| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | imasring.u | . . . 4
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) | 
| 2 |  | imasring.v | . . . 4
⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | 
| 3 |  | imasring.f | . . . 4
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) | 
| 4 |  | imasring.r | . . . 4
⊢ (𝜑 → 𝑅 ∈ Ring) | 
| 5 | 1, 2, 3, 4 | imasbas 17558 | . . 3
⊢ (𝜑 → 𝐵 = (Base‘𝑈)) | 
| 6 |  | eqidd 2737 | . . 3
⊢ (𝜑 → (+g‘𝑈) = (+g‘𝑈)) | 
| 7 |  | eqidd 2737 | . . 3
⊢ (𝜑 → (.r‘𝑈) = (.r‘𝑈)) | 
| 8 |  | imasring.p | . . . . . 6
⊢  + =
(+g‘𝑅) | 
| 9 | 8 | a1i 11 | . . . . 5
⊢ (𝜑 → + =
(+g‘𝑅)) | 
| 10 |  | imasring.e1 | . . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) | 
| 11 |  | ringgrp 20236 | . . . . . 6
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | 
| 12 | 4, 11 | syl 17 | . . . . 5
⊢ (𝜑 → 𝑅 ∈ Grp) | 
| 13 |  | eqid 2736 | . . . . 5
⊢
(0g‘𝑅) = (0g‘𝑅) | 
| 14 | 1, 2, 9, 3, 10, 12, 13 | imasgrp 19075 | . . . 4
⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘(0g‘𝑅)) = (0g‘𝑈))) | 
| 15 | 14 | simpld 494 | . . 3
⊢ (𝜑 → 𝑈 ∈ Grp) | 
| 16 |  | imasring.e2 | . . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) | 
| 17 |  | imasring.t | . . . . 5
⊢  · =
(.r‘𝑅) | 
| 18 |  | eqid 2736 | . . . . 5
⊢
(.r‘𝑈) = (.r‘𝑈) | 
| 19 | 4 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑅 ∈ Ring) | 
| 20 |  | simprl 770 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑢 ∈ 𝑉) | 
| 21 | 2 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑉 = (Base‘𝑅)) | 
| 22 | 20, 21 | eleqtrd 2842 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑢 ∈ (Base‘𝑅)) | 
| 23 |  | simprr 772 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ 𝑉) | 
| 24 | 23, 21 | eleqtrd 2842 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ (Base‘𝑅)) | 
| 25 |  | eqid 2736 | . . . . . . . . 9
⊢
(Base‘𝑅) =
(Base‘𝑅) | 
| 26 | 25, 17 | ringcl 20248 | . . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 · 𝑣) ∈ (Base‘𝑅)) | 
| 27 | 19, 22, 24, 26 | syl3anc 1372 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 · 𝑣) ∈ (Base‘𝑅)) | 
| 28 | 27, 21 | eleqtrrd 2843 | . . . . . 6
⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 · 𝑣) ∈ 𝑉) | 
| 29 | 28 | caovclg 7626 | . . . . 5
⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉) | 
| 30 | 3, 16, 1, 2, 4, 17,
18, 29 | imasmulf 17582 | . . . 4
⊢ (𝜑 → (.r‘𝑈):(𝐵 × 𝐵)⟶𝐵) | 
| 31 |  | fovcdm 7604 | . . . 4
⊢
(((.r‘𝑈):(𝐵 × 𝐵)⟶𝐵 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(.r‘𝑈)𝑣) ∈ 𝐵) | 
| 32 | 30, 31 | syl3an1 1163 | . . 3
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(.r‘𝑈)𝑣) ∈ 𝐵) | 
| 33 |  | forn 6822 | . . . . . . . . . 10
⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵) | 
| 34 | 3, 33 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → ran 𝐹 = 𝐵) | 
| 35 | 34 | eleq2d 2826 | . . . . . . . 8
⊢ (𝜑 → (𝑢 ∈ ran 𝐹 ↔ 𝑢 ∈ 𝐵)) | 
| 36 | 34 | eleq2d 2826 | . . . . . . . 8
⊢ (𝜑 → (𝑣 ∈ ran 𝐹 ↔ 𝑣 ∈ 𝐵)) | 
| 37 | 34 | eleq2d 2826 | . . . . . . . 8
⊢ (𝜑 → (𝑤 ∈ ran 𝐹 ↔ 𝑤 ∈ 𝐵)) | 
| 38 | 35, 36, 37 | 3anbi123d 1437 | . . . . . . 7
⊢ (𝜑 → ((𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹) ↔ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))) | 
| 39 |  | fofn 6821 | . . . . . . . . 9
⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉) | 
| 40 | 3, 39 | syl 17 | . . . . . . . 8
⊢ (𝜑 → 𝐹 Fn 𝑉) | 
| 41 |  | fvelrnb 6968 | . . . . . . . . 9
⊢ (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢)) | 
| 42 |  | fvelrnb 6968 | . . . . . . . . 9
⊢ (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣)) | 
| 43 |  | fvelrnb 6968 | . . . . . . . . 9
⊢ (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤)) | 
| 44 | 41, 42, 43 | 3anbi123d 1437 | . . . . . . . 8
⊢ (𝐹 Fn 𝑉 → ((𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹) ↔ (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 ∧ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣 ∧ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤))) | 
| 45 | 40, 44 | syl 17 | . . . . . . 7
⊢ (𝜑 → ((𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹) ↔ (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 ∧ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣 ∧ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤))) | 
| 46 | 38, 45 | bitr3d 281 | . . . . . 6
⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ↔ (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 ∧ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣 ∧ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤))) | 
| 47 |  | 3reeanv 3229 | . . . . . 6
⊢
(∃𝑥 ∈
𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) ↔ (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 ∧ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣 ∧ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤)) | 
| 48 | 46, 47 | bitr4di 289 | . . . . 5
⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤))) | 
| 49 | 4 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑅 ∈ Ring) | 
| 50 |  | simp2 1137 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ 𝑉) | 
| 51 | 2 | 3ad2ant1 1133 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑉 = (Base‘𝑅)) | 
| 52 | 50, 51 | eleqtrd 2842 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅)) | 
| 53 | 52 | 3adant3r3 1184 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ (Base‘𝑅)) | 
| 54 |  | simp3 1138 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉) | 
| 55 | 54, 51 | eleqtrd 2842 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ (Base‘𝑅)) | 
| 56 | 55 | 3adant3r3 1184 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝑅)) | 
| 57 |  | simpr3 1196 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉) | 
| 58 | 2 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑉 = (Base‘𝑅)) | 
| 59 | 57, 58 | eleqtrd 2842 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ (Base‘𝑅)) | 
| 60 | 25, 17 | ringass 20251 | . . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) | 
| 61 | 49, 53, 56, 59, 60 | syl13anc 1373 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) | 
| 62 | 61 | fveq2d 6909 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 · 𝑦) · 𝑧)) = (𝐹‘(𝑥 · (𝑦 · 𝑧)))) | 
| 63 |  | simpl 482 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝜑) | 
| 64 | 28 | caovclg 7626 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ 𝑉) | 
| 65 | 64 | 3adantr3 1171 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ 𝑉) | 
| 66 | 3, 16, 1, 2, 4, 17,
18 | imasmulval 17581 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 · 𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑦))(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧))) | 
| 67 | 63, 65, 57, 66 | syl3anc 1372 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 · 𝑦))(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧))) | 
| 68 |  | simpr1 1194 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ 𝑉) | 
| 69 | 28 | caovclg 7626 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 · 𝑧) ∈ 𝑉) | 
| 70 | 69 | 3adantr1 1169 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 · 𝑧) ∈ 𝑉) | 
| 71 | 3, 16, 1, 2, 4, 17,
18 | imasmulval 17581 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑦 · 𝑧) ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧)))) | 
| 72 | 63, 68, 70, 71 | syl3anc 1372 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧)))) | 
| 73 | 62, 67, 72 | 3eqtr4d 2786 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 · 𝑦))(.r‘𝑈)(𝐹‘𝑧)) = ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 · 𝑧)))) | 
| 74 | 3, 16, 1, 2, 4, 17,
18 | imasmulval 17581 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 · 𝑦))) | 
| 75 | 74 | 3adant3r3 1184 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 · 𝑦))) | 
| 76 | 75 | oveq1d 7447 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦))(.r‘𝑈)(𝐹‘𝑧)) = ((𝐹‘(𝑥 · 𝑦))(.r‘𝑈)(𝐹‘𝑧))) | 
| 77 | 3, 16, 1, 2, 4, 17,
18 | imasmulval 17581 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 · 𝑧))) | 
| 78 | 77 | 3adant3r1 1182 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 · 𝑧))) | 
| 79 | 78 | oveq2d 7448 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧))) = ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 · 𝑧)))) | 
| 80 | 73, 76, 79 | 3eqtr4d 2786 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦))(.r‘𝑈)(𝐹‘𝑧)) = ((𝐹‘𝑥)(.r‘𝑈)((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧)))) | 
| 81 |  | simp1 1136 | . . . . . . . . . . . . 13
⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑥) = 𝑢) | 
| 82 |  | simp2 1137 | . . . . . . . . . . . . 13
⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑦) = 𝑣) | 
| 83 | 81, 82 | oveq12d 7450 | . . . . . . . . . . . 12
⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦)) = (𝑢(.r‘𝑈)𝑣)) | 
| 84 |  | simp3 1138 | . . . . . . . . . . . 12
⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑧) = 𝑤) | 
| 85 | 83, 84 | oveq12d 7450 | . . . . . . . . . . 11
⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦))(.r‘𝑈)(𝐹‘𝑧)) = ((𝑢(.r‘𝑈)𝑣)(.r‘𝑈)𝑤)) | 
| 86 | 82, 84 | oveq12d 7450 | . . . . . . . . . . . 12
⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧)) = (𝑣(.r‘𝑈)𝑤)) | 
| 87 | 81, 86 | oveq12d 7450 | . . . . . . . . . . 11
⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(.r‘𝑈)((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧))) = (𝑢(.r‘𝑈)(𝑣(.r‘𝑈)𝑤))) | 
| 88 | 85, 87 | eqeq12d 2752 | . . . . . . . . . 10
⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦))(.r‘𝑈)(𝐹‘𝑧)) = ((𝐹‘𝑥)(.r‘𝑈)((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧))) ↔ ((𝑢(.r‘𝑈)𝑣)(.r‘𝑈)𝑤) = (𝑢(.r‘𝑈)(𝑣(.r‘𝑈)𝑤)))) | 
| 89 | 80, 88 | syl5ibcom 245 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(.r‘𝑈)𝑣)(.r‘𝑈)𝑤) = (𝑢(.r‘𝑈)(𝑣(.r‘𝑈)𝑤)))) | 
| 90 | 89 | 3exp2 1354 | . . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑉 → (𝑧 ∈ 𝑉 → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(.r‘𝑈)𝑣)(.r‘𝑈)𝑤) = (𝑢(.r‘𝑈)(𝑣(.r‘𝑈)𝑤))))))) | 
| 91 | 90 | imp32 418 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑧 ∈ 𝑉 → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(.r‘𝑈)𝑣)(.r‘𝑈)𝑤) = (𝑢(.r‘𝑈)(𝑣(.r‘𝑈)𝑤))))) | 
| 92 | 91 | rexlimdv 3152 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(.r‘𝑈)𝑣)(.r‘𝑈)𝑤) = (𝑢(.r‘𝑈)(𝑣(.r‘𝑈)𝑤)))) | 
| 93 | 92 | rexlimdvva 3212 | . . . . 5
⊢ (𝜑 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(.r‘𝑈)𝑣)(.r‘𝑈)𝑤) = (𝑢(.r‘𝑈)(𝑣(.r‘𝑈)𝑤)))) | 
| 94 | 48, 93 | sylbid 240 | . . . 4
⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝑢(.r‘𝑈)𝑣)(.r‘𝑈)𝑤) = (𝑢(.r‘𝑈)(𝑣(.r‘𝑈)𝑤)))) | 
| 95 | 94 | imp 406 | . . 3
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(.r‘𝑈)𝑣)(.r‘𝑈)𝑤) = (𝑢(.r‘𝑈)(𝑣(.r‘𝑈)𝑤))) | 
| 96 | 25, 8, 17 | ringdi 20259 | . . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) | 
| 97 | 49, 53, 56, 59, 96 | syl13anc 1373 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) | 
| 98 | 97 | fveq2d 6909 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘(𝑥 · (𝑦 + 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧)))) | 
| 99 | 25, 8 | ringacl 20276 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑅 ∈ Ring ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 + 𝑣) ∈ (Base‘𝑅)) | 
| 100 | 19, 22, 24, 99 | syl3anc 1372 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 + 𝑣) ∈ (Base‘𝑅)) | 
| 101 | 100, 21 | eleqtrrd 2843 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 + 𝑣) ∈ 𝑉) | 
| 102 | 101 | caovclg 7626 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉) | 
| 103 | 102 | 3adantr1 1169 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉) | 
| 104 | 3, 16, 1, 2, 4, 17,
18 | imasmulval 17581 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧)))) | 
| 105 | 63, 68, 103, 104 | syl3anc 1372 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧)))) | 
| 106 | 28 | caovclg 7626 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · 𝑧) ∈ 𝑉) | 
| 107 | 106 | 3adantr2 1170 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · 𝑧) ∈ 𝑉) | 
| 108 |  | eqid 2736 | . . . . . . . . . . . . . 14
⊢
(+g‘𝑈) = (+g‘𝑈) | 
| 109 | 3, 10, 1, 2, 4, 8, 108 | imasaddval 17578 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 · 𝑦) ∈ 𝑉 ∧ (𝑥 · 𝑧) ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑦))(+g‘𝑈)(𝐹‘(𝑥 · 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧)))) | 
| 110 | 63, 65, 107, 109 | syl3anc 1372 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 · 𝑦))(+g‘𝑈)(𝐹‘(𝑥 · 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧)))) | 
| 111 | 98, 105, 110 | 3eqtr4d 2786 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 + 𝑧))) = ((𝐹‘(𝑥 · 𝑦))(+g‘𝑈)(𝐹‘(𝑥 · 𝑧)))) | 
| 112 | 3, 10, 1, 2, 4, 8, 108 | imasaddval 17578 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧))) | 
| 113 | 112 | 3adant3r1 1182 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧))) | 
| 114 | 113 | oveq2d 7448 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧))) = ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 + 𝑧)))) | 
| 115 | 3, 16, 1, 2, 4, 17,
18 | imasmulval 17581 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑥 · 𝑧))) | 
| 116 | 115 | 3adant3r2 1183 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑥 · 𝑧))) | 
| 117 | 75, 116 | oveq12d 7450 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦))(+g‘𝑈)((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧))) = ((𝐹‘(𝑥 · 𝑦))(+g‘𝑈)(𝐹‘(𝑥 · 𝑧)))) | 
| 118 | 111, 114,
117 | 3eqtr4d 2786 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧))) = (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦))(+g‘𝑈)((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧)))) | 
| 119 | 82, 84 | oveq12d 7450 | . . . . . . . . . . . 12
⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧)) = (𝑣(+g‘𝑈)𝑤)) | 
| 120 | 81, 119 | oveq12d 7450 | . . . . . . . . . . 11
⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(.r‘𝑈)((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧))) = (𝑢(.r‘𝑈)(𝑣(+g‘𝑈)𝑤))) | 
| 121 | 81, 84 | oveq12d 7450 | . . . . . . . . . . . 12
⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧)) = (𝑢(.r‘𝑈)𝑤)) | 
| 122 | 83, 121 | oveq12d 7450 | . . . . . . . . . . 11
⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦))(+g‘𝑈)((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧))) = ((𝑢(.r‘𝑈)𝑣)(+g‘𝑈)(𝑢(.r‘𝑈)𝑤))) | 
| 123 | 120, 122 | eqeq12d 2752 | . . . . . . . . . 10
⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (((𝐹‘𝑥)(.r‘𝑈)((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧))) = (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦))(+g‘𝑈)((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧))) ↔ (𝑢(.r‘𝑈)(𝑣(+g‘𝑈)𝑤)) = ((𝑢(.r‘𝑈)𝑣)(+g‘𝑈)(𝑢(.r‘𝑈)𝑤)))) | 
| 124 | 118, 123 | syl5ibcom 245 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝑢(.r‘𝑈)(𝑣(+g‘𝑈)𝑤)) = ((𝑢(.r‘𝑈)𝑣)(+g‘𝑈)(𝑢(.r‘𝑈)𝑤)))) | 
| 125 | 124 | 3exp2 1354 | . . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑉 → (𝑧 ∈ 𝑉 → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝑢(.r‘𝑈)(𝑣(+g‘𝑈)𝑤)) = ((𝑢(.r‘𝑈)𝑣)(+g‘𝑈)(𝑢(.r‘𝑈)𝑤))))))) | 
| 126 | 125 | imp32 418 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑧 ∈ 𝑉 → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝑢(.r‘𝑈)(𝑣(+g‘𝑈)𝑤)) = ((𝑢(.r‘𝑈)𝑣)(+g‘𝑈)(𝑢(.r‘𝑈)𝑤))))) | 
| 127 | 126 | rexlimdv 3152 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝑢(.r‘𝑈)(𝑣(+g‘𝑈)𝑤)) = ((𝑢(.r‘𝑈)𝑣)(+g‘𝑈)(𝑢(.r‘𝑈)𝑤)))) | 
| 128 | 127 | rexlimdvva 3212 | . . . . 5
⊢ (𝜑 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝑢(.r‘𝑈)(𝑣(+g‘𝑈)𝑤)) = ((𝑢(.r‘𝑈)𝑣)(+g‘𝑈)(𝑢(.r‘𝑈)𝑤)))) | 
| 129 | 48, 128 | sylbid 240 | . . . 4
⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → (𝑢(.r‘𝑈)(𝑣(+g‘𝑈)𝑤)) = ((𝑢(.r‘𝑈)𝑣)(+g‘𝑈)(𝑢(.r‘𝑈)𝑤)))) | 
| 130 | 129 | imp 406 | . . 3
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝑈)(𝑣(+g‘𝑈)𝑤)) = ((𝑢(.r‘𝑈)𝑣)(+g‘𝑈)(𝑢(.r‘𝑈)𝑤))) | 
| 131 | 25, 8, 17 | ringdir 20260 | . . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) | 
| 132 | 49, 53, 56, 59, 131 | syl13anc 1373 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) | 
| 133 | 132 | fveq2d 6909 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) · 𝑧)) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧)))) | 
| 134 | 101 | caovclg 7626 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉) | 
| 135 | 134 | 3adantr3 1171 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉) | 
| 136 | 3, 16, 1, 2, 4, 17,
18 | imasmulval 17581 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘(𝑥 + 𝑦))(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 + 𝑦) · 𝑧))) | 
| 137 | 63, 135, 57, 136 | syl3anc 1372 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 + 𝑦))(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 + 𝑦) · 𝑧))) | 
| 138 | 3, 10, 1, 2, 4, 8, 108 | imasaddval 17578 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 · 𝑧) ∈ 𝑉 ∧ (𝑦 · 𝑧) ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑧))(+g‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧)))) | 
| 139 | 63, 107, 70, 138 | syl3anc 1372 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 · 𝑧))(+g‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧)))) | 
| 140 | 133, 137,
139 | 3eqtr4d 2786 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 + 𝑦))(.r‘𝑈)(𝐹‘𝑧)) = ((𝐹‘(𝑥 · 𝑧))(+g‘𝑈)(𝐹‘(𝑦 · 𝑧)))) | 
| 141 | 3, 10, 1, 2, 4, 8, 108 | imasaddval 17578 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦))) | 
| 142 | 141 | 3adant3r3 1184 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦))) | 
| 143 | 142 | oveq1d 7447 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦))(.r‘𝑈)(𝐹‘𝑧)) = ((𝐹‘(𝑥 + 𝑦))(.r‘𝑈)(𝐹‘𝑧))) | 
| 144 | 116, 78 | oveq12d 7450 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧))(+g‘𝑈)((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧))) = ((𝐹‘(𝑥 · 𝑧))(+g‘𝑈)(𝐹‘(𝑦 · 𝑧)))) | 
| 145 | 140, 143,
144 | 3eqtr4d 2786 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦))(.r‘𝑈)(𝐹‘𝑧)) = (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧))(+g‘𝑈)((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧)))) | 
| 146 | 81, 82 | oveq12d 7450 | . . . . . . . . . . . 12
⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) = (𝑢(+g‘𝑈)𝑣)) | 
| 147 | 146, 84 | oveq12d 7450 | . . . . . . . . . . 11
⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦))(.r‘𝑈)(𝐹‘𝑧)) = ((𝑢(+g‘𝑈)𝑣)(.r‘𝑈)𝑤)) | 
| 148 | 121, 86 | oveq12d 7450 | . . . . . . . . . . 11
⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧))(+g‘𝑈)((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧))) = ((𝑢(.r‘𝑈)𝑤)(+g‘𝑈)(𝑣(.r‘𝑈)𝑤))) | 
| 149 | 147, 148 | eqeq12d 2752 | . . . . . . . . . 10
⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦))(.r‘𝑈)(𝐹‘𝑧)) = (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧))(+g‘𝑈)((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧))) ↔ ((𝑢(+g‘𝑈)𝑣)(.r‘𝑈)𝑤) = ((𝑢(.r‘𝑈)𝑤)(+g‘𝑈)(𝑣(.r‘𝑈)𝑤)))) | 
| 150 | 145, 149 | syl5ibcom 245 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+g‘𝑈)𝑣)(.r‘𝑈)𝑤) = ((𝑢(.r‘𝑈)𝑤)(+g‘𝑈)(𝑣(.r‘𝑈)𝑤)))) | 
| 151 | 150 | 3exp2 1354 | . . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑉 → (𝑧 ∈ 𝑉 → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+g‘𝑈)𝑣)(.r‘𝑈)𝑤) = ((𝑢(.r‘𝑈)𝑤)(+g‘𝑈)(𝑣(.r‘𝑈)𝑤))))))) | 
| 152 | 151 | imp32 418 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑧 ∈ 𝑉 → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+g‘𝑈)𝑣)(.r‘𝑈)𝑤) = ((𝑢(.r‘𝑈)𝑤)(+g‘𝑈)(𝑣(.r‘𝑈)𝑤))))) | 
| 153 | 152 | rexlimdv 3152 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+g‘𝑈)𝑣)(.r‘𝑈)𝑤) = ((𝑢(.r‘𝑈)𝑤)(+g‘𝑈)(𝑣(.r‘𝑈)𝑤)))) | 
| 154 | 153 | rexlimdvva 3212 | . . . . 5
⊢ (𝜑 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+g‘𝑈)𝑣)(.r‘𝑈)𝑤) = ((𝑢(.r‘𝑈)𝑤)(+g‘𝑈)(𝑣(.r‘𝑈)𝑤)))) | 
| 155 | 48, 154 | sylbid 240 | . . . 4
⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝑢(+g‘𝑈)𝑣)(.r‘𝑈)𝑤) = ((𝑢(.r‘𝑈)𝑤)(+g‘𝑈)(𝑣(.r‘𝑈)𝑤)))) | 
| 156 | 155 | imp 406 | . . 3
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+g‘𝑈)𝑣)(.r‘𝑈)𝑤) = ((𝑢(.r‘𝑈)𝑤)(+g‘𝑈)(𝑣(.r‘𝑈)𝑤))) | 
| 157 |  | fof 6819 | . . . . 5
⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵) | 
| 158 | 3, 157 | syl 17 | . . . 4
⊢ (𝜑 → 𝐹:𝑉⟶𝐵) | 
| 159 |  | imasring.o | . . . . . . 7
⊢  1 =
(1r‘𝑅) | 
| 160 | 25, 159 | ringidcl 20263 | . . . . . 6
⊢ (𝑅 ∈ Ring → 1 ∈
(Base‘𝑅)) | 
| 161 | 4, 160 | syl 17 | . . . . 5
⊢ (𝜑 → 1 ∈ (Base‘𝑅)) | 
| 162 | 161, 2 | eleqtrrd 2843 | . . . 4
⊢ (𝜑 → 1 ∈ 𝑉) | 
| 163 | 158, 162 | ffvelcdmd 7104 | . . 3
⊢ (𝜑 → (𝐹‘ 1 ) ∈ 𝐵) | 
| 164 | 40, 41 | syl 17 | . . . . . 6
⊢ (𝜑 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢)) | 
| 165 | 35, 164 | bitr3d 281 | . . . . 5
⊢ (𝜑 → (𝑢 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢)) | 
| 166 |  | simpl 482 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝜑) | 
| 167 | 162 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 1 ∈ 𝑉) | 
| 168 |  | simpr 484 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉) | 
| 169 | 3, 16, 1, 2, 4, 17,
18 | imasmulval 17581 | . . . . . . . . 9
⊢ ((𝜑 ∧ 1 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 1
)(.r‘𝑈)(𝐹‘𝑥)) = (𝐹‘( 1 · 𝑥))) | 
| 170 | 166, 167,
168, 169 | syl3anc 1372 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 1
)(.r‘𝑈)(𝐹‘𝑥)) = (𝐹‘( 1 · 𝑥))) | 
| 171 | 2 | eleq2d 2826 | . . . . . . . . . . 11
⊢ (𝜑 → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑅))) | 
| 172 | 171 | biimpa 476 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅)) | 
| 173 | 25, 17, 159 | ringlidm 20267 | . . . . . . . . . 10
⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ( 1 · 𝑥) = 𝑥) | 
| 174 | 4, 172, 173 | syl2an2r 685 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 1 · 𝑥) = 𝑥) | 
| 175 | 174 | fveq2d 6909 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 1 · 𝑥)) = (𝐹‘𝑥)) | 
| 176 | 170, 175 | eqtrd 2776 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 1
)(.r‘𝑈)(𝐹‘𝑥)) = (𝐹‘𝑥)) | 
| 177 |  | oveq2 7440 | . . . . . . . 8
⊢ ((𝐹‘𝑥) = 𝑢 → ((𝐹‘ 1
)(.r‘𝑈)(𝐹‘𝑥)) = ((𝐹‘ 1
)(.r‘𝑈)𝑢)) | 
| 178 |  | id 22 | . . . . . . . 8
⊢ ((𝐹‘𝑥) = 𝑢 → (𝐹‘𝑥) = 𝑢) | 
| 179 | 177, 178 | eqeq12d 2752 | . . . . . . 7
⊢ ((𝐹‘𝑥) = 𝑢 → (((𝐹‘ 1
)(.r‘𝑈)(𝐹‘𝑥)) = (𝐹‘𝑥) ↔ ((𝐹‘ 1
)(.r‘𝑈)𝑢) = 𝑢)) | 
| 180 | 176, 179 | syl5ibcom 245 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥) = 𝑢 → ((𝐹‘ 1
)(.r‘𝑈)𝑢) = 𝑢)) | 
| 181 | 180 | rexlimdva 3154 | . . . . 5
⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 → ((𝐹‘ 1
)(.r‘𝑈)𝑢) = 𝑢)) | 
| 182 | 165, 181 | sylbid 240 | . . . 4
⊢ (𝜑 → (𝑢 ∈ 𝐵 → ((𝐹‘ 1
)(.r‘𝑈)𝑢) = 𝑢)) | 
| 183 | 182 | imp 406 | . . 3
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → ((𝐹‘ 1
)(.r‘𝑈)𝑢) = 𝑢) | 
| 184 | 3, 16, 1, 2, 4, 17,
18 | imasmulval 17581 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 1 ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘ 1 )) = (𝐹‘(𝑥 · 1 ))) | 
| 185 | 167, 184 | mpd3an3 1463 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘ 1 )) = (𝐹‘(𝑥 · 1 ))) | 
| 186 | 25, 17, 159 | ringridm 20268 | . . . . . . . . . 10
⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 · 1 ) = 𝑥) | 
| 187 | 4, 172, 186 | syl2an2r 685 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑥 · 1 ) = 𝑥) | 
| 188 | 187 | fveq2d 6909 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(𝑥 · 1 )) = (𝐹‘𝑥)) | 
| 189 | 185, 188 | eqtrd 2776 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘ 1 )) = (𝐹‘𝑥)) | 
| 190 |  | oveq1 7439 | . . . . . . . 8
⊢ ((𝐹‘𝑥) = 𝑢 → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘ 1 )) = (𝑢(.r‘𝑈)(𝐹‘ 1 ))) | 
| 191 | 190, 178 | eqeq12d 2752 | . . . . . . 7
⊢ ((𝐹‘𝑥) = 𝑢 → (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘ 1 )) = (𝐹‘𝑥) ↔ (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢)) | 
| 192 | 189, 191 | syl5ibcom 245 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥) = 𝑢 → (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢)) | 
| 193 | 192 | rexlimdva 3154 | . . . . 5
⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 → (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢)) | 
| 194 | 165, 193 | sylbid 240 | . . . 4
⊢ (𝜑 → (𝑢 ∈ 𝐵 → (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢)) | 
| 195 | 194 | imp 406 | . . 3
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢) | 
| 196 | 5, 6, 7, 15, 32, 95, 130, 156, 163, 183, 195 | isringd 20289 | . 2
⊢ (𝜑 → 𝑈 ∈ Ring) | 
| 197 | 163, 5 | eleqtrd 2842 | . . . 4
⊢ (𝜑 → (𝐹‘ 1 ) ∈ (Base‘𝑈)) | 
| 198 | 5 | eleq2d 2826 | . . . . . 6
⊢ (𝜑 → (𝑢 ∈ 𝐵 ↔ 𝑢 ∈ (Base‘𝑈))) | 
| 199 | 182, 194 | jcad 512 | . . . . . 6
⊢ (𝜑 → (𝑢 ∈ 𝐵 → (((𝐹‘ 1
)(.r‘𝑈)𝑢) = 𝑢 ∧ (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢))) | 
| 200 | 198, 199 | sylbird 260 | . . . . 5
⊢ (𝜑 → (𝑢 ∈ (Base‘𝑈) → (((𝐹‘ 1
)(.r‘𝑈)𝑢) = 𝑢 ∧ (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢))) | 
| 201 | 200 | ralrimiv 3144 | . . . 4
⊢ (𝜑 → ∀𝑢 ∈ (Base‘𝑈)(((𝐹‘ 1
)(.r‘𝑈)𝑢) = 𝑢 ∧ (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢)) | 
| 202 |  | eqid 2736 | . . . . . 6
⊢
(Base‘𝑈) =
(Base‘𝑈) | 
| 203 |  | eqid 2736 | . . . . . 6
⊢
(1r‘𝑈) = (1r‘𝑈) | 
| 204 | 202, 18, 203 | isringid 20269 | . . . . 5
⊢ (𝑈 ∈ Ring → (((𝐹‘ 1 ) ∈ (Base‘𝑈) ∧ ∀𝑢 ∈ (Base‘𝑈)(((𝐹‘ 1
)(.r‘𝑈)𝑢) = 𝑢 ∧ (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢)) ↔ (1r‘𝑈) = (𝐹‘ 1 ))) | 
| 205 | 196, 204 | syl 17 | . . . 4
⊢ (𝜑 → (((𝐹‘ 1 ) ∈ (Base‘𝑈) ∧ ∀𝑢 ∈ (Base‘𝑈)(((𝐹‘ 1
)(.r‘𝑈)𝑢) = 𝑢 ∧ (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢)) ↔ (1r‘𝑈) = (𝐹‘ 1 ))) | 
| 206 | 197, 201,
205 | mpbi2and 712 | . . 3
⊢ (𝜑 → (1r‘𝑈) = (𝐹‘ 1 )) | 
| 207 | 206 | eqcomd 2742 | . 2
⊢ (𝜑 → (𝐹‘ 1 ) =
(1r‘𝑈)) | 
| 208 | 196, 207 | jca 511 | 1
⊢ (𝜑 → (𝑈 ∈ Ring ∧ (𝐹‘ 1 ) =
(1r‘𝑈))) |