| Step | Hyp | Ref
 | Expression | 
| 1 |   | mhmrcl2 13096 | 
. . . . 5
⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑆 ∈ Mnd) | 
| 2 |   | mhmrcl1 13095 | 
. . . . 5
⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑅 ∈ Mnd) | 
| 3 | 1, 2 | jca 306 | 
. . . 4
⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝑆 ∈ Mnd ∧ 𝑅 ∈ Mnd)) | 
| 4 | 3 | adantr 276 | 
. . 3
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (𝑆 ∈ Mnd ∧ 𝑅 ∈ Mnd)) | 
| 5 |   | f1ocnv 5517 | 
. . . . . 6
⊢ (𝐹:𝐵–1-1-onto→𝐶 → ◡𝐹:𝐶–1-1-onto→𝐵) | 
| 6 | 5 | adantl 277 | 
. . . . 5
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹:𝐶–1-1-onto→𝐵) | 
| 7 |   | f1of 5504 | 
. . . . 5
⊢ (◡𝐹:𝐶–1-1-onto→𝐵 → ◡𝐹:𝐶⟶𝐵) | 
| 8 | 6, 7 | syl 14 | 
. . . 4
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹:𝐶⟶𝐵) | 
| 9 |   | simpll 527 | 
. . . . . . . 8
⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐹 ∈ (𝑅 MndHom 𝑆)) | 
| 10 | 8 | adantr 276 | 
. . . . . . . . 9
⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ◡𝐹:𝐶⟶𝐵) | 
| 11 |   | simprl 529 | 
. . . . . . . . 9
⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ 𝐶) | 
| 12 | 10, 11 | ffvelcdmd 5698 | 
. . . . . . . 8
⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (◡𝐹‘𝑥) ∈ 𝐵) | 
| 13 |   | simprr 531 | 
. . . . . . . . 9
⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ 𝐶) | 
| 14 | 10, 13 | ffvelcdmd 5698 | 
. . . . . . . 8
⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (◡𝐹‘𝑦) ∈ 𝐵) | 
| 15 |   | mhmf1o.b | 
. . . . . . . . 9
⊢ 𝐵 = (Base‘𝑅) | 
| 16 |   | eqid 2196 | 
. . . . . . . . 9
⊢
(+g‘𝑅) = (+g‘𝑅) | 
| 17 |   | eqid 2196 | 
. . . . . . . . 9
⊢
(+g‘𝑆) = (+g‘𝑆) | 
| 18 | 15, 16, 17 | mhmlin 13099 | 
. . . . . . . 8
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ (◡𝐹‘𝑥) ∈ 𝐵 ∧ (◡𝐹‘𝑦) ∈ 𝐵) → (𝐹‘((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦))) = ((𝐹‘(◡𝐹‘𝑥))(+g‘𝑆)(𝐹‘(◡𝐹‘𝑦)))) | 
| 19 | 9, 12, 14, 18 | syl3anc 1249 | 
. . . . . . 7
⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦))) = ((𝐹‘(◡𝐹‘𝑥))(+g‘𝑆)(𝐹‘(◡𝐹‘𝑦)))) | 
| 20 |   | simpr 110 | 
. . . . . . . . . 10
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹:𝐵–1-1-onto→𝐶) | 
| 21 | 20 | adantr 276 | 
. . . . . . . . 9
⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐹:𝐵–1-1-onto→𝐶) | 
| 22 |   | f1ocnvfv2 5825 | 
. . . . . . . . 9
⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑥 ∈ 𝐶) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥) | 
| 23 | 21, 11, 22 | syl2anc 411 | 
. . . . . . . 8
⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥) | 
| 24 |   | f1ocnvfv2 5825 | 
. . . . . . . . 9
⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑦 ∈ 𝐶) → (𝐹‘(◡𝐹‘𝑦)) = 𝑦) | 
| 25 | 21, 13, 24 | syl2anc 411 | 
. . . . . . . 8
⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘(◡𝐹‘𝑦)) = 𝑦) | 
| 26 | 23, 25 | oveq12d 5940 | 
. . . . . . 7
⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘(◡𝐹‘𝑥))(+g‘𝑆)(𝐹‘(◡𝐹‘𝑦))) = (𝑥(+g‘𝑆)𝑦)) | 
| 27 | 19, 26 | eqtrd 2229 | 
. . . . . 6
⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦))) = (𝑥(+g‘𝑆)𝑦)) | 
| 28 | 2 | adantr 276 | 
. . . . . . . . 9
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝑅 ∈ Mnd) | 
| 29 | 28 | adantr 276 | 
. . . . . . . 8
⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑅 ∈ Mnd) | 
| 30 | 15, 16 | mndcl 13064 | 
. . . . . . . 8
⊢ ((𝑅 ∈ Mnd ∧ (◡𝐹‘𝑥) ∈ 𝐵 ∧ (◡𝐹‘𝑦) ∈ 𝐵) → ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)) ∈ 𝐵) | 
| 31 | 29, 12, 14, 30 | syl3anc 1249 | 
. . . . . . 7
⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)) ∈ 𝐵) | 
| 32 |   | f1ocnvfv 5826 | 
. . . . . . 7
⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)) ∈ 𝐵) → ((𝐹‘((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦))) = (𝑥(+g‘𝑆)𝑦) → (◡𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)))) | 
| 33 | 21, 31, 32 | syl2anc 411 | 
. . . . . 6
⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦))) = (𝑥(+g‘𝑆)𝑦) → (◡𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)))) | 
| 34 | 27, 33 | mpd 13 | 
. . . . 5
⊢ (((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (◡𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦))) | 
| 35 | 34 | ralrimivva 2579 | 
. . . 4
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (◡𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦))) | 
| 36 |   | eqid 2196 | 
. . . . . . . . 9
⊢
(0g‘𝑅) = (0g‘𝑅) | 
| 37 |   | eqid 2196 | 
. . . . . . . . 9
⊢
(0g‘𝑆) = (0g‘𝑆) | 
| 38 | 36, 37 | mhm0 13100 | 
. . . . . . . 8
⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆)) | 
| 39 | 38 | adantr 276 | 
. . . . . . 7
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆)) | 
| 40 | 39 | eqcomd 2202 | 
. . . . . 6
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) →
(0g‘𝑆) =
(𝐹‘(0g‘𝑅))) | 
| 41 | 40 | fveq2d 5562 | 
. . . . 5
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹‘(0g‘𝑆)) = (◡𝐹‘(𝐹‘(0g‘𝑅)))) | 
| 42 | 15, 36 | mndidcl 13071 | 
. . . . . . . 8
⊢ (𝑅 ∈ Mnd →
(0g‘𝑅)
∈ 𝐵) | 
| 43 | 2, 42 | syl 14 | 
. . . . . . 7
⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (0g‘𝑅) ∈ 𝐵) | 
| 44 | 43 | adantr 276 | 
. . . . . 6
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) →
(0g‘𝑅)
∈ 𝐵) | 
| 45 |   | f1ocnvfv1 5824 | 
. . . . . 6
⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧
(0g‘𝑅)
∈ 𝐵) → (◡𝐹‘(𝐹‘(0g‘𝑅))) = (0g‘𝑅)) | 
| 46 | 20, 44, 45 | syl2anc 411 | 
. . . . 5
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹‘(𝐹‘(0g‘𝑅))) = (0g‘𝑅)) | 
| 47 | 41, 46 | eqtrd 2229 | 
. . . 4
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹‘(0g‘𝑆)) = (0g‘𝑅)) | 
| 48 | 8, 35, 47 | 3jca 1179 | 
. . 3
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹:𝐶⟶𝐵 ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (◡𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)) ∧ (◡𝐹‘(0g‘𝑆)) = (0g‘𝑅))) | 
| 49 |   | mhmf1o.c | 
. . . 4
⊢ 𝐶 = (Base‘𝑆) | 
| 50 | 49, 15, 17, 16, 37, 36 | ismhm 13093 | 
. . 3
⊢ (◡𝐹 ∈ (𝑆 MndHom 𝑅) ↔ ((𝑆 ∈ Mnd ∧ 𝑅 ∈ Mnd) ∧ (◡𝐹:𝐶⟶𝐵 ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (◡𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑅)(◡𝐹‘𝑦)) ∧ (◡𝐹‘(0g‘𝑆)) = (0g‘𝑅)))) | 
| 51 | 4, 48, 50 | sylanbrc 417 | 
. 2
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ (𝑆 MndHom 𝑅)) | 
| 52 | 15, 49 | mhmf 13097 | 
. . . . 5
⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹:𝐵⟶𝐶) | 
| 53 | 52 | adantr 276 | 
. . . 4
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹:𝐵⟶𝐶) | 
| 54 | 53 | ffnd 5408 | 
. . 3
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹 Fn 𝐵) | 
| 55 | 49, 15 | mhmf 13097 | 
. . . . 5
⊢ (◡𝐹 ∈ (𝑆 MndHom 𝑅) → ◡𝐹:𝐶⟶𝐵) | 
| 56 | 55 | adantl 277 | 
. . . 4
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MndHom 𝑅)) → ◡𝐹:𝐶⟶𝐵) | 
| 57 | 56 | ffnd 5408 | 
. . 3
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MndHom 𝑅)) → ◡𝐹 Fn 𝐶) | 
| 58 |   | dff1o4 5512 | 
. . 3
⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹 Fn 𝐵 ∧ ◡𝐹 Fn 𝐶)) | 
| 59 | 54, 57, 58 | sylanbrc 417 | 
. 2
⊢ ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 MndHom 𝑅)) → 𝐹:𝐵–1-1-onto→𝐶) | 
| 60 | 51, 59 | impbida 596 | 
1
⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 MndHom 𝑅))) |