Step | Hyp | Ref
| Expression |
1 | | eqid 2738 |
. . . . . 6
⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) |
2 | | mplind.i |
. . . . . 6
⊢ (𝜑 → 𝐼 ∈ 𝑊) |
3 | | mplind.r |
. . . . . 6
⊢ (𝜑 → 𝑅 ∈ CRing) |
4 | 1, 2, 3 | psrassa 20786 |
. . . . 5
⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ AssAlg) |
5 | | inss2 4118 |
. . . . . 6
⊢ (𝐻 ∩ 𝐵) ⊆ 𝐵 |
6 | | mplind.sy |
. . . . . . . 8
⊢ 𝑌 = (𝐼 mPoly 𝑅) |
7 | | mplind.sb |
. . . . . . . 8
⊢ 𝐵 = (Base‘𝑌) |
8 | | crngring 19421 |
. . . . . . . . 9
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) |
9 | 3, 8 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑅 ∈ Ring) |
10 | 1, 6, 7, 2, 9 | mplsubrg 20814 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ (SubRing‘(𝐼 mPwSer 𝑅))) |
11 | | eqid 2738 |
. . . . . . . 8
⊢
(Base‘(𝐼
mPwSer 𝑅)) =
(Base‘(𝐼 mPwSer 𝑅)) |
12 | 11 | subrgss 19648 |
. . . . . . 7
⊢ (𝐵 ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅))) |
13 | 10, 12 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅))) |
14 | 5, 13 | sstrid 3886 |
. . . . 5
⊢ (𝜑 → (𝐻 ∩ 𝐵) ⊆ (Base‘(𝐼 mPwSer 𝑅))) |
15 | | mplind.sv |
. . . . . . . . 9
⊢ 𝑉 = (𝐼 mVar 𝑅) |
16 | 6, 15, 7, 2, 9 | mvrf2 20865 |
. . . . . . . 8
⊢ (𝜑 → 𝑉:𝐼⟶𝐵) |
17 | 16 | ffnd 6499 |
. . . . . . 7
⊢ (𝜑 → 𝑉 Fn 𝐼) |
18 | | mplind.v |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) ∈ 𝐻) |
19 | 18 | ralrimiva 3096 |
. . . . . . 7
⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ 𝐻) |
20 | | fnfvrnss 6888 |
. . . . . . 7
⊢ ((𝑉 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ 𝐻) → ran 𝑉 ⊆ 𝐻) |
21 | 17, 19, 20 | syl2anc 587 |
. . . . . 6
⊢ (𝜑 → ran 𝑉 ⊆ 𝐻) |
22 | 16 | frnd 6506 |
. . . . . 6
⊢ (𝜑 → ran 𝑉 ⊆ 𝐵) |
23 | 21, 22 | ssind 4121 |
. . . . 5
⊢ (𝜑 → ran 𝑉 ⊆ (𝐻 ∩ 𝐵)) |
24 | | eqid 2738 |
. . . . . 6
⊢
(AlgSpan‘(𝐼
mPwSer 𝑅)) =
(AlgSpan‘(𝐼 mPwSer
𝑅)) |
25 | 24, 11 | aspss 20683 |
. . . . 5
⊢ (((𝐼 mPwSer 𝑅) ∈ AssAlg ∧ (𝐻 ∩ 𝐵) ⊆ (Base‘(𝐼 mPwSer 𝑅)) ∧ ran 𝑉 ⊆ (𝐻 ∩ 𝐵)) → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) ⊆ ((AlgSpan‘(𝐼 mPwSer 𝑅))‘(𝐻 ∩ 𝐵))) |
26 | 4, 14, 23, 25 | syl3anc 1372 |
. . . 4
⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) ⊆ ((AlgSpan‘(𝐼 mPwSer 𝑅))‘(𝐻 ∩ 𝐵))) |
27 | 6, 1, 15, 24, 2, 3 | mplbas2 20846 |
. . . . 5
⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = (Base‘𝑌)) |
28 | 27, 7 | eqtr4di 2791 |
. . . 4
⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = 𝐵) |
29 | 5 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → (𝐻 ∩ 𝐵) ⊆ 𝐵) |
30 | 6 | mplassa 20830 |
. . . . . . . . . . . . . 14
⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ CRing) → 𝑌 ∈ AssAlg) |
31 | 2, 3, 30 | syl2anc 587 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑌 ∈ AssAlg) |
32 | | mplind.sc |
. . . . . . . . . . . . . 14
⊢ 𝐶 = (algSc‘𝑌) |
33 | | eqid 2738 |
. . . . . . . . . . . . . 14
⊢
(Scalar‘𝑌) =
(Scalar‘𝑌) |
34 | 32, 33 | asclrhm 20697 |
. . . . . . . . . . . . 13
⊢ (𝑌 ∈ AssAlg → 𝐶 ∈ ((Scalar‘𝑌) RingHom 𝑌)) |
35 | 31, 34 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐶 ∈ ((Scalar‘𝑌) RingHom 𝑌)) |
36 | | eqid 2738 |
. . . . . . . . . . . . 13
⊢
(1r‘(Scalar‘𝑌)) =
(1r‘(Scalar‘𝑌)) |
37 | | eqid 2738 |
. . . . . . . . . . . . 13
⊢
(1r‘𝑌) = (1r‘𝑌) |
38 | 36, 37 | rhm1 19597 |
. . . . . . . . . . . 12
⊢ (𝐶 ∈ ((Scalar‘𝑌) RingHom 𝑌) → (𝐶‘(1r‘(Scalar‘𝑌))) = (1r‘𝑌)) |
39 | 35, 38 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐶‘(1r‘(Scalar‘𝑌))) = (1r‘𝑌)) |
40 | | fveq2 6668 |
. . . . . . . . . . . . 13
⊢ (𝑥 =
(1r‘(Scalar‘𝑌)) → (𝐶‘𝑥) = (𝐶‘(1r‘(Scalar‘𝑌)))) |
41 | 40 | eleq1d 2817 |
. . . . . . . . . . . 12
⊢ (𝑥 =
(1r‘(Scalar‘𝑌)) → ((𝐶‘𝑥) ∈ 𝐻 ↔ (𝐶‘(1r‘(Scalar‘𝑌))) ∈ 𝐻)) |
42 | | mplind.s |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝐶‘𝑥) ∈ 𝐻) |
43 | 42 | ralrimiva 3096 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∀𝑥 ∈ 𝐾 (𝐶‘𝑥) ∈ 𝐻) |
44 | 6, 2, 3 | mplsca 20820 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑅 = (Scalar‘𝑌)) |
45 | 44, 9 | eqeltrrd 2834 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (Scalar‘𝑌) ∈ Ring) |
46 | | eqid 2738 |
. . . . . . . . . . . . . . 15
⊢
(Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌)) |
47 | 46, 36 | ringidcl 19433 |
. . . . . . . . . . . . . 14
⊢
((Scalar‘𝑌)
∈ Ring → (1r‘(Scalar‘𝑌)) ∈ (Base‘(Scalar‘𝑌))) |
48 | 45, 47 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 →
(1r‘(Scalar‘𝑌)) ∈ (Base‘(Scalar‘𝑌))) |
49 | | mplind.sk |
. . . . . . . . . . . . . 14
⊢ 𝐾 = (Base‘𝑅) |
50 | 44 | fveq2d 6672 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (Base‘𝑅) =
(Base‘(Scalar‘𝑌))) |
51 | 49, 50 | syl5eq 2785 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑌))) |
52 | 48, 51 | eleqtrrd 2836 |
. . . . . . . . . . . 12
⊢ (𝜑 →
(1r‘(Scalar‘𝑌)) ∈ 𝐾) |
53 | 41, 43, 52 | rspcdva 3526 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐶‘(1r‘(Scalar‘𝑌))) ∈ 𝐻) |
54 | 39, 53 | eqeltrrd 2834 |
. . . . . . . . . 10
⊢ (𝜑 → (1r‘𝑌) ∈ 𝐻) |
55 | | assaring 20670 |
. . . . . . . . . . . 12
⊢ (𝑌 ∈ AssAlg → 𝑌 ∈ Ring) |
56 | 31, 55 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑌 ∈ Ring) |
57 | 7, 37 | ringidcl 19433 |
. . . . . . . . . . 11
⊢ (𝑌 ∈ Ring →
(1r‘𝑌)
∈ 𝐵) |
58 | 56, 57 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (1r‘𝑌) ∈ 𝐵) |
59 | 54, 58 | elind 4082 |
. . . . . . . . 9
⊢ (𝜑 → (1r‘𝑌) ∈ (𝐻 ∩ 𝐵)) |
60 | 59 | ne0d 4222 |
. . . . . . . 8
⊢ (𝜑 → (𝐻 ∩ 𝐵) ≠ ∅) |
61 | | elinel1 4083 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 ∈ (𝐻 ∩ 𝐵) → 𝑧 ∈ 𝐻) |
62 | | elinel1 4083 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 ∈ (𝐻 ∩ 𝐵) → 𝑤 ∈ 𝐻) |
63 | 61, 62 | anim12i 616 |
. . . . . . . . . . . . . 14
⊢ ((𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵)) → (𝑧 ∈ 𝐻 ∧ 𝑤 ∈ 𝐻)) |
64 | | mplind.p |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 + 𝑦) ∈ 𝐻) |
65 | 64 | caovclg 7350 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐻 ∧ 𝑤 ∈ 𝐻)) → (𝑧 + 𝑤) ∈ 𝐻) |
66 | 63, 65 | sylan2 596 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧 + 𝑤) ∈ 𝐻) |
67 | | assalmod 20669 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑌 ∈ AssAlg → 𝑌 ∈ LMod) |
68 | 31, 67 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑌 ∈ LMod) |
69 | | lmodgrp 19753 |
. . . . . . . . . . . . . . . 16
⊢ (𝑌 ∈ LMod → 𝑌 ∈ Grp) |
70 | 68, 69 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑌 ∈ Grp) |
71 | 70 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑌 ∈ Grp) |
72 | | simprl 771 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑧 ∈ (𝐻 ∩ 𝐵)) |
73 | 72 | elin2d 4087 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑧 ∈ 𝐵) |
74 | | simprr 773 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ (𝐻 ∩ 𝐵)) |
75 | 74 | elin2d 4087 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ 𝐵) |
76 | | mplind.sp |
. . . . . . . . . . . . . . 15
⊢ + =
(+g‘𝑌) |
77 | 7, 76 | grpcl 18220 |
. . . . . . . . . . . . . 14
⊢ ((𝑌 ∈ Grp ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → (𝑧 + 𝑤) ∈ 𝐵) |
78 | 71, 73, 75, 77 | syl3anc 1372 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧 + 𝑤) ∈ 𝐵) |
79 | 66, 78 | elind 4082 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵)) |
80 | 79 | anassrs 471 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵)) → (𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵)) |
81 | 80 | ralrimiva 3096 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵)) |
82 | | mplind.st |
. . . . . . . . . . . . 13
⊢ · =
(.r‘𝑌) |
83 | | eqid 2738 |
. . . . . . . . . . . . 13
⊢
(invg‘𝑌) = (invg‘𝑌) |
84 | 56 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑌 ∈ Ring) |
85 | | simpr 488 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑧 ∈ (𝐻 ∩ 𝐵)) |
86 | 85 | elin2d 4087 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑧 ∈ 𝐵) |
87 | 7, 82, 37, 83, 84, 86 | ringnegl 19459 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → (((invg‘𝑌)‘(1r‘𝑌)) · 𝑧) = ((invg‘𝑌)‘𝑧)) |
88 | | simpl 486 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝜑) |
89 | | rhmghm 19592 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐶 ∈ ((Scalar‘𝑌) RingHom 𝑌) → 𝐶 ∈ ((Scalar‘𝑌) GrpHom 𝑌)) |
90 | 35, 89 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐶 ∈ ((Scalar‘𝑌) GrpHom 𝑌)) |
91 | | eqid 2738 |
. . . . . . . . . . . . . . . . . 18
⊢
(invg‘(Scalar‘𝑌)) =
(invg‘(Scalar‘𝑌)) |
92 | 46, 91, 83 | ghminv 18476 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐶 ∈ ((Scalar‘𝑌) GrpHom 𝑌) ∧
(1r‘(Scalar‘𝑌)) ∈ (Base‘(Scalar‘𝑌))) → (𝐶‘((invg‘(Scalar‘𝑌))‘(1r‘(Scalar‘𝑌)))) = ((invg‘𝑌)‘(𝐶‘(1r‘(Scalar‘𝑌))))) |
93 | 90, 48, 92 | syl2anc 587 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐶‘((invg‘(Scalar‘𝑌))‘(1r‘(Scalar‘𝑌)))) = ((invg‘𝑌)‘(𝐶‘(1r‘(Scalar‘𝑌))))) |
94 | 39 | fveq2d 6672 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
((invg‘𝑌)‘(𝐶‘(1r‘(Scalar‘𝑌)))) =
((invg‘𝑌)‘(1r‘𝑌))) |
95 | 93, 94 | eqtrd 2773 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐶‘((invg‘(Scalar‘𝑌))‘(1r‘(Scalar‘𝑌)))) = ((invg‘𝑌)‘(1r‘𝑌))) |
96 | | fveq2 6668 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 =
((invg‘(Scalar‘𝑌))‘(1r‘(Scalar‘𝑌))) → (𝐶‘𝑥) = (𝐶‘((invg‘(Scalar‘𝑌))‘(1r‘(Scalar‘𝑌))))) |
97 | 96 | eleq1d 2817 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 =
((invg‘(Scalar‘𝑌))‘(1r‘(Scalar‘𝑌))) → ((𝐶‘𝑥) ∈ 𝐻 ↔ (𝐶‘((invg‘(Scalar‘𝑌))‘(1r‘(Scalar‘𝑌)))) ∈ 𝐻)) |
98 | | ringgrp 19414 |
. . . . . . . . . . . . . . . . . . 19
⊢
((Scalar‘𝑌)
∈ Ring → (Scalar‘𝑌) ∈ Grp) |
99 | 45, 98 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (Scalar‘𝑌) ∈ Grp) |
100 | 46, 91 | grpinvcl 18262 |
. . . . . . . . . . . . . . . . . 18
⊢
(((Scalar‘𝑌)
∈ Grp ∧ (1r‘(Scalar‘𝑌)) ∈ (Base‘(Scalar‘𝑌))) →
((invg‘(Scalar‘𝑌))‘(1r‘(Scalar‘𝑌))) ∈
(Base‘(Scalar‘𝑌))) |
101 | 99, 48, 100 | syl2anc 587 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 →
((invg‘(Scalar‘𝑌))‘(1r‘(Scalar‘𝑌))) ∈
(Base‘(Scalar‘𝑌))) |
102 | 101, 51 | eleqtrrd 2836 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
((invg‘(Scalar‘𝑌))‘(1r‘(Scalar‘𝑌))) ∈ 𝐾) |
103 | 97, 43, 102 | rspcdva 3526 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐶‘((invg‘(Scalar‘𝑌))‘(1r‘(Scalar‘𝑌)))) ∈ 𝐻) |
104 | 95, 103 | eqeltrrd 2834 |
. . . . . . . . . . . . . 14
⊢ (𝜑 →
((invg‘𝑌)‘(1r‘𝑌)) ∈ 𝐻) |
105 | 104 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ((invg‘𝑌)‘(1r‘𝑌)) ∈ 𝐻) |
106 | 85 | elin1d 4086 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑧 ∈ 𝐻) |
107 | | mplind.t |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 · 𝑦) ∈ 𝐻) |
108 | 107 | caovclg 7350 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧
(((invg‘𝑌)‘(1r‘𝑌)) ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → (((invg‘𝑌)‘(1r‘𝑌)) · 𝑧) ∈ 𝐻) |
109 | 88, 105, 106, 108 | syl12anc 836 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → (((invg‘𝑌)‘(1r‘𝑌)) · 𝑧) ∈ 𝐻) |
110 | 87, 109 | eqeltrrd 2834 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ((invg‘𝑌)‘𝑧) ∈ 𝐻) |
111 | 70 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑌 ∈ Grp) |
112 | 7, 83 | grpinvcl 18262 |
. . . . . . . . . . . 12
⊢ ((𝑌 ∈ Grp ∧ 𝑧 ∈ 𝐵) → ((invg‘𝑌)‘𝑧) ∈ 𝐵) |
113 | 111, 86, 112 | syl2anc 587 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ((invg‘𝑌)‘𝑧) ∈ 𝐵) |
114 | 110, 113 | elind 4082 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ((invg‘𝑌)‘𝑧) ∈ (𝐻 ∩ 𝐵)) |
115 | 81, 114 | jca 515 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → (∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵) ∧ ((invg‘𝑌)‘𝑧) ∈ (𝐻 ∩ 𝐵))) |
116 | 115 | ralrimiva 3096 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑧 ∈ (𝐻 ∩ 𝐵)(∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵) ∧ ((invg‘𝑌)‘𝑧) ∈ (𝐻 ∩ 𝐵))) |
117 | 7, 76, 83 | issubg2 18405 |
. . . . . . . . 9
⊢ (𝑌 ∈ Grp → ((𝐻 ∩ 𝐵) ∈ (SubGrp‘𝑌) ↔ ((𝐻 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐻 ∩ 𝐵) ≠ ∅ ∧ ∀𝑧 ∈ (𝐻 ∩ 𝐵)(∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵) ∧ ((invg‘𝑌)‘𝑧) ∈ (𝐻 ∩ 𝐵))))) |
118 | 70, 117 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ((𝐻 ∩ 𝐵) ∈ (SubGrp‘𝑌) ↔ ((𝐻 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐻 ∩ 𝐵) ≠ ∅ ∧ ∀𝑧 ∈ (𝐻 ∩ 𝐵)(∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵) ∧ ((invg‘𝑌)‘𝑧) ∈ (𝐻 ∩ 𝐵))))) |
119 | 29, 60, 116, 118 | mpbir3and 1343 |
. . . . . . 7
⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (SubGrp‘𝑌)) |
120 | | elinel1 4083 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐻 ∩ 𝐵) → 𝑥 ∈ 𝐻) |
121 | | elinel1 4083 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ (𝐻 ∩ 𝐵) → 𝑦 ∈ 𝐻) |
122 | 120, 121 | anim12i 616 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵)) → (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) |
123 | 122, 107 | sylan2 596 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → (𝑥 · 𝑦) ∈ 𝐻) |
124 | 56 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑌 ∈ Ring) |
125 | | simprl 771 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑥 ∈ (𝐻 ∩ 𝐵)) |
126 | 125 | elin2d 4087 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑥 ∈ 𝐵) |
127 | | simprr 773 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑦 ∈ (𝐻 ∩ 𝐵)) |
128 | 127 | elin2d 4087 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑦 ∈ 𝐵) |
129 | 7, 82 | ringcl 19426 |
. . . . . . . . . 10
⊢ ((𝑌 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵) |
130 | 124, 126,
128, 129 | syl3anc 1372 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → (𝑥 · 𝑦) ∈ 𝐵) |
131 | 123, 130 | elind 4082 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → (𝑥 · 𝑦) ∈ (𝐻 ∩ 𝐵)) |
132 | 131 | ralrimivva 3103 |
. . . . . . 7
⊢ (𝜑 → ∀𝑥 ∈ (𝐻 ∩ 𝐵)∀𝑦 ∈ (𝐻 ∩ 𝐵)(𝑥 · 𝑦) ∈ (𝐻 ∩ 𝐵)) |
133 | 7, 37, 82 | issubrg2 19667 |
. . . . . . . 8
⊢ (𝑌 ∈ Ring → ((𝐻 ∩ 𝐵) ∈ (SubRing‘𝑌) ↔ ((𝐻 ∩ 𝐵) ∈ (SubGrp‘𝑌) ∧ (1r‘𝑌) ∈ (𝐻 ∩ 𝐵) ∧ ∀𝑥 ∈ (𝐻 ∩ 𝐵)∀𝑦 ∈ (𝐻 ∩ 𝐵)(𝑥 · 𝑦) ∈ (𝐻 ∩ 𝐵)))) |
134 | 56, 133 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ((𝐻 ∩ 𝐵) ∈ (SubRing‘𝑌) ↔ ((𝐻 ∩ 𝐵) ∈ (SubGrp‘𝑌) ∧ (1r‘𝑌) ∈ (𝐻 ∩ 𝐵) ∧ ∀𝑥 ∈ (𝐻 ∩ 𝐵)∀𝑦 ∈ (𝐻 ∩ 𝐵)(𝑥 · 𝑦) ∈ (𝐻 ∩ 𝐵)))) |
135 | 119, 59, 132, 134 | mpbir3and 1343 |
. . . . . 6
⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (SubRing‘𝑌)) |
136 | 6, 1, 7 | mplval2 20805 |
. . . . . . . 8
⊢ 𝑌 = ((𝐼 mPwSer 𝑅) ↾s 𝐵) |
137 | 136 | subsubrg 19673 |
. . . . . . 7
⊢ (𝐵 ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → ((𝐻 ∩ 𝐵) ∈ (SubRing‘𝑌) ↔ ((𝐻 ∩ 𝐵) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) ∧ (𝐻 ∩ 𝐵) ⊆ 𝐵))) |
138 | 137 | simprbda 502 |
. . . . . 6
⊢ ((𝐵 ∈ (SubRing‘(𝐼 mPwSer 𝑅)) ∧ (𝐻 ∩ 𝐵) ∈ (SubRing‘𝑌)) → (𝐻 ∩ 𝐵) ∈ (SubRing‘(𝐼 mPwSer 𝑅))) |
139 | 10, 135, 138 | syl2anc 587 |
. . . . 5
⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (SubRing‘(𝐼 mPwSer 𝑅))) |
140 | | assalmod 20669 |
. . . . . . 7
⊢ ((𝐼 mPwSer 𝑅) ∈ AssAlg → (𝐼 mPwSer 𝑅) ∈ LMod) |
141 | 4, 140 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ LMod) |
142 | 1, 6, 7, 2, 9 | mpllss 20812 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ (LSubSp‘(𝐼 mPwSer 𝑅))) |
143 | 31 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑌 ∈ AssAlg) |
144 | | simprl 771 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑧 ∈ (Base‘(Scalar‘𝑌))) |
145 | | simprr 773 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ (𝐻 ∩ 𝐵)) |
146 | 145 | elin2d 4087 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ 𝐵) |
147 | | eqid 2738 |
. . . . . . . . . . . 12
⊢ (
·𝑠 ‘𝑌) = ( ·𝑠
‘𝑌) |
148 | 32, 33, 46, 7, 82, 147 | asclmul1 20692 |
. . . . . . . . . . 11
⊢ ((𝑌 ∈ AssAlg ∧ 𝑧 ∈
(Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ 𝐵) → ((𝐶‘𝑧) · 𝑤) = (𝑧( ·𝑠
‘𝑌)𝑤)) |
149 | 143, 144,
146, 148 | syl3anc 1372 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → ((𝐶‘𝑧) · 𝑤) = (𝑧( ·𝑠
‘𝑌)𝑤)) |
150 | | fveq2 6668 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑧 → (𝐶‘𝑥) = (𝐶‘𝑧)) |
151 | 150 | eleq1d 2817 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑧 → ((𝐶‘𝑥) ∈ 𝐻 ↔ (𝐶‘𝑧) ∈ 𝐻)) |
152 | 43 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → ∀𝑥 ∈ 𝐾 (𝐶‘𝑥) ∈ 𝐻) |
153 | 51 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝐾 = (Base‘(Scalar‘𝑌))) |
154 | 144, 153 | eleqtrrd 2836 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑧 ∈ 𝐾) |
155 | 151, 152,
154 | rspcdva 3526 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝐶‘𝑧) ∈ 𝐻) |
156 | 145 | elin1d 4086 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ 𝐻) |
157 | 155, 156 | jca 515 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → ((𝐶‘𝑧) ∈ 𝐻 ∧ 𝑤 ∈ 𝐻)) |
158 | 107 | caovclg 7350 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝐶‘𝑧) ∈ 𝐻 ∧ 𝑤 ∈ 𝐻)) → ((𝐶‘𝑧) · 𝑤) ∈ 𝐻) |
159 | 157, 158 | syldan 594 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → ((𝐶‘𝑧) · 𝑤) ∈ 𝐻) |
160 | 149, 159 | eqeltrrd 2834 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧( ·𝑠
‘𝑌)𝑤) ∈ 𝐻) |
161 | 68 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑌 ∈ LMod) |
162 | 7, 33, 147, 46 | lmodvscl 19763 |
. . . . . . . . . 10
⊢ ((𝑌 ∈ LMod ∧ 𝑧 ∈
(Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ 𝐵) → (𝑧( ·𝑠
‘𝑌)𝑤) ∈ 𝐵) |
163 | 161, 144,
146, 162 | syl3anc 1372 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧( ·𝑠
‘𝑌)𝑤) ∈ 𝐵) |
164 | 160, 163 | elind 4082 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧( ·𝑠
‘𝑌)𝑤) ∈ (𝐻 ∩ 𝐵)) |
165 | 164 | ralrimivva 3103 |
. . . . . . 7
⊢ (𝜑 → ∀𝑧 ∈ (Base‘(Scalar‘𝑌))∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧( ·𝑠
‘𝑌)𝑤) ∈ (𝐻 ∩ 𝐵)) |
166 | | eqid 2738 |
. . . . . . . . 9
⊢
(LSubSp‘𝑌) =
(LSubSp‘𝑌) |
167 | 33, 46, 7, 147, 166 | islss4 19846 |
. . . . . . . 8
⊢ (𝑌 ∈ LMod → ((𝐻 ∩ 𝐵) ∈ (LSubSp‘𝑌) ↔ ((𝐻 ∩ 𝐵) ∈ (SubGrp‘𝑌) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝑌))∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧( ·𝑠
‘𝑌)𝑤) ∈ (𝐻 ∩ 𝐵)))) |
168 | 68, 167 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ((𝐻 ∩ 𝐵) ∈ (LSubSp‘𝑌) ↔ ((𝐻 ∩ 𝐵) ∈ (SubGrp‘𝑌) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝑌))∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧( ·𝑠
‘𝑌)𝑤) ∈ (𝐻 ∩ 𝐵)))) |
169 | 119, 165,
168 | mpbir2and 713 |
. . . . . 6
⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (LSubSp‘𝑌)) |
170 | | eqid 2738 |
. . . . . . . 8
⊢
(LSubSp‘(𝐼
mPwSer 𝑅)) =
(LSubSp‘(𝐼 mPwSer
𝑅)) |
171 | 136, 170,
166 | lsslss 19845 |
. . . . . . 7
⊢ (((𝐼 mPwSer 𝑅) ∈ LMod ∧ 𝐵 ∈ (LSubSp‘(𝐼 mPwSer 𝑅))) → ((𝐻 ∩ 𝐵) ∈ (LSubSp‘𝑌) ↔ ((𝐻 ∩ 𝐵) ∈ (LSubSp‘(𝐼 mPwSer 𝑅)) ∧ (𝐻 ∩ 𝐵) ⊆ 𝐵))) |
172 | 171 | simprbda 502 |
. . . . . 6
⊢ ((((𝐼 mPwSer 𝑅) ∈ LMod ∧ 𝐵 ∈ (LSubSp‘(𝐼 mPwSer 𝑅))) ∧ (𝐻 ∩ 𝐵) ∈ (LSubSp‘𝑌)) → (𝐻 ∩ 𝐵) ∈ (LSubSp‘(𝐼 mPwSer 𝑅))) |
173 | 141, 142,
169, 172 | syl21anc 837 |
. . . . 5
⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (LSubSp‘(𝐼 mPwSer 𝑅))) |
174 | 24, 11, 170 | aspid 20681 |
. . . . 5
⊢ (((𝐼 mPwSer 𝑅) ∈ AssAlg ∧ (𝐻 ∩ 𝐵) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) ∧ (𝐻 ∩ 𝐵) ∈ (LSubSp‘(𝐼 mPwSer 𝑅))) → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘(𝐻 ∩ 𝐵)) = (𝐻 ∩ 𝐵)) |
175 | 4, 139, 173, 174 | syl3anc 1372 |
. . . 4
⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘(𝐻 ∩ 𝐵)) = (𝐻 ∩ 𝐵)) |
176 | 26, 28, 175 | 3sstr3d 3921 |
. . 3
⊢ (𝜑 → 𝐵 ⊆ (𝐻 ∩ 𝐵)) |
177 | | mplind.x |
. . 3
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
178 | 176, 177 | sseldd 3876 |
. 2
⊢ (𝜑 → 𝑋 ∈ (𝐻 ∩ 𝐵)) |
179 | 178 | elin1d 4086 |
1
⊢ (𝜑 → 𝑋 ∈ 𝐻) |