Step | Hyp | Ref
| Expression |
1 | | lclkrlem2e.k |
. . . . . 6
β’ (π β (πΎ β HL β§ π β π»)) |
2 | 1 | adantr 482 |
. . . . 5
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β (πΎ β HL β§ π β π»)) |
3 | | lclkrlem2e.x |
. . . . . . . . 9
β’ (π β π β (π β { 0 })) |
4 | 3 | eldifad 3926 |
. . . . . . . 8
β’ (π β π β π) |
5 | 4 | snssd 4773 |
. . . . . . 7
β’ (π β {π} β π) |
6 | 5 | adantr 482 |
. . . . . 6
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β {π} β π) |
7 | | lclkrlem2e.h |
. . . . . . 7
β’ π» = (LHypβπΎ) |
8 | | eqid 2733 |
. . . . . . 7
β’
((DIsoHβπΎ)βπ) = ((DIsoHβπΎ)βπ) |
9 | | lclkrlem2e.u |
. . . . . . 7
β’ π = ((DVecHβπΎ)βπ) |
10 | | lclkrlem2e.v |
. . . . . . 7
β’ π = (Baseβπ) |
11 | | lclkrlem2e.o |
. . . . . . 7
β’ β₯ =
((ocHβπΎ)βπ) |
12 | 7, 8, 9, 10, 11 | dochcl 39866 |
. . . . . 6
β’ (((πΎ β HL β§ π β π») β§ {π} β π) β ( β₯ β{π}) β ran
((DIsoHβπΎ)βπ)) |
13 | 2, 6, 12 | syl2anc 585 |
. . . . 5
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β ( β₯ β{π}) β ran
((DIsoHβπΎ)βπ)) |
14 | 7, 8, 11 | dochoc 39880 |
. . . . 5
β’ (((πΎ β HL β§ π β π») β§ ( β₯ β{π}) β ran
((DIsoHβπΎ)βπ)) β ( β₯ β( β₯
β( β₯ β{π}))) = ( β₯ β{π})) |
15 | 2, 13, 14 | syl2anc 585 |
. . . 4
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β ( β₯ β( β₯
β( β₯ β{π}))) = ( β₯ β{π})) |
16 | | lclkrlem2e.le |
. . . . . . . 8
β’ (π β (πΏβπΈ) = ( β₯ β{π})) |
17 | 16 | adantr 482 |
. . . . . . 7
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β (πΏβπΈ) = ( β₯ β{π})) |
18 | | inidm 4182 |
. . . . . . . . . . 11
β’ ((πΏβπΈ) β© (πΏβπΈ)) = (πΏβπΈ) |
19 | | lclkrlem2e.ne |
. . . . . . . . . . . 12
β’ (π β (πΏβπΈ) = (πΏβπΊ)) |
20 | 19 | ineq2d 4176 |
. . . . . . . . . . 11
β’ (π β ((πΏβπΈ) β© (πΏβπΈ)) = ((πΏβπΈ) β© (πΏβπΊ))) |
21 | 18, 20 | eqtr3id 2787 |
. . . . . . . . . 10
β’ (π β (πΏβπΈ) = ((πΏβπΈ) β© (πΏβπΊ))) |
22 | | lclkrlem2e.f |
. . . . . . . . . . 11
β’ πΉ = (LFnlβπ) |
23 | | lclkrlem2e.l |
. . . . . . . . . . 11
β’ πΏ = (LKerβπ) |
24 | | lclkrlem2e.d |
. . . . . . . . . . 11
β’ π· = (LDualβπ) |
25 | | lclkrlem2e.p |
. . . . . . . . . . 11
β’ + =
(+gβπ·) |
26 | 7, 9, 1 | dvhlmod 39623 |
. . . . . . . . . . 11
β’ (π β π β LMod) |
27 | | lclkrlem2e.e |
. . . . . . . . . . 11
β’ (π β πΈ β πΉ) |
28 | | lclkrlem2e.g |
. . . . . . . . . . 11
β’ (π β πΊ β πΉ) |
29 | 22, 23, 24, 25, 26, 27, 28 | lkrin 37676 |
. . . . . . . . . 10
β’ (π β ((πΏβπΈ) β© (πΏβπΊ)) β (πΏβ(πΈ + πΊ))) |
30 | 21, 29 | eqsstrd 3986 |
. . . . . . . . 9
β’ (π β (πΏβπΈ) β (πΏβ(πΈ + πΊ))) |
31 | 30 | adantr 482 |
. . . . . . . 8
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β (πΏβπΈ) β (πΏβ(πΈ + πΊ))) |
32 | | eqid 2733 |
. . . . . . . . 9
β’
(LSHypβπ) =
(LSHypβπ) |
33 | 7, 9, 1 | dvhlvec 39622 |
. . . . . . . . . 10
β’ (π β π β LVec) |
34 | 33 | adantr 482 |
. . . . . . . . 9
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β π β LVec) |
35 | | eqid 2733 |
. . . . . . . . . . . . 13
β’
(LSpanβπ) =
(LSpanβπ) |
36 | 7, 9, 11, 10, 35, 1, 5 | dochocsp 39892 |
. . . . . . . . . . . 12
β’ (π β ( β₯
β((LSpanβπ)β{π})) = ( β₯ β{π})) |
37 | 36 | adantr 482 |
. . . . . . . . . . 11
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β ( β₯
β((LSpanβπ)β{π})) = ( β₯ β{π})) |
38 | 17, 37 | eqtr4d 2776 |
. . . . . . . . . 10
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β (πΏβπΈ) = ( β₯
β((LSpanβπ)β{π}))) |
39 | | eqid 2733 |
. . . . . . . . . . 11
β’
(LSAtomsβπ) =
(LSAtomsβπ) |
40 | | lclkrlem2e.z |
. . . . . . . . . . . . 13
β’ 0 =
(0gβπ) |
41 | 10, 35, 40, 39, 26, 3 | lsatlspsn 37505 |
. . . . . . . . . . . 12
β’ (π β ((LSpanβπ)β{π}) β (LSAtomsβπ)) |
42 | 41 | adantr 482 |
. . . . . . . . . . 11
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β ((LSpanβπ)β{π}) β (LSAtomsβπ)) |
43 | 7, 9, 11, 39, 32, 2, 42 | dochsatshp 39964 |
. . . . . . . . . 10
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β ( β₯
β((LSpanβπ)β{π})) β (LSHypβπ)) |
44 | 38, 43 | eqeltrd 2834 |
. . . . . . . . 9
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β (πΏβπΈ) β (LSHypβπ)) |
45 | | simpr 486 |
. . . . . . . . 9
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β (πΏβ(πΈ + πΊ)) β (LSHypβπ)) |
46 | 32, 34, 44, 45 | lshpcmp 37500 |
. . . . . . . 8
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β ((πΏβπΈ) β (πΏβ(πΈ + πΊ)) β (πΏβπΈ) = (πΏβ(πΈ + πΊ)))) |
47 | 31, 46 | mpbid 231 |
. . . . . . 7
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β (πΏβπΈ) = (πΏβ(πΈ + πΊ))) |
48 | 17, 47 | eqtr3d 2775 |
. . . . . 6
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β ( β₯ β{π}) = (πΏβ(πΈ + πΊ))) |
49 | 48 | fveq2d 6850 |
. . . . 5
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β ( β₯ β( β₯
β{π})) = ( β₯
β(πΏβ(πΈ + πΊ)))) |
50 | 49 | fveq2d 6850 |
. . . 4
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β ( β₯ β( β₯
β( β₯ β{π}))) = ( β₯ β( β₯
β(πΏβ(πΈ + πΊ))))) |
51 | 15, 50, 48 | 3eqtr3d 2781 |
. . 3
β’ ((π β§ (πΏβ(πΈ + πΊ)) β (LSHypβπ)) β ( β₯ β( β₯
β(πΏβ(πΈ + πΊ)))) = (πΏβ(πΈ + πΊ))) |
52 | 51 | ex 414 |
. 2
β’ (π β ((πΏβ(πΈ + πΊ)) β (LSHypβπ) β ( β₯ β( β₯
β(πΏβ(πΈ + πΊ)))) = (πΏβ(πΈ + πΊ)))) |
53 | 7, 9, 11, 10, 1 | dochoc1 39874 |
. . 3
β’ (π β ( β₯ β( β₯
βπ)) = π) |
54 | | 2fveq3 6851 |
. . . 4
β’ ((πΏβ(πΈ + πΊ)) = π β ( β₯ β( β₯
β(πΏβ(πΈ + πΊ)))) = ( β₯ β( β₯
βπ))) |
55 | | id 22 |
. . . 4
β’ ((πΏβ(πΈ + πΊ)) = π β (πΏβ(πΈ + πΊ)) = π) |
56 | 54, 55 | eqeq12d 2749 |
. . 3
β’ ((πΏβ(πΈ + πΊ)) = π β (( β₯ β( β₯
β(πΏβ(πΈ + πΊ)))) = (πΏβ(πΈ + πΊ)) β ( β₯ β( β₯
βπ)) = π)) |
57 | 53, 56 | syl5ibrcom 247 |
. 2
β’ (π β ((πΏβ(πΈ + πΊ)) = π β ( β₯ β( β₯
β(πΏβ(πΈ + πΊ)))) = (πΏβ(πΈ + πΊ)))) |
58 | 22, 24, 25, 26, 27, 28 | ldualvaddcl 37642 |
. . 3
β’ (π β (πΈ + πΊ) β πΉ) |
59 | 10, 32, 22, 23, 33, 58 | lkrshpor 37619 |
. 2
β’ (π β ((πΏβ(πΈ + πΊ)) β (LSHypβπ) β¨ (πΏβ(πΈ + πΊ)) = π)) |
60 | 52, 57, 59 | mpjaod 859 |
1
β’ (π β ( β₯ β( β₯
β(πΏβ(πΈ + πΊ)))) = (πΏβ(πΈ + πΊ))) |