| Step | Hyp | Ref
| Expression |
| 1 | | mxidlirredi.x |
. . 3
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| 2 | | mxidlirredi.r |
. . . . . 6
⊢ (𝜑 → 𝑅 ∈ IDomn) |
| 3 | 2 | idomringd 20728 |
. . . . 5
⊢ (𝜑 → 𝑅 ∈ Ring) |
| 4 | | mxidlirredi.1 |
. . . . 5
⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) |
| 5 | | mxidlirredi.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝑅) |
| 6 | 5 | mxidlnr 33492 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ 𝐵) |
| 7 | 3, 4, 6 | syl2anc 584 |
. . . 4
⊢ (𝜑 → 𝑀 ≠ 𝐵) |
| 8 | | eqid 2737 |
. . . . . 6
⊢
(Unit‘𝑅) =
(Unit‘𝑅) |
| 9 | | mxidlirredi.k |
. . . . . 6
⊢ 𝐾 = (RSpan‘𝑅) |
| 10 | | mxidlirredi.m |
. . . . . 6
⊢ 𝑀 = (𝐾‘{𝑋}) |
| 11 | 8, 9, 10, 5, 1, 2 | unitpidl1 33452 |
. . . . 5
⊢ (𝜑 → (𝑀 = 𝐵 ↔ 𝑋 ∈ (Unit‘𝑅))) |
| 12 | 11 | necon3abid 2977 |
. . . 4
⊢ (𝜑 → (𝑀 ≠ 𝐵 ↔ ¬ 𝑋 ∈ (Unit‘𝑅))) |
| 13 | 7, 12 | mpbid 232 |
. . 3
⊢ (𝜑 → ¬ 𝑋 ∈ (Unit‘𝑅)) |
| 14 | 1, 13 | eldifd 3962 |
. 2
⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ (Unit‘𝑅))) |
| 15 | 3 | ad3antrrr 730 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) → 𝑅 ∈ Ring) |
| 16 | 4 | ad3antrrr 730 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) → 𝑀 ∈ (MaxIdeal‘𝑅)) |
| 17 | | simplr 769 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) |
| 18 | 17 | eldifad 3963 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ 𝐵) |
| 19 | 18 | snssd 4809 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) → {𝑔} ⊆ 𝐵) |
| 20 | | eqid 2737 |
. . . . . . . . . 10
⊢
(LIdeal‘𝑅) =
(LIdeal‘𝑅) |
| 21 | 9, 5, 20 | rspcl 21245 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → (𝐾‘{𝑔}) ∈ (LIdeal‘𝑅)) |
| 22 | 15, 19, 21 | syl2anc 584 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) → (𝐾‘{𝑔}) ∈ (LIdeal‘𝑅)) |
| 23 | 3 | ad4antr 732 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ Ring) |
| 24 | 23 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(.r‘𝑅)𝑋)) → 𝑅 ∈ Ring) |
| 25 | | simp-5r 786 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(.r‘𝑅)𝑋)) → 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) |
| 26 | 25 | eldifad 3963 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(.r‘𝑅)𝑋)) → 𝑔 ∈ 𝐵) |
| 27 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(.r‘𝑅) = (.r‘𝑅) |
| 28 | | simplr 769 |
. . . . . . . . . . . . . 14
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(.r‘𝑅)𝑋)) → 𝑞 ∈ 𝐵) |
| 29 | | simp-6r 788 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(.r‘𝑅)𝑋)) → 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) |
| 30 | 29 | eldifad 3963 |
. . . . . . . . . . . . . 14
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(.r‘𝑅)𝑋)) → 𝑓 ∈ 𝐵) |
| 31 | 5, 27, 24, 28, 30 | ringcld 20257 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(.r‘𝑅)𝑋)) → (𝑞(.r‘𝑅)𝑓) ∈ 𝐵) |
| 32 | | oveq1 7438 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = (𝑞(.r‘𝑅)𝑓) → (𝑦(.r‘𝑅)𝑔) = ((𝑞(.r‘𝑅)𝑓)(.r‘𝑅)𝑔)) |
| 33 | 32 | eqeq2d 2748 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑞(.r‘𝑅)𝑓) → (𝑥 = (𝑦(.r‘𝑅)𝑔) ↔ 𝑥 = ((𝑞(.r‘𝑅)𝑓)(.r‘𝑅)𝑔))) |
| 34 | 33 | adantl 481 |
. . . . . . . . . . . . 13
⊢
((((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(.r‘𝑅)𝑋)) ∧ 𝑦 = (𝑞(.r‘𝑅)𝑓)) → (𝑥 = (𝑦(.r‘𝑅)𝑔) ↔ 𝑥 = ((𝑞(.r‘𝑅)𝑓)(.r‘𝑅)𝑔))) |
| 35 | | simp-4r 784 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(.r‘𝑅)𝑋)) → (𝑓(.r‘𝑅)𝑔) = 𝑋) |
| 36 | 35 | oveq2d 7447 |
. . . . . . . . . . . . . 14
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(.r‘𝑅)𝑋)) → (𝑞(.r‘𝑅)(𝑓(.r‘𝑅)𝑔)) = (𝑞(.r‘𝑅)𝑋)) |
| 37 | 5, 27, 24, 28, 30, 26 | ringassd 20254 |
. . . . . . . . . . . . . 14
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(.r‘𝑅)𝑋)) → ((𝑞(.r‘𝑅)𝑓)(.r‘𝑅)𝑔) = (𝑞(.r‘𝑅)(𝑓(.r‘𝑅)𝑔))) |
| 38 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(.r‘𝑅)𝑋)) → 𝑥 = (𝑞(.r‘𝑅)𝑋)) |
| 39 | 36, 37, 38 | 3eqtr4rd 2788 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(.r‘𝑅)𝑋)) → 𝑥 = ((𝑞(.r‘𝑅)𝑓)(.r‘𝑅)𝑔)) |
| 40 | 31, 34, 39 | rspcedvd 3624 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(.r‘𝑅)𝑋)) → ∃𝑦 ∈ 𝐵 𝑥 = (𝑦(.r‘𝑅)𝑔)) |
| 41 | 5, 27, 9 | elrspsn 21250 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ Ring ∧ 𝑔 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝑔}) ↔ ∃𝑦 ∈ 𝐵 𝑥 = (𝑦(.r‘𝑅)𝑔))) |
| 42 | 41 | biimpar 477 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝑔 ∈ 𝐵) ∧ ∃𝑦 ∈ 𝐵 𝑥 = (𝑦(.r‘𝑅)𝑔)) → 𝑥 ∈ (𝐾‘{𝑔})) |
| 43 | 24, 26, 40, 42 | syl21anc 838 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) ∧ 𝑞 ∈ 𝐵) ∧ 𝑥 = (𝑞(.r‘𝑅)𝑋)) → 𝑥 ∈ (𝐾‘{𝑔})) |
| 44 | 1 | ad4antr 732 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑋 ∈ 𝐵) |
| 45 | | simpr 484 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ 𝑀) |
| 46 | 45, 10 | eleqtrdi 2851 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ (𝐾‘{𝑋})) |
| 47 | 5, 27, 9 | elrspsn 21250 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝑋}) ↔ ∃𝑞 ∈ 𝐵 𝑥 = (𝑞(.r‘𝑅)𝑋))) |
| 48 | 47 | biimpa 476 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ (𝐾‘{𝑋})) → ∃𝑞 ∈ 𝐵 𝑥 = (𝑞(.r‘𝑅)𝑋)) |
| 49 | 23, 44, 46, 48 | syl21anc 838 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → ∃𝑞 ∈ 𝐵 𝑥 = (𝑞(.r‘𝑅)𝑋)) |
| 50 | 43, 49 | r19.29a 3162 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ (𝐾‘{𝑔})) |
| 51 | 50 | ex 412 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) → (𝑥 ∈ 𝑀 → 𝑥 ∈ (𝐾‘{𝑔}))) |
| 52 | 51 | ssrdv 3989 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) → 𝑀 ⊆ (𝐾‘{𝑔})) |
| 53 | 9, 5 | rspssid 21246 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → {𝑔} ⊆ (𝐾‘{𝑔})) |
| 54 | | vex 3484 |
. . . . . . . . . . . 12
⊢ 𝑔 ∈ V |
| 55 | 54 | snss 4785 |
. . . . . . . . . . 11
⊢ (𝑔 ∈ (𝐾‘{𝑔}) ↔ {𝑔} ⊆ (𝐾‘{𝑔})) |
| 56 | 53, 55 | sylibr 234 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → 𝑔 ∈ (𝐾‘{𝑔})) |
| 57 | 15, 19, 56 | syl2anc 584 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ (𝐾‘{𝑔})) |
| 58 | | df-idom 20696 |
. . . . . . . . . . . . . . 15
⊢ IDomn =
(CRing ∩ Domn) |
| 59 | 2, 58 | eleqtrdi 2851 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn)) |
| 60 | 59 | elin1d 4204 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑅 ∈ CRing) |
| 61 | 60 | ad6antr 736 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → 𝑅 ∈ CRing) |
| 62 | | simplr 769 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → 𝑟 ∈ 𝐵) |
| 63 | | simp-6r 788 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) |
| 64 | 63 | eldifad 3963 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → 𝑓 ∈ 𝐵) |
| 65 | | mxidlirredi.0 |
. . . . . . . . . . . . . 14
⊢ 0 =
(0g‘𝑅) |
| 66 | 15 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑅 ∈ Ring) |
| 67 | 66 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → 𝑅 ∈ Ring) |
| 68 | 5, 27, 67, 62, 64 | ringcld 20257 |
. . . . . . . . . . . . . 14
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → (𝑟(.r‘𝑅)𝑓) ∈ 𝐵) |
| 69 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢
(1r‘𝑅) = (1r‘𝑅) |
| 70 | 5, 69 | ringidcl 20262 |
. . . . . . . . . . . . . . . 16
⊢ (𝑅 ∈ Ring →
(1r‘𝑅)
∈ 𝐵) |
| 71 | 3, 70 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (1r‘𝑅) ∈ 𝐵) |
| 72 | 71 | ad6antr 736 |
. . . . . . . . . . . . . 14
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → (1r‘𝑅) ∈ 𝐵) |
| 73 | 18 | ad3antrrr 730 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → 𝑔 ∈ 𝐵) |
| 74 | | simpr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) ∧ 𝑔 = 0 ) → 𝑔 = 0 ) |
| 75 | 74 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) ∧ 𝑔 = 0 ) → (𝑓(.r‘𝑅)𝑔) = (𝑓(.r‘𝑅) 0 )) |
| 76 | | simp-5r 786 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) ∧ 𝑔 = 0 ) → (𝑓(.r‘𝑅)𝑔) = 𝑋) |
| 77 | 66 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) ∧ 𝑔 = 0 ) → 𝑅 ∈ Ring) |
| 78 | 64 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) ∧ 𝑔 = 0 ) → 𝑓 ∈ 𝐵) |
| 79 | 5, 27, 65 | ringrz 20291 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ 𝐵) → (𝑓(.r‘𝑅) 0 ) = 0 ) |
| 80 | 77, 78, 79 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) ∧ 𝑔 = 0 ) → (𝑓(.r‘𝑅) 0 ) = 0 ) |
| 81 | 75, 76, 80 | 3eqtr3d 2785 |
. . . . . . . . . . . . . . . . 17
⊢
((((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) ∧ 𝑔 = 0 ) → 𝑋 = 0 ) |
| 82 | | mxidlirredi.y |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑋 ≠ 0 ) |
| 83 | 82 | neneqd 2945 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ¬ 𝑋 = 0 ) |
| 84 | 83 | ad7antr 738 |
. . . . . . . . . . . . . . . . 17
⊢
((((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) ∧ 𝑔 = 0 ) → ¬ 𝑋 = 0 ) |
| 85 | 81, 84 | pm2.65da 817 |
. . . . . . . . . . . . . . . 16
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → ¬ 𝑔 = 0 ) |
| 86 | 85 | neqned 2947 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → 𝑔 ≠ 0 ) |
| 87 | | eldifsn 4786 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 ∈ (𝐵 ∖ { 0 }) ↔ (𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 )) |
| 88 | 73, 86, 87 | sylanbrc 583 |
. . . . . . . . . . . . . 14
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → 𝑔 ∈ (𝐵 ∖ { 0 })) |
| 89 | 2 | ad6antr 736 |
. . . . . . . . . . . . . 14
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → 𝑅 ∈ IDomn) |
| 90 | 5, 27, 69, 67, 73 | ringlidmd 20269 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → ((1r‘𝑅)(.r‘𝑅)𝑔) = 𝑔) |
| 91 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → 𝑔 = (𝑟(.r‘𝑅)𝑋)) |
| 92 | 5, 27, 67, 62, 64, 73 | ringassd 20254 |
. . . . . . . . . . . . . . . 16
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → ((𝑟(.r‘𝑅)𝑓)(.r‘𝑅)𝑔) = (𝑟(.r‘𝑅)(𝑓(.r‘𝑅)𝑔))) |
| 93 | | simp-4r 784 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → (𝑓(.r‘𝑅)𝑔) = 𝑋) |
| 94 | 93 | oveq2d 7447 |
. . . . . . . . . . . . . . . 16
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → (𝑟(.r‘𝑅)(𝑓(.r‘𝑅)𝑔)) = (𝑟(.r‘𝑅)𝑋)) |
| 95 | 92, 94 | eqtr2d 2778 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → (𝑟(.r‘𝑅)𝑋) = ((𝑟(.r‘𝑅)𝑓)(.r‘𝑅)𝑔)) |
| 96 | 90, 91, 95 | 3eqtrrd 2782 |
. . . . . . . . . . . . . 14
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → ((𝑟(.r‘𝑅)𝑓)(.r‘𝑅)𝑔) = ((1r‘𝑅)(.r‘𝑅)𝑔)) |
| 97 | 5, 65, 27, 68, 72, 88, 89, 96 | idomrcan 33282 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → (𝑟(.r‘𝑅)𝑓) = (1r‘𝑅)) |
| 98 | 8, 69 | 1unit 20374 |
. . . . . . . . . . . . . . 15
⊢ (𝑅 ∈ Ring →
(1r‘𝑅)
∈ (Unit‘𝑅)) |
| 99 | 3, 98 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1r‘𝑅) ∈ (Unit‘𝑅)) |
| 100 | 99 | ad6antr 736 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → (1r‘𝑅) ∈ (Unit‘𝑅)) |
| 101 | 97, 100 | eqeltrd 2841 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → (𝑟(.r‘𝑅)𝑓) ∈ (Unit‘𝑅)) |
| 102 | 8, 27, 5 | unitmulclb 20381 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ CRing ∧ 𝑟 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵) → ((𝑟(.r‘𝑅)𝑓) ∈ (Unit‘𝑅) ↔ (𝑟 ∈ (Unit‘𝑅) ∧ 𝑓 ∈ (Unit‘𝑅)))) |
| 103 | 102 | simplbda 499 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ CRing ∧ 𝑟 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵) ∧ (𝑟(.r‘𝑅)𝑓) ∈ (Unit‘𝑅)) → 𝑓 ∈ (Unit‘𝑅)) |
| 104 | 61, 62, 64, 101, 103 | syl31anc 1375 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) ∧ 𝑟 ∈ 𝐵) ∧ 𝑔 = (𝑟(.r‘𝑅)𝑋)) → 𝑓 ∈ (Unit‘𝑅)) |
| 105 | 1 | ad4antr 732 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑋 ∈ 𝐵) |
| 106 | | simpr 484 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑔 ∈ 𝑀) |
| 107 | 106, 10 | eleqtrdi 2851 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑔 ∈ (𝐾‘{𝑋})) |
| 108 | 5, 27, 9 | elrspsn 21250 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑔 ∈ (𝐾‘{𝑋}) ↔ ∃𝑟 ∈ 𝐵 𝑔 = (𝑟(.r‘𝑅)𝑋))) |
| 109 | 108 | biimpa 476 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑔 ∈ (𝐾‘{𝑋})) → ∃𝑟 ∈ 𝐵 𝑔 = (𝑟(.r‘𝑅)𝑋)) |
| 110 | 66, 105, 107, 109 | syl21anc 838 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → ∃𝑟 ∈ 𝐵 𝑔 = (𝑟(.r‘𝑅)𝑋)) |
| 111 | 104, 110 | r19.29a 3162 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑓 ∈ (Unit‘𝑅)) |
| 112 | | simp-4r 784 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) |
| 113 | 112 | eldifbd 3964 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) ∧ 𝑔 ∈ 𝑀) → ¬ 𝑓 ∈ (Unit‘𝑅)) |
| 114 | 111, 113 | pm2.65da 817 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) → ¬ 𝑔 ∈ 𝑀) |
| 115 | 57, 114 | eldifd 3962 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ ((𝐾‘{𝑔}) ∖ 𝑀)) |
| 116 | 5, 15, 16, 22, 52, 115 | mxidlmaxv 33496 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) → (𝐾‘{𝑔}) = 𝐵) |
| 117 | | eqid 2737 |
. . . . . . . 8
⊢ (𝐾‘{𝑔}) = (𝐾‘{𝑔}) |
| 118 | 2 | ad3antrrr 730 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) → 𝑅 ∈ IDomn) |
| 119 | 8, 9, 117, 5, 18, 118 | unitpidl1 33452 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) → ((𝐾‘{𝑔}) = 𝐵 ↔ 𝑔 ∈ (Unit‘𝑅))) |
| 120 | 116, 119 | mpbid 232 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) → 𝑔 ∈ (Unit‘𝑅)) |
| 121 | 17 | eldifbd 3964 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r‘𝑅)𝑔) = 𝑋) → ¬ 𝑔 ∈ (Unit‘𝑅)) |
| 122 | 120, 121 | pm2.65da 817 |
. . . . 5
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) → ¬ (𝑓(.r‘𝑅)𝑔) = 𝑋) |
| 123 | 122 | anasss 466 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))) → ¬ (𝑓(.r‘𝑅)𝑔) = 𝑋) |
| 124 | 123 | neqned 2947 |
. . 3
⊢ ((𝜑 ∧ (𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))) → (𝑓(.r‘𝑅)𝑔) ≠ 𝑋) |
| 125 | 124 | ralrimivva 3202 |
. 2
⊢ (𝜑 → ∀𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))∀𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))(𝑓(.r‘𝑅)𝑔) ≠ 𝑋) |
| 126 | | eqid 2737 |
. . 3
⊢
(Irred‘𝑅) =
(Irred‘𝑅) |
| 127 | | eqid 2737 |
. . 3
⊢ (𝐵 ∖ (Unit‘𝑅)) = (𝐵 ∖ (Unit‘𝑅)) |
| 128 | 5, 8, 126, 127, 27 | isirred 20419 |
. 2
⊢ (𝑋 ∈ (Irred‘𝑅) ↔ (𝑋 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ ∀𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))∀𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))(𝑓(.r‘𝑅)𝑔) ≠ 𝑋)) |
| 129 | 14, 125, 128 | sylanbrc 583 |
1
⊢ (𝜑 → 𝑋 ∈ (Irred‘𝑅)) |