| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | sraassab.s | . . . . . . . 8
⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑊)) | 
| 2 |  | eqid 2736 | . . . . . . . . 9
⊢
(Base‘𝑊) =
(Base‘𝑊) | 
| 3 | 2 | subrgss 20573 | . . . . . . . 8
⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊)) | 
| 4 | 1, 3 | syl 17 | . . . . . . 7
⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) | 
| 5 | 4 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝐴 ∈ AssAlg) → 𝑆 ⊆ (Base‘𝑊)) | 
| 6 | 5 | sselda 3982 | . . . . 5
⊢ (((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ (Base‘𝑊)) | 
| 7 |  | simpllr 775 | . . . . . . . 8
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → 𝐴 ∈ AssAlg) | 
| 8 |  | eqid 2736 | . . . . . . . . . . . . . 14
⊢ (𝑊 ↾s 𝑆) = (𝑊 ↾s 𝑆) | 
| 9 | 8 | subrgbas 20582 | . . . . . . . . . . . . 13
⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 = (Base‘(𝑊 ↾s 𝑆))) | 
| 10 | 1, 9 | syl 17 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝑆 = (Base‘(𝑊 ↾s 𝑆))) | 
| 11 |  | sraassab.a | . . . . . . . . . . . . . . 15
⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆) | 
| 12 | 11 | a1i 11 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) | 
| 13 | 12, 4 | srasca 21184 | . . . . . . . . . . . . 13
⊢ (𝜑 → (𝑊 ↾s 𝑆) = (Scalar‘𝐴)) | 
| 14 | 13 | fveq2d 6909 | . . . . . . . . . . . 12
⊢ (𝜑 → (Base‘(𝑊 ↾s 𝑆)) =
(Base‘(Scalar‘𝐴))) | 
| 15 | 10, 14 | eqtrd 2776 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑆 = (Base‘(Scalar‘𝐴))) | 
| 16 | 15 | eqimssd 4039 | . . . . . . . . . 10
⊢ (𝜑 → 𝑆 ⊆ (Base‘(Scalar‘𝐴))) | 
| 17 | 16 | sselda 3982 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ (Base‘(Scalar‘𝐴))) | 
| 18 | 17 | ad4ant13 751 | . . . . . . . 8
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → 𝑦 ∈ (Base‘(Scalar‘𝐴))) | 
| 19 | 12, 4 | srabase 21178 | . . . . . . . . . . 11
⊢ (𝜑 → (Base‘𝑊) = (Base‘𝐴)) | 
| 20 | 19 | eqimssd 4039 | . . . . . . . . . 10
⊢ (𝜑 → (Base‘𝑊) ⊆ (Base‘𝐴)) | 
| 21 | 20 | ad2antrr 726 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) → (Base‘𝑊) ⊆ (Base‘𝐴)) | 
| 22 | 21 | sselda 3982 | . . . . . . . 8
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝐴)) | 
| 23 |  | sraassab.w | . . . . . . . . . . 11
⊢ (𝜑 → 𝑊 ∈ Ring) | 
| 24 |  | eqid 2736 | . . . . . . . . . . . 12
⊢
(1r‘𝑊) = (1r‘𝑊) | 
| 25 | 2, 24 | ringidcl 20263 | . . . . . . . . . . 11
⊢ (𝑊 ∈ Ring →
(1r‘𝑊)
∈ (Base‘𝑊)) | 
| 26 | 23, 25 | syl 17 | . . . . . . . . . 10
⊢ (𝜑 → (1r‘𝑊) ∈ (Base‘𝑊)) | 
| 27 | 26, 19 | eleqtrd 2842 | . . . . . . . . 9
⊢ (𝜑 → (1r‘𝑊) ∈ (Base‘𝐴)) | 
| 28 | 27 | ad3antrrr 730 | . . . . . . . 8
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → (1r‘𝑊) ∈ (Base‘𝐴)) | 
| 29 |  | eqid 2736 | . . . . . . . . 9
⊢
(Base‘𝐴) =
(Base‘𝐴) | 
| 30 |  | eqid 2736 | . . . . . . . . 9
⊢
(Scalar‘𝐴) =
(Scalar‘𝐴) | 
| 31 |  | eqid 2736 | . . . . . . . . 9
⊢
(Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴)) | 
| 32 |  | eqid 2736 | . . . . . . . . 9
⊢ (
·𝑠 ‘𝐴) = ( ·𝑠
‘𝐴) | 
| 33 |  | eqid 2736 | . . . . . . . . 9
⊢
(.r‘𝐴) = (.r‘𝐴) | 
| 34 | 29, 30, 31, 32, 33 | assaassr 21880 | . . . . . . . 8
⊢ ((𝐴 ∈ AssAlg ∧ (𝑦 ∈
(Base‘(Scalar‘𝐴)) ∧ 𝑥 ∈ (Base‘𝐴) ∧ (1r‘𝑊) ∈ (Base‘𝐴))) → (𝑥(.r‘𝐴)(𝑦( ·𝑠
‘𝐴)(1r‘𝑊))) = (𝑦( ·𝑠
‘𝐴)(𝑥(.r‘𝐴)(1r‘𝑊)))) | 
| 35 | 7, 18, 22, 28, 34 | syl13anc 1373 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥(.r‘𝐴)(𝑦( ·𝑠
‘𝐴)(1r‘𝑊))) = (𝑦( ·𝑠
‘𝐴)(𝑥(.r‘𝐴)(1r‘𝑊)))) | 
| 36 | 12, 4 | sramulr 21182 | . . . . . . . . . 10
⊢ (𝜑 → (.r‘𝑊) = (.r‘𝐴)) | 
| 37 | 36 | ad3antrrr 730 | . . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → (.r‘𝑊) = (.r‘𝐴)) | 
| 38 | 37 | oveqd 7449 | . . . . . . . 8
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥(.r‘𝑊)(𝑦( ·𝑠
‘𝐴)(1r‘𝑊))) = (𝑥(.r‘𝐴)(𝑦( ·𝑠
‘𝐴)(1r‘𝑊)))) | 
| 39 | 12, 4 | sravsca 21186 | . . . . . . . . . . . 12
⊢ (𝜑 → (.r‘𝑊) = (
·𝑠 ‘𝐴)) | 
| 40 | 39 | ad3antrrr 730 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → (.r‘𝑊) = (
·𝑠 ‘𝐴)) | 
| 41 | 40 | oveqd 7449 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑦(.r‘𝑊)(1r‘𝑊)) = (𝑦( ·𝑠
‘𝐴)(1r‘𝑊))) | 
| 42 |  | eqid 2736 | . . . . . . . . . . 11
⊢
(.r‘𝑊) = (.r‘𝑊) | 
| 43 | 23 | ad3antrrr 730 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → 𝑊 ∈ Ring) | 
| 44 | 6 | adantr 480 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → 𝑦 ∈ (Base‘𝑊)) | 
| 45 | 2, 42, 24, 43, 44 | ringridmd 20271 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑦(.r‘𝑊)(1r‘𝑊)) = 𝑦) | 
| 46 | 41, 45 | eqtr3d 2778 | . . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑦( ·𝑠
‘𝐴)(1r‘𝑊)) = 𝑦) | 
| 47 | 46 | oveq2d 7448 | . . . . . . . 8
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥(.r‘𝑊)(𝑦( ·𝑠
‘𝐴)(1r‘𝑊))) = (𝑥(.r‘𝑊)𝑦)) | 
| 48 | 38, 47 | eqtr3d 2778 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥(.r‘𝐴)(𝑦( ·𝑠
‘𝐴)(1r‘𝑊))) = (𝑥(.r‘𝑊)𝑦)) | 
| 49 | 40 | oveqd 7449 | . . . . . . . 8
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑦(.r‘𝑊)(𝑥(.r‘𝐴)(1r‘𝑊))) = (𝑦( ·𝑠
‘𝐴)(𝑥(.r‘𝐴)(1r‘𝑊)))) | 
| 50 | 37 | oveqd 7449 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥(.r‘𝑊)(1r‘𝑊)) = (𝑥(.r‘𝐴)(1r‘𝑊))) | 
| 51 |  | simpr 484 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝑊)) | 
| 52 | 2, 42, 24, 43, 51 | ringridmd 20271 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥(.r‘𝑊)(1r‘𝑊)) = 𝑥) | 
| 53 | 50, 52 | eqtr3d 2778 | . . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑥(.r‘𝐴)(1r‘𝑊)) = 𝑥) | 
| 54 | 53 | oveq2d 7448 | . . . . . . . 8
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑦(.r‘𝑊)(𝑥(.r‘𝐴)(1r‘𝑊))) = (𝑦(.r‘𝑊)𝑥)) | 
| 55 | 49, 54 | eqtr3d 2778 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑦( ·𝑠
‘𝐴)(𝑥(.r‘𝐴)(1r‘𝑊))) = (𝑦(.r‘𝑊)𝑥)) | 
| 56 | 35, 48, 55 | 3eqtr3rd 2785 | . . . . . 6
⊢ ((((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑦(.r‘𝑊)𝑥) = (𝑥(.r‘𝑊)𝑦)) | 
| 57 | 56 | ralrimiva 3145 | . . . . 5
⊢ (((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) → ∀𝑥 ∈ (Base‘𝑊)(𝑦(.r‘𝑊)𝑥) = (𝑥(.r‘𝑊)𝑦)) | 
| 58 |  | eqid 2736 | . . . . . . 7
⊢
(mulGrp‘𝑊) =
(mulGrp‘𝑊) | 
| 59 | 58, 2 | mgpbas 20143 | . . . . . 6
⊢
(Base‘𝑊) =
(Base‘(mulGrp‘𝑊)) | 
| 60 | 58, 42 | mgpplusg 20142 | . . . . . 6
⊢
(.r‘𝑊) = (+g‘(mulGrp‘𝑊)) | 
| 61 |  | sraassab.z | . . . . . 6
⊢ 𝑍 =
(Cntr‘(mulGrp‘𝑊)) | 
| 62 | 59, 60, 61 | elcntr 19349 | . . . . 5
⊢ (𝑦 ∈ 𝑍 ↔ (𝑦 ∈ (Base‘𝑊) ∧ ∀𝑥 ∈ (Base‘𝑊)(𝑦(.r‘𝑊)𝑥) = (𝑥(.r‘𝑊)𝑦))) | 
| 63 | 6, 57, 62 | sylanbrc 583 | . . . 4
⊢ (((𝜑 ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑍) | 
| 64 | 63 | ex 412 | . . 3
⊢ ((𝜑 ∧ 𝐴 ∈ AssAlg) → (𝑦 ∈ 𝑆 → 𝑦 ∈ 𝑍)) | 
| 65 | 64 | ssrdv 3988 | . 2
⊢ ((𝜑 ∧ 𝐴 ∈ AssAlg) → 𝑆 ⊆ 𝑍) | 
| 66 | 19 | adantr 480 | . . 3
⊢ ((𝜑 ∧ 𝑆 ⊆ 𝑍) → (Base‘𝑊) = (Base‘𝐴)) | 
| 67 | 13 | adantr 480 | . . 3
⊢ ((𝜑 ∧ 𝑆 ⊆ 𝑍) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴)) | 
| 68 | 10 | adantr 480 | . . 3
⊢ ((𝜑 ∧ 𝑆 ⊆ 𝑍) → 𝑆 = (Base‘(𝑊 ↾s 𝑆))) | 
| 69 | 39 | adantr 480 | . . 3
⊢ ((𝜑 ∧ 𝑆 ⊆ 𝑍) → (.r‘𝑊) = (
·𝑠 ‘𝐴)) | 
| 70 | 36 | adantr 480 | . . 3
⊢ ((𝜑 ∧ 𝑆 ⊆ 𝑍) → (.r‘𝑊) = (.r‘𝐴)) | 
| 71 | 11 | sralmod 21195 | . . . . 5
⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod) | 
| 72 | 1, 71 | syl 17 | . . . 4
⊢ (𝜑 → 𝐴 ∈ LMod) | 
| 73 | 72 | adantr 480 | . . 3
⊢ ((𝜑 ∧ 𝑆 ⊆ 𝑍) → 𝐴 ∈ LMod) | 
| 74 | 11, 2 | sraring 21194 | . . . . 5
⊢ ((𝑊 ∈ Ring ∧ 𝑆 ⊆ (Base‘𝑊)) → 𝐴 ∈ Ring) | 
| 75 | 23, 4, 74 | syl2anc 584 | . . . 4
⊢ (𝜑 → 𝐴 ∈ Ring) | 
| 76 | 75 | adantr 480 | . . 3
⊢ ((𝜑 ∧ 𝑆 ⊆ 𝑍) → 𝐴 ∈ Ring) | 
| 77 | 23 | ad2antrr 726 | . . . 4
⊢ (((𝜑 ∧ 𝑆 ⊆ 𝑍) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring) | 
| 78 | 4 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑆 ⊆ 𝑍) → 𝑆 ⊆ (Base‘𝑊)) | 
| 79 | 78 | sselda 3982 | . . . . 5
⊢ (((𝜑 ∧ 𝑆 ⊆ 𝑍) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝑊)) | 
| 80 | 79 | 3ad2antr1 1188 | . . . 4
⊢ (((𝜑 ∧ 𝑆 ⊆ 𝑍) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊)) | 
| 81 |  | simpr2 1195 | . . . 4
⊢ (((𝜑 ∧ 𝑆 ⊆ 𝑍) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊)) | 
| 82 |  | simpr3 1196 | . . . 4
⊢ (((𝜑 ∧ 𝑆 ⊆ 𝑍) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊)) | 
| 83 | 2, 42, 77, 80, 81, 82 | ringassd 20255 | . . 3
⊢ (((𝜑 ∧ 𝑆 ⊆ 𝑍) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r‘𝑊)𝑦)(.r‘𝑊)𝑧) = (𝑥(.r‘𝑊)(𝑦(.r‘𝑊)𝑧))) | 
| 84 |  | ssel2 3977 | . . . . . . . 8
⊢ ((𝑆 ⊆ 𝑍 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑍) | 
| 85 | 84 | ad2ant2lr 748 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑆 ⊆ 𝑍) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑥 ∈ 𝑍) | 
| 86 |  | simprr 772 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑆 ⊆ 𝑍) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊)) | 
| 87 | 59, 60, 61 | cntri 19351 | . . . . . . 7
⊢ ((𝑥 ∈ 𝑍 ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(.r‘𝑊)𝑦) = (𝑦(.r‘𝑊)𝑥)) | 
| 88 | 85, 86, 87 | syl2anc 584 | . . . . . 6
⊢ (((𝜑 ∧ 𝑆 ⊆ 𝑍) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(.r‘𝑊)𝑦) = (𝑦(.r‘𝑊)𝑥)) | 
| 89 | 88 | 3adantr3 1171 | . . . . 5
⊢ (((𝜑 ∧ 𝑆 ⊆ 𝑍) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥(.r‘𝑊)𝑦) = (𝑦(.r‘𝑊)𝑥)) | 
| 90 | 89 | oveq1d 7447 | . . . 4
⊢ (((𝜑 ∧ 𝑆 ⊆ 𝑍) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r‘𝑊)𝑦)(.r‘𝑊)𝑧) = ((𝑦(.r‘𝑊)𝑥)(.r‘𝑊)𝑧)) | 
| 91 | 2, 42, 77, 81, 80, 82 | ringassd 20255 | . . . 4
⊢ (((𝜑 ∧ 𝑆 ⊆ 𝑍) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑦(.r‘𝑊)𝑥)(.r‘𝑊)𝑧) = (𝑦(.r‘𝑊)(𝑥(.r‘𝑊)𝑧))) | 
| 92 | 90, 83, 91 | 3eqtr3rd 2785 | . . 3
⊢ (((𝜑 ∧ 𝑆 ⊆ 𝑍) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑦(.r‘𝑊)(𝑥(.r‘𝑊)𝑧)) = (𝑥(.r‘𝑊)(𝑦(.r‘𝑊)𝑧))) | 
| 93 | 66, 67, 68, 69, 70, 73, 76, 83, 92 | isassad 21886 | . 2
⊢ ((𝜑 ∧ 𝑆 ⊆ 𝑍) → 𝐴 ∈ AssAlg) | 
| 94 | 65, 93 | impbida 800 | 1
⊢ (𝜑 → (𝐴 ∈ AssAlg ↔ 𝑆 ⊆ 𝑍)) |