| Step | Hyp | Ref
| Expression |
| 1 | | imacrhmcl.h |
. . . 4
⊢ (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁)) |
| 2 | | imacrhmcl.s |
. . . 4
⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑀)) |
| 3 | | rhmima 20604 |
. . . 4
⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑆 ∈ (SubRing‘𝑀)) → (𝐹 “ 𝑆) ∈ (SubRing‘𝑁)) |
| 4 | 1, 2, 3 | syl2anc 584 |
. . 3
⊢ (𝜑 → (𝐹 “ 𝑆) ∈ (SubRing‘𝑁)) |
| 5 | | imacrhmcl.c |
. . . 4
⊢ 𝐶 = (𝑁 ↾s (𝐹 “ 𝑆)) |
| 6 | 5 | subrgring 20574 |
. . 3
⊢ ((𝐹 “ 𝑆) ∈ (SubRing‘𝑁) → 𝐶 ∈ Ring) |
| 7 | 4, 6 | syl 17 |
. 2
⊢ (𝜑 → 𝐶 ∈ Ring) |
| 8 | 5 | ressbasss2 17286 |
. . . . . 6
⊢
(Base‘𝐶)
⊆ (𝐹 “ 𝑆) |
| 9 | 8 | sseli 3979 |
. . . . 5
⊢ (𝑥 ∈ (Base‘𝐶) → 𝑥 ∈ (𝐹 “ 𝑆)) |
| 10 | 8 | sseli 3979 |
. . . . 5
⊢ (𝑦 ∈ (Base‘𝐶) → 𝑦 ∈ (𝐹 “ 𝑆)) |
| 11 | 9, 10 | anim12i 613 |
. . . 4
⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) |
| 12 | | eqid 2737 |
. . . . . . . . . 10
⊢
(Base‘𝑀) =
(Base‘𝑀) |
| 13 | | eqid 2737 |
. . . . . . . . . 10
⊢
(Base‘𝑁) =
(Base‘𝑁) |
| 14 | 12, 13 | rhmf 20485 |
. . . . . . . . 9
⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁)) |
| 15 | 1, 14 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐹:(Base‘𝑀)⟶(Base‘𝑁)) |
| 16 | 15 | ffund 6740 |
. . . . . . 7
⊢ (𝜑 → Fun 𝐹) |
| 17 | | fvelima 6974 |
. . . . . . 7
⊢ ((Fun
𝐹 ∧ 𝑥 ∈ (𝐹 “ 𝑆)) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥) |
| 18 | 16, 17 | sylan 580 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 “ 𝑆)) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥) |
| 19 | 18 | adantrr 717 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) → ∃𝑎 ∈ 𝑆 (𝐹‘𝑎) = 𝑥) |
| 20 | | fvelima 6974 |
. . . . . . . . 9
⊢ ((Fun
𝐹 ∧ 𝑦 ∈ (𝐹 “ 𝑆)) → ∃𝑏 ∈ 𝑆 (𝐹‘𝑏) = 𝑦) |
| 21 | 16, 20 | sylan 580 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝑆)) → ∃𝑏 ∈ 𝑆 (𝐹‘𝑏) = 𝑦) |
| 22 | 21 | adantrl 716 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) → ∃𝑏 ∈ 𝑆 (𝐹‘𝑏) = 𝑦) |
| 23 | 22 | adantr 480 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → ∃𝑏 ∈ 𝑆 (𝐹‘𝑏) = 𝑦) |
| 24 | | imacrhmcl.m |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ CRing) |
| 25 | 24 | ad3antrrr 730 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑀 ∈ CRing) |
| 26 | 12 | subrgss 20572 |
. . . . . . . . . . . . 13
⊢ (𝑆 ∈ (SubRing‘𝑀) → 𝑆 ⊆ (Base‘𝑀)) |
| 27 | 2, 26 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑀)) |
| 28 | 27 | ad3antrrr 730 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑆 ⊆ (Base‘𝑀)) |
| 29 | | simplrl 777 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑎 ∈ 𝑆) |
| 30 | 28, 29 | sseldd 3984 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑎 ∈ (Base‘𝑀)) |
| 31 | | simprl 771 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑏 ∈ 𝑆) |
| 32 | 28, 31 | sseldd 3984 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝑏 ∈ (Base‘𝑀)) |
| 33 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(.r‘𝑀) = (.r‘𝑀) |
| 34 | 12, 33 | crngcom 20248 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ CRing ∧ 𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀)) → (𝑎(.r‘𝑀)𝑏) = (𝑏(.r‘𝑀)𝑎)) |
| 35 | 25, 30, 32, 34 | syl3anc 1373 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (𝑎(.r‘𝑀)𝑏) = (𝑏(.r‘𝑀)𝑎)) |
| 36 | 35 | fveq2d 6910 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (𝐹‘(𝑎(.r‘𝑀)𝑏)) = (𝐹‘(𝑏(.r‘𝑀)𝑎))) |
| 37 | 1 | ad3antrrr 730 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → 𝐹 ∈ (𝑀 RingHom 𝑁)) |
| 38 | | eqid 2737 |
. . . . . . . . . 10
⊢
(.r‘𝑁) = (.r‘𝑁) |
| 39 | 12, 33, 38 | rhmmul 20486 |
. . . . . . . . 9
⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀)) → (𝐹‘(𝑎(.r‘𝑀)𝑏)) = ((𝐹‘𝑎)(.r‘𝑁)(𝐹‘𝑏))) |
| 40 | 37, 30, 32, 39 | syl3anc 1373 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (𝐹‘(𝑎(.r‘𝑀)𝑏)) = ((𝐹‘𝑎)(.r‘𝑁)(𝐹‘𝑏))) |
| 41 | 12, 33, 38 | rhmmul 20486 |
. . . . . . . . 9
⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑏 ∈ (Base‘𝑀) ∧ 𝑎 ∈ (Base‘𝑀)) → (𝐹‘(𝑏(.r‘𝑀)𝑎)) = ((𝐹‘𝑏)(.r‘𝑁)(𝐹‘𝑎))) |
| 42 | 37, 32, 30, 41 | syl3anc 1373 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (𝐹‘(𝑏(.r‘𝑀)𝑎)) = ((𝐹‘𝑏)(.r‘𝑁)(𝐹‘𝑎))) |
| 43 | 36, 40, 42 | 3eqtr3d 2785 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → ((𝐹‘𝑎)(.r‘𝑁)(𝐹‘𝑏)) = ((𝐹‘𝑏)(.r‘𝑁)(𝐹‘𝑎))) |
| 44 | | imaexg 7935 |
. . . . . . . . . 10
⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → (𝐹 “ 𝑆) ∈ V) |
| 45 | 5, 38 | ressmulr 17351 |
. . . . . . . . . 10
⊢ ((𝐹 “ 𝑆) ∈ V → (.r‘𝑁) = (.r‘𝐶)) |
| 46 | 1, 44, 45 | 3syl 18 |
. . . . . . . . 9
⊢ (𝜑 → (.r‘𝑁) = (.r‘𝐶)) |
| 47 | 46 | ad3antrrr 730 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (.r‘𝑁) = (.r‘𝐶)) |
| 48 | | simplrr 778 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (𝐹‘𝑎) = 𝑥) |
| 49 | | simprr 773 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (𝐹‘𝑏) = 𝑦) |
| 50 | 47, 48, 49 | oveq123d 7452 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → ((𝐹‘𝑎)(.r‘𝑁)(𝐹‘𝑏)) = (𝑥(.r‘𝐶)𝑦)) |
| 51 | 47, 49, 48 | oveq123d 7452 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → ((𝐹‘𝑏)(.r‘𝑁)(𝐹‘𝑎)) = (𝑦(.r‘𝐶)𝑥)) |
| 52 | 43, 50, 51 | 3eqtr3d 2785 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) ∧ (𝑏 ∈ 𝑆 ∧ (𝐹‘𝑏) = 𝑦)) → (𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥)) |
| 53 | 23, 52 | rexlimddv 3161 |
. . . . 5
⊢ (((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ (𝐹‘𝑎) = 𝑥)) → (𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥)) |
| 54 | 19, 53 | rexlimddv 3161 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐹 “ 𝑆) ∧ 𝑦 ∈ (𝐹 “ 𝑆))) → (𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥)) |
| 55 | 11, 54 | sylan2 593 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥)) |
| 56 | 55 | ralrimivva 3202 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥)) |
| 57 | | eqid 2737 |
. . 3
⊢
(Base‘𝐶) =
(Base‘𝐶) |
| 58 | | eqid 2737 |
. . 3
⊢
(.r‘𝐶) = (.r‘𝐶) |
| 59 | 57, 58 | iscrng2 20249 |
. 2
⊢ (𝐶 ∈ CRing ↔ (𝐶 ∈ Ring ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥))) |
| 60 | 7, 56, 59 | sylanbrc 583 |
1
⊢ (𝜑 → 𝐶 ∈ CRing) |