑅) ∧ ◡𝐹 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑅))))) | 
| 32 | 3, 30, 31 | sylanbrc 583 | . 2
⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ (𝑆 RngHom 𝑅)) | 
| 33 | 7, 8 | rnghmf 20448 | . . . . 5
⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹:𝐵⟶𝐶) | 
| 34 | 33 | adantr 480 | . . . 4
⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → 𝐹:𝐵⟶𝐶) | 
| 35 | 34 | ffnd 6737 | . . 3
⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → 𝐹 Fn 𝐵) | 
| 36 | 8, 7 | rnghmf 20448 | . . . . 5
⊢ (◡𝐹 ∈ (𝑆 RngHom 𝑅) → ◡𝐹:𝐶⟶𝐵) | 
| 37 | 36 | adantl 481 | . . . 4
⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → ◡𝐹:𝐶⟶𝐵) | 
| 38 | 37 | ffnd 6737 | . . 3
⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → ◡𝐹 Fn 𝐶) | 
| 39 |  | dff1o4 6856 | . . 3
⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹 Fn 𝐵 ∧ ◡𝐹 Fn 𝐶)) | 
| 40 | 35, 38, 39 | sylanbrc 583 | . 2
⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → 𝐹:𝐵–1-1-onto→𝐶) | 
| 41 | 32, 40 | impbida 801 | 1
⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 RngHom 𝑅))) |