| Step | Hyp | Ref
| Expression |
| 1 | | simpllr 776 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → 𝑟 ∈ (Base‘𝑅)) |
| 2 | | ovex 7464 |
. . . . . 6
⊢ (𝑅 ~QG 𝑀) ∈ V |
| 3 | 2 | ecelqsi 8813 |
. . . . 5
⊢ (𝑟 ∈ (Base‘𝑅) → [𝑟](𝑅 ~QG 𝑀) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀))) |
| 4 | 1, 3 | syl 17 |
. . . 4
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → [𝑟](𝑅 ~QG 𝑀) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀))) |
| 5 | | qsdrng.q |
. . . . . . 7
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀)) |
| 6 | 5 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))) |
| 7 | | eqid 2737 |
. . . . . . 7
⊢
(Base‘𝑅) =
(Base‘𝑅) |
| 8 | 7 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅)) |
| 9 | | ovexd 7466 |
. . . . . 6
⊢ (𝜑 → (𝑅 ~QG 𝑀) ∈ V) |
| 10 | | qsdrng.r |
. . . . . 6
⊢ (𝜑 → 𝑅 ∈ NzRing) |
| 11 | 6, 8, 9, 10 | qusbas 17590 |
. . . . 5
⊢ (𝜑 → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = (Base‘𝑄)) |
| 12 | 11 | ad3antrrr 730 |
. . . 4
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = (Base‘𝑄)) |
| 13 | 4, 12 | eleqtrd 2843 |
. . 3
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → [𝑟](𝑅 ~QG 𝑀) ∈ (Base‘𝑄)) |
| 14 | | oveq1 7438 |
. . . . 5
⊢ (𝑣 = [𝑟](𝑅 ~QG 𝑀) → (𝑣(.r‘𝑄)[𝑋](𝑅 ~QG 𝑀)) = ([𝑟](𝑅 ~QG 𝑀)(.r‘𝑄)[𝑋](𝑅 ~QG 𝑀))) |
| 15 | 14 | eqeq1d 2739 |
. . . 4
⊢ (𝑣 = [𝑟](𝑅 ~QG 𝑀) → ((𝑣(.r‘𝑄)[𝑋](𝑅 ~QG 𝑀)) = (1r‘𝑄) ↔ ([𝑟](𝑅 ~QG 𝑀)(.r‘𝑄)[𝑋](𝑅 ~QG 𝑀)) = (1r‘𝑄))) |
| 16 | 15 | adantl 481 |
. . 3
⊢
(((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → ((𝑣(.r‘𝑄)[𝑋](𝑅 ~QG 𝑀)) = (1r‘𝑄) ↔ ([𝑟](𝑅 ~QG 𝑀)(.r‘𝑄)[𝑋](𝑅 ~QG 𝑀)) = (1r‘𝑄))) |
| 17 | | eqid 2737 |
. . . . . 6
⊢
(.r‘𝑅) = (.r‘𝑅) |
| 18 | | eqid 2737 |
. . . . . 6
⊢
(.r‘𝑄) = (.r‘𝑄) |
| 19 | | nzrring 20516 |
. . . . . . . 8
⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) |
| 20 | 10, 19 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ Ring) |
| 21 | 20 | ad3antrrr 730 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → 𝑅 ∈ Ring) |
| 22 | | qsdrngi.1 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) |
| 23 | 7 | mxidlidl 33491 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅)) |
| 24 | 20, 22, 23 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅)) |
| 25 | | qsdrng.0 |
. . . . . . . . . . . 12
⊢ 𝑂 =
(oppr‘𝑅) |
| 26 | 25 | opprring 20347 |
. . . . . . . . . . 11
⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring) |
| 27 | 20, 26 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑂 ∈ Ring) |
| 28 | | qsdrngi.2 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑂)) |
| 29 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(Base‘𝑂) =
(Base‘𝑂) |
| 30 | 29 | mxidlidl 33491 |
. . . . . . . . . 10
⊢ ((𝑂 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑂)) → 𝑀 ∈ (LIdeal‘𝑂)) |
| 31 | 27, 28, 30 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑂)) |
| 32 | 24, 31 | elind 4200 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ((LIdeal‘𝑅) ∩ (LIdeal‘𝑂))) |
| 33 | | eqid 2737 |
. . . . . . . . 9
⊢
(LIdeal‘𝑅) =
(LIdeal‘𝑅) |
| 34 | | eqid 2737 |
. . . . . . . . 9
⊢
(LIdeal‘𝑂) =
(LIdeal‘𝑂) |
| 35 | | eqid 2737 |
. . . . . . . . 9
⊢
(2Ideal‘𝑅) =
(2Ideal‘𝑅) |
| 36 | 33, 25, 34, 35 | 2idlval 21261 |
. . . . . . . 8
⊢
(2Ideal‘𝑅) =
((LIdeal‘𝑅) ∩
(LIdeal‘𝑂)) |
| 37 | 32, 36 | eleqtrrdi 2852 |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈ (2Ideal‘𝑅)) |
| 38 | 37 | ad3antrrr 730 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → 𝑀 ∈ (2Ideal‘𝑅)) |
| 39 | | qsdrngilem.1 |
. . . . . . 7
⊢ (𝜑 → 𝑋 ∈ (Base‘𝑅)) |
| 40 | 39 | ad3antrrr 730 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → 𝑋 ∈ (Base‘𝑅)) |
| 41 | 5, 7, 17, 18, 21, 38, 1, 40 | qusmul2idl 21289 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → ([𝑟](𝑅 ~QG 𝑀)(.r‘𝑄)[𝑋](𝑅 ~QG 𝑀)) = [(𝑟(.r‘𝑅)𝑋)](𝑅 ~QG 𝑀)) |
| 42 | | lidlnsg 21258 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) → 𝑀 ∈ (NrmSGrp‘𝑅)) |
| 43 | 20, 24, 42 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ (NrmSGrp‘𝑅)) |
| 44 | | nsgsubg 19176 |
. . . . . . . 8
⊢ (𝑀 ∈ (NrmSGrp‘𝑅) → 𝑀 ∈ (SubGrp‘𝑅)) |
| 45 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑅 ~QG 𝑀) = (𝑅 ~QG 𝑀) |
| 46 | 7, 45 | eqger 19196 |
. . . . . . . 8
⊢ (𝑀 ∈ (SubGrp‘𝑅) → (𝑅 ~QG 𝑀) Er (Base‘𝑅)) |
| 47 | 43, 44, 46 | 3syl 18 |
. . . . . . 7
⊢ (𝜑 → (𝑅 ~QG 𝑀) Er (Base‘𝑅)) |
| 48 | 47 | ad3antrrr 730 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → (𝑅 ~QG 𝑀) Er (Base‘𝑅)) |
| 49 | 7, 33 | lidlss 21222 |
. . . . . . . . 9
⊢ (𝑀 ∈ (LIdeal‘𝑅) → 𝑀 ⊆ (Base‘𝑅)) |
| 50 | 24, 49 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ⊆ (Base‘𝑅)) |
| 51 | 50 | ad3antrrr 730 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → 𝑀 ⊆ (Base‘𝑅)) |
| 52 | 7, 17, 21, 1, 40 | ringcld 20257 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → (𝑟(.r‘𝑅)𝑋) ∈ (Base‘𝑅)) |
| 53 | | eqid 2737 |
. . . . . . . . . 10
⊢
(1r‘𝑅) = (1r‘𝑅) |
| 54 | 7, 53 | ringidcl 20262 |
. . . . . . . . 9
⊢ (𝑅 ∈ Ring →
(1r‘𝑅)
∈ (Base‘𝑅)) |
| 55 | 20, 54 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅)) |
| 56 | 55 | ad3antrrr 730 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → (1r‘𝑅) ∈ (Base‘𝑅)) |
| 57 | | simpr 484 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) |
| 58 | 57 | oveq2d 7447 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → (((invg‘𝑅)‘(𝑟(.r‘𝑅)𝑋))(+g‘𝑅)(1r‘𝑅)) = (((invg‘𝑅)‘(𝑟(.r‘𝑅)𝑋))(+g‘𝑅)((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚))) |
| 59 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(+g‘𝑅) = (+g‘𝑅) |
| 60 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(0g‘𝑅) = (0g‘𝑅) |
| 61 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(invg‘𝑅) = (invg‘𝑅) |
| 62 | 20 | ringgrpd 20239 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑅 ∈ Grp) |
| 63 | 62 | ad3antrrr 730 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → 𝑅 ∈ Grp) |
| 64 | 7, 59, 60, 61, 63, 52 | grplinvd 19012 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → (((invg‘𝑅)‘(𝑟(.r‘𝑅)𝑋))(+g‘𝑅)(𝑟(.r‘𝑅)𝑋)) = (0g‘𝑅)) |
| 65 | 64 | oveq1d 7446 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → ((((invg‘𝑅)‘(𝑟(.r‘𝑅)𝑋))(+g‘𝑅)(𝑟(.r‘𝑅)𝑋))(+g‘𝑅)𝑚) = ((0g‘𝑅)(+g‘𝑅)𝑚)) |
| 66 | 7, 61, 63, 52 | grpinvcld 19006 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → ((invg‘𝑅)‘(𝑟(.r‘𝑅)𝑋)) ∈ (Base‘𝑅)) |
| 67 | | simplr 769 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → 𝑚 ∈ 𝑀) |
| 68 | 51, 67 | sseldd 3984 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → 𝑚 ∈ (Base‘𝑅)) |
| 69 | 7, 59, 63, 66, 52, 68 | grpassd 18963 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → ((((invg‘𝑅)‘(𝑟(.r‘𝑅)𝑋))(+g‘𝑅)(𝑟(.r‘𝑅)𝑋))(+g‘𝑅)𝑚) = (((invg‘𝑅)‘(𝑟(.r‘𝑅)𝑋))(+g‘𝑅)((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚))) |
| 70 | 7, 59, 60, 63, 68 | grplidd 18987 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → ((0g‘𝑅)(+g‘𝑅)𝑚) = 𝑚) |
| 71 | 65, 69, 70 | 3eqtr3d 2785 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → (((invg‘𝑅)‘(𝑟(.r‘𝑅)𝑋))(+g‘𝑅)((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) = 𝑚) |
| 72 | 58, 71 | eqtrd 2777 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → (((invg‘𝑅)‘(𝑟(.r‘𝑅)𝑋))(+g‘𝑅)(1r‘𝑅)) = 𝑚) |
| 73 | 72, 67 | eqeltrd 2841 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → (((invg‘𝑅)‘(𝑟(.r‘𝑅)𝑋))(+g‘𝑅)(1r‘𝑅)) ∈ 𝑀) |
| 74 | 7, 61, 59, 45 | eqgval 19195 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ⊆ (Base‘𝑅)) → ((𝑟(.r‘𝑅)𝑋)(𝑅 ~QG 𝑀)(1r‘𝑅) ↔ ((𝑟(.r‘𝑅)𝑋) ∈ (Base‘𝑅) ∧ (1r‘𝑅) ∈ (Base‘𝑅) ∧
(((invg‘𝑅)‘(𝑟(.r‘𝑅)𝑋))(+g‘𝑅)(1r‘𝑅)) ∈ 𝑀))) |
| 75 | 74 | biimpar 477 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ⊆ (Base‘𝑅)) ∧ ((𝑟(.r‘𝑅)𝑋) ∈ (Base‘𝑅) ∧ (1r‘𝑅) ∈ (Base‘𝑅) ∧
(((invg‘𝑅)‘(𝑟(.r‘𝑅)𝑋))(+g‘𝑅)(1r‘𝑅)) ∈ 𝑀)) → (𝑟(.r‘𝑅)𝑋)(𝑅 ~QG 𝑀)(1r‘𝑅)) |
| 76 | 21, 51, 52, 56, 73, 75 | syl23anc 1379 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → (𝑟(.r‘𝑅)𝑋)(𝑅 ~QG 𝑀)(1r‘𝑅)) |
| 77 | 48, 76 | erthi 8798 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → [(𝑟(.r‘𝑅)𝑋)](𝑅 ~QG 𝑀) = [(1r‘𝑅)](𝑅 ~QG 𝑀)) |
| 78 | 41, 77 | eqtrd 2777 |
. . . 4
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → ([𝑟](𝑅 ~QG 𝑀)(.r‘𝑄)[𝑋](𝑅 ~QG 𝑀)) = [(1r‘𝑅)](𝑅 ~QG 𝑀)) |
| 79 | 5, 35, 53 | qus1 21284 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (2Ideal‘𝑅)) → (𝑄 ∈ Ring ∧
[(1r‘𝑅)](𝑅 ~QG 𝑀) = (1r‘𝑄))) |
| 80 | 79 | simprd 495 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (2Ideal‘𝑅)) →
[(1r‘𝑅)](𝑅 ~QG 𝑀) = (1r‘𝑄)) |
| 81 | 21, 38, 80 | syl2anc 584 |
. . . 4
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → [(1r‘𝑅)](𝑅 ~QG 𝑀) = (1r‘𝑄)) |
| 82 | 78, 81 | eqtrd 2777 |
. . 3
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → ([𝑟](𝑅 ~QG 𝑀)(.r‘𝑄)[𝑋](𝑅 ~QG 𝑀)) = (1r‘𝑄)) |
| 83 | 13, 16, 82 | rspcedvd 3624 |
. 2
⊢ ((((𝜑 ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑚 ∈ 𝑀) ∧ (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) → ∃𝑣 ∈ (Base‘𝑄)(𝑣(.r‘𝑄)[𝑋](𝑅 ~QG 𝑀)) = (1r‘𝑄)) |
| 84 | 39 | snssd 4809 |
. . . . . . 7
⊢ (𝜑 → {𝑋} ⊆ (Base‘𝑅)) |
| 85 | 50, 84 | unssd 4192 |
. . . . . 6
⊢ (𝜑 → (𝑀 ∪ {𝑋}) ⊆ (Base‘𝑅)) |
| 86 | | eqid 2737 |
. . . . . . 7
⊢
(RSpan‘𝑅) =
(RSpan‘𝑅) |
| 87 | 86, 7, 33 | rspcl 21245 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∪ {𝑋}) ⊆ (Base‘𝑅)) → ((RSpan‘𝑅)‘(𝑀 ∪ {𝑋})) ∈ (LIdeal‘𝑅)) |
| 88 | 20, 85, 87 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → ((RSpan‘𝑅)‘(𝑀 ∪ {𝑋})) ∈ (LIdeal‘𝑅)) |
| 89 | 86, 7 | rspssid 21246 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∪ {𝑋}) ⊆ (Base‘𝑅)) → (𝑀 ∪ {𝑋}) ⊆ ((RSpan‘𝑅)‘(𝑀 ∪ {𝑋}))) |
| 90 | 20, 85, 89 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (𝑀 ∪ {𝑋}) ⊆ ((RSpan‘𝑅)‘(𝑀 ∪ {𝑋}))) |
| 91 | 90 | unssad 4193 |
. . . . 5
⊢ (𝜑 → 𝑀 ⊆ ((RSpan‘𝑅)‘(𝑀 ∪ {𝑋}))) |
| 92 | 90 | unssbd 4194 |
. . . . . . 7
⊢ (𝜑 → {𝑋} ⊆ ((RSpan‘𝑅)‘(𝑀 ∪ {𝑋}))) |
| 93 | | snssg 4783 |
. . . . . . . 8
⊢ (𝑋 ∈ (Base‘𝑅) → (𝑋 ∈ ((RSpan‘𝑅)‘(𝑀 ∪ {𝑋})) ↔ {𝑋} ⊆ ((RSpan‘𝑅)‘(𝑀 ∪ {𝑋})))) |
| 94 | 93 | biimpar 477 |
. . . . . . 7
⊢ ((𝑋 ∈ (Base‘𝑅) ∧ {𝑋} ⊆ ((RSpan‘𝑅)‘(𝑀 ∪ {𝑋}))) → 𝑋 ∈ ((RSpan‘𝑅)‘(𝑀 ∪ {𝑋}))) |
| 95 | 39, 92, 94 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈ ((RSpan‘𝑅)‘(𝑀 ∪ {𝑋}))) |
| 96 | | qsdrngilem.2 |
. . . . . 6
⊢ (𝜑 → ¬ 𝑋 ∈ 𝑀) |
| 97 | 95, 96 | eldifd 3962 |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ (((RSpan‘𝑅)‘(𝑀 ∪ {𝑋})) ∖ 𝑀)) |
| 98 | 7, 20, 22, 88, 91, 97 | mxidlmaxv 33496 |
. . . 4
⊢ (𝜑 → ((RSpan‘𝑅)‘(𝑀 ∪ {𝑋})) = (Base‘𝑅)) |
| 99 | 55, 98 | eleqtrrd 2844 |
. . 3
⊢ (𝜑 → (1r‘𝑅) ∈ ((RSpan‘𝑅)‘(𝑀 ∪ {𝑋}))) |
| 100 | 39, 96 | eldifd 3962 |
. . . 4
⊢ (𝜑 → 𝑋 ∈ ((Base‘𝑅) ∖ 𝑀)) |
| 101 | 86, 7, 60, 17, 20, 59, 24, 100 | elrspunsn 33457 |
. . 3
⊢ (𝜑 →
((1r‘𝑅)
∈ ((RSpan‘𝑅)‘(𝑀 ∪ {𝑋})) ↔ ∃𝑟 ∈ (Base‘𝑅)∃𝑚 ∈ 𝑀 (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚))) |
| 102 | 99, 101 | mpbid 232 |
. 2
⊢ (𝜑 → ∃𝑟 ∈ (Base‘𝑅)∃𝑚 ∈ 𝑀 (1r‘𝑅) = ((𝑟(.r‘𝑅)𝑋)(+g‘𝑅)𝑚)) |
| 103 | 83, 102 | r19.29vva 3216 |
1
⊢ (𝜑 → ∃𝑣 ∈ (Base‘𝑄)(𝑣(.r‘𝑄)[𝑋](𝑅 ~QG 𝑀)) = (1r‘𝑄)) |