Proof of Theorem subrgunit
| Step | Hyp | Ref
 | Expression | 
| 1 |   | subrgugrp.1 | 
. . . . 5
⊢ 𝑆 = (𝑅 ↾s 𝐴) | 
| 2 |   | subrgugrp.2 | 
. . . . 5
⊢ 𝑈 = (Unit‘𝑅) | 
| 3 |   | subrgugrp.3 | 
. . . . 5
⊢ 𝑉 = (Unit‘𝑆) | 
| 4 | 1, 2, 3 | subrguss 13792 | 
. . . 4
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈) | 
| 5 | 4 | sselda 3183 | 
. . 3
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑈) | 
| 6 | 1 | subrgbas 13786 | 
. . . . 5
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆)) | 
| 7 | 6 | adantr 276 | 
. . . 4
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝐴 = (Base‘𝑆)) | 
| 8 | 3 | a1i 9 | 
. . . 4
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝑉 = (Unit‘𝑆)) | 
| 9 | 1 | subrgring 13780 | 
. . . . . 6
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring) | 
| 10 |   | ringsrg 13603 | 
. . . . . 6
⊢ (𝑆 ∈ Ring → 𝑆 ∈ SRing) | 
| 11 | 9, 10 | syl 14 | 
. . . . 5
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ SRing) | 
| 12 | 11 | adantr 276 | 
. . . 4
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝑆 ∈ SRing) | 
| 13 |   | simpr 110 | 
. . . 4
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) | 
| 14 | 7, 8, 12, 13 | unitcld 13664 | 
. . 3
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐴) | 
| 15 |   | eqid 2196 | 
. . . . . 6
⊢
(invr‘𝑆) = (invr‘𝑆) | 
| 16 |   | eqid 2196 | 
. . . . . 6
⊢
(Base‘𝑆) =
(Base‘𝑆) | 
| 17 | 3, 15, 16 | ringinvcl 13681 | 
. . . . 5
⊢ ((𝑆 ∈ Ring ∧ 𝑋 ∈ 𝑉) → ((invr‘𝑆)‘𝑋) ∈ (Base‘𝑆)) | 
| 18 | 9, 17 | sylan 283 | 
. . . 4
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → ((invr‘𝑆)‘𝑋) ∈ (Base‘𝑆)) | 
| 19 |   | subrgunit.4 | 
. . . . 5
⊢ 𝐼 = (invr‘𝑅) | 
| 20 | 1, 19, 3, 15 | subrginv 13793 | 
. . . 4
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → (𝐼‘𝑋) = ((invr‘𝑆)‘𝑋)) | 
| 21 | 18, 20, 7 | 3eltr4d 2280 | 
. . 3
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → (𝐼‘𝑋) ∈ 𝐴) | 
| 22 | 5, 14, 21 | 3jca 1179 | 
. 2
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) | 
| 23 |   | eqidd 2197 | 
. . . . 5
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (Base‘𝑆) = (Base‘𝑆)) | 
| 24 |   | eqidd 2197 | 
. . . . 5
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (∥r‘𝑆) =
(∥r‘𝑆)) | 
| 25 | 11 | adantr 276 | 
. . . . 5
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑆 ∈ SRing) | 
| 26 |   | eqidd 2197 | 
. . . . 5
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (.r‘𝑆) = (.r‘𝑆)) | 
| 27 |   | simpr2 1006 | 
. . . . . 6
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋 ∈ 𝐴) | 
| 28 | 6 | adantr 276 | 
. . . . . 6
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝐴 = (Base‘𝑆)) | 
| 29 | 27, 28 | eleqtrd 2275 | 
. . . . 5
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋 ∈ (Base‘𝑆)) | 
| 30 |   | simpr3 1007 | 
. . . . . 6
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝐼‘𝑋) ∈ 𝐴) | 
| 31 | 30, 28 | eleqtrd 2275 | 
. . . . 5
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝐼‘𝑋) ∈ (Base‘𝑆)) | 
| 32 | 23, 24, 25, 26, 29, 31 | dvdsrmuld 13652 | 
. . . 4
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋(∥r‘𝑆)((𝐼‘𝑋)(.r‘𝑆)𝑋)) | 
| 33 |   | subrgrcl 13782 | 
. . . . . 6
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | 
| 34 |   | simpr1 1005 | 
. . . . . 6
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋 ∈ 𝑈) | 
| 35 |   | eqid 2196 | 
. . . . . . 7
⊢
(.r‘𝑅) = (.r‘𝑅) | 
| 36 |   | eqid 2196 | 
. . . . . . 7
⊢
(1r‘𝑅) = (1r‘𝑅) | 
| 37 | 2, 19, 35, 36 | unitlinv 13682 | 
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → ((𝐼‘𝑋)(.r‘𝑅)𝑋) = (1r‘𝑅)) | 
| 38 | 33, 34, 37 | syl2an2r 595 | 
. . . . 5
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘𝑅)𝑋) = (1r‘𝑅)) | 
| 39 | 1, 35 | ressmulrg 12822 | 
. . . . . . . 8
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑅 ∈ Ring) →
(.r‘𝑅) =
(.r‘𝑆)) | 
| 40 | 33, 39 | mpdan 421 | 
. . . . . . 7
⊢ (𝐴 ∈ (SubRing‘𝑅) →
(.r‘𝑅) =
(.r‘𝑆)) | 
| 41 | 40 | adantr 276 | 
. . . . . 6
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (.r‘𝑅) = (.r‘𝑆)) | 
| 42 | 41 | oveqd 5939 | 
. . . . 5
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘𝑅)𝑋) = ((𝐼‘𝑋)(.r‘𝑆)𝑋)) | 
| 43 | 1, 36 | subrg1 13787 | 
. . . . . 6
⊢ (𝐴 ∈ (SubRing‘𝑅) →
(1r‘𝑅) =
(1r‘𝑆)) | 
| 44 | 43 | adantr 276 | 
. . . . 5
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (1r‘𝑅) = (1r‘𝑆)) | 
| 45 | 38, 42, 44 | 3eqtr3d 2237 | 
. . . 4
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘𝑆)𝑋) = (1r‘𝑆)) | 
| 46 | 32, 45 | breqtrd 4059 | 
. . 3
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋(∥r‘𝑆)(1r‘𝑆)) | 
| 47 | 9 | adantr 276 | 
. . . . . 6
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑆 ∈ Ring) | 
| 48 |   | eqid 2196 | 
. . . . . . 7
⊢
(oppr‘𝑆) = (oppr‘𝑆) | 
| 49 | 48, 16 | opprbasg 13631 | 
. . . . . 6
⊢ (𝑆 ∈ Ring →
(Base‘𝑆) =
(Base‘(oppr‘𝑆))) | 
| 50 | 47, 49 | syl 14 | 
. . . . 5
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (Base‘𝑆) =
(Base‘(oppr‘𝑆))) | 
| 51 |   | eqidd 2197 | 
. . . . 5
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) →
(∥r‘(oppr‘𝑆)) =
(∥r‘(oppr‘𝑆))) | 
| 52 | 48 | opprring 13635 | 
. . . . . 6
⊢ (𝑆 ∈ Ring →
(oppr‘𝑆) ∈ Ring) | 
| 53 |   | ringsrg 13603 | 
. . . . . 6
⊢
((oppr‘𝑆) ∈ Ring →
(oppr‘𝑆) ∈ SRing) | 
| 54 | 47, 52, 53 | 3syl 17 | 
. . . . 5
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) →
(oppr‘𝑆) ∈ SRing) | 
| 55 |   | eqidd 2197 | 
. . . . 5
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) →
(.r‘(oppr‘𝑆)) =
(.r‘(oppr‘𝑆))) | 
| 56 | 50, 51, 54, 55, 29, 31 | dvdsrmuld 13652 | 
. . . 4
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋(∥r‘(oppr‘𝑆))((𝐼‘𝑋)(.r‘(oppr‘𝑆))𝑋)) | 
| 57 |   | eqid 2196 | 
. . . . . . 7
⊢
(.r‘𝑆) = (.r‘𝑆) | 
| 58 |   | eqid 2196 | 
. . . . . . 7
⊢
(.r‘(oppr‘𝑆)) =
(.r‘(oppr‘𝑆)) | 
| 59 | 16, 57, 48, 58 | opprmulg 13627 | 
. . . . . 6
⊢ ((𝑆 ∈ Ring ∧ (𝐼‘𝑋) ∈ (Base‘𝑆) ∧ 𝑋 ∈ (Base‘𝑆)) → ((𝐼‘𝑋)(.r‘(oppr‘𝑆))𝑋) = (𝑋(.r‘𝑆)(𝐼‘𝑋))) | 
| 60 | 47, 31, 29, 59 | syl3anc 1249 | 
. . . . 5
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘(oppr‘𝑆))𝑋) = (𝑋(.r‘𝑆)(𝐼‘𝑋))) | 
| 61 | 2, 19, 35, 36 | unitrinv 13683 | 
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (1r‘𝑅)) | 
| 62 | 33, 34, 61 | syl2an2r 595 | 
. . . . . 6
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (1r‘𝑅)) | 
| 63 | 41 | oveqd 5939 | 
. . . . . 6
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (𝑋(.r‘𝑆)(𝐼‘𝑋))) | 
| 64 | 62, 63, 44 | 3eqtr3d 2237 | 
. . . . 5
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝑋(.r‘𝑆)(𝐼‘𝑋)) = (1r‘𝑆)) | 
| 65 | 60, 64 | eqtrd 2229 | 
. . . 4
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘(oppr‘𝑆))𝑋) = (1r‘𝑆)) | 
| 66 | 56, 65 | breqtrd 4059 | 
. . 3
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋(∥r‘(oppr‘𝑆))(1r‘𝑆)) | 
| 67 | 3 | a1i 9 | 
. . . . 5
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 = (Unit‘𝑆)) | 
| 68 |   | eqidd 2197 | 
. . . . 5
⊢ (𝐴 ∈ (SubRing‘𝑅) →
(1r‘𝑆) =
(1r‘𝑆)) | 
| 69 |   | eqidd 2197 | 
. . . . 5
⊢ (𝐴 ∈ (SubRing‘𝑅) →
(∥r‘𝑆) = (∥r‘𝑆)) | 
| 70 |   | eqidd 2197 | 
. . . . 5
⊢ (𝐴 ∈ (SubRing‘𝑅) →
(oppr‘𝑆) = (oppr‘𝑆)) | 
| 71 |   | eqidd 2197 | 
. . . . 5
⊢ (𝐴 ∈ (SubRing‘𝑅) →
(∥r‘(oppr‘𝑆)) =
(∥r‘(oppr‘𝑆))) | 
| 72 | 67, 68, 69, 70, 71, 11 | isunitd 13662 | 
. . . 4
⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑋 ∈ 𝑉 ↔ (𝑋(∥r‘𝑆)(1r‘𝑆) ∧ 𝑋(∥r‘(oppr‘𝑆))(1r‘𝑆)))) | 
| 73 | 72 | adantr 276 | 
. . 3
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝑋 ∈ 𝑉 ↔ (𝑋(∥r‘𝑆)(1r‘𝑆) ∧ 𝑋(∥r‘(oppr‘𝑆))(1r‘𝑆)))) | 
| 74 | 46, 66, 73 | mpbir2and 946 | 
. 2
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋 ∈ 𝑉) | 
| 75 | 22, 74 | impbida 596 | 
1
⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴))) |