Step | Hyp | Ref
| Expression |
1 | | simp1 1136 |
. . 3
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β (πΎ β HL β§ π β π»)) |
2 | | dicvaddcl.l |
. . . . . . 7
β’ β€ =
(leβπΎ) |
3 | | dicvaddcl.a |
. . . . . . 7
β’ π΄ = (AtomsβπΎ) |
4 | | dicvaddcl.h |
. . . . . . 7
β’ π» = (LHypβπΎ) |
5 | | dicvaddcl.i |
. . . . . . 7
β’ πΌ = ((DIsoCβπΎ)βπ) |
6 | | dicvaddcl.u |
. . . . . . 7
β’ π = ((DVecHβπΎ)βπ) |
7 | | eqid 2731 |
. . . . . . 7
β’
(Baseβπ) =
(Baseβπ) |
8 | 2, 3, 4, 5, 6, 7 | dicssdvh 39755 |
. . . . . 6
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β (πΌβπ) β (Baseβπ)) |
9 | | eqid 2731 |
. . . . . . . . 9
β’
((LTrnβπΎ)βπ) = ((LTrnβπΎ)βπ) |
10 | | eqid 2731 |
. . . . . . . . 9
β’
((TEndoβπΎ)βπ) = ((TEndoβπΎ)βπ) |
11 | 4, 9, 10, 6, 7 | dvhvbase 39656 |
. . . . . . . 8
β’ ((πΎ β HL β§ π β π») β (Baseβπ) = (((LTrnβπΎ)βπ) Γ ((TEndoβπΎ)βπ))) |
12 | 11 | eqcomd 2737 |
. . . . . . 7
β’ ((πΎ β HL β§ π β π») β (((LTrnβπΎ)βπ) Γ ((TEndoβπΎ)βπ)) = (Baseβπ)) |
13 | 12 | adantr 481 |
. . . . . 6
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β (((LTrnβπΎ)βπ) Γ ((TEndoβπΎ)βπ)) = (Baseβπ)) |
14 | 8, 13 | sseqtrrd 4003 |
. . . . 5
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β (πΌβπ) β (((LTrnβπΎ)βπ) Γ ((TEndoβπΎ)βπ))) |
15 | 14 | 3adant3 1132 |
. . . 4
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β (πΌβπ) β (((LTrnβπΎ)βπ) Γ ((TEndoβπΎ)βπ))) |
16 | | simp3l 1201 |
. . . 4
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β π β (πΌβπ)) |
17 | 15, 16 | sseldd 3963 |
. . 3
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β π β (((LTrnβπΎ)βπ) Γ ((TEndoβπΎ)βπ))) |
18 | | simp3r 1202 |
. . . 4
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β π β (πΌβπ)) |
19 | 15, 18 | sseldd 3963 |
. . 3
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β π β (((LTrnβπΎ)βπ) Γ ((TEndoβπΎ)βπ))) |
20 | | eqid 2731 |
. . . 4
β’
(Scalarβπ) =
(Scalarβπ) |
21 | | dicvaddcl.p |
. . . 4
β’ + =
(+gβπ) |
22 | | eqid 2731 |
. . . 4
β’
(+gβ(Scalarβπ)) =
(+gβ(Scalarβπ)) |
23 | 4, 9, 10, 6, 20, 21, 22 | dvhvadd 39661 |
. . 3
β’ (((πΎ β HL β§ π β π») β§ (π β (((LTrnβπΎ)βπ) Γ ((TEndoβπΎ)βπ)) β§ π β (((LTrnβπΎ)βπ) Γ ((TEndoβπΎ)βπ)))) β (π + π) = β¨((1st βπ) β (1st
βπ)),
((2nd βπ)(+gβ(Scalarβπ))(2nd βπ))β©) |
24 | 1, 17, 19, 23 | syl12anc 835 |
. 2
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β (π + π) = β¨((1st βπ) β (1st
βπ)),
((2nd βπ)(+gβ(Scalarβπ))(2nd βπ))β©) |
25 | 2, 3, 4, 10, 5 | dicelval2nd 39758 |
. . . . . 6
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ π β (πΌβπ)) β (2nd βπ) β ((TEndoβπΎ)βπ)) |
26 | 25 | 3adant3r 1181 |
. . . . 5
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β (2nd βπ) β ((TEndoβπΎ)βπ)) |
27 | 2, 3, 4, 10, 5 | dicelval2nd 39758 |
. . . . . 6
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ π β (πΌβπ)) β (2nd βπ) β ((TEndoβπΎ)βπ)) |
28 | 27 | 3adant3l 1180 |
. . . . 5
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β (2nd βπ) β ((TEndoβπΎ)βπ)) |
29 | | eqid 2731 |
. . . . . . . 8
β’
(ocβπΎ) =
(ocβπΎ) |
30 | 2, 29, 3, 4 | lhpocnel 38587 |
. . . . . . 7
β’ ((πΎ β HL β§ π β π») β (((ocβπΎ)βπ) β π΄ β§ Β¬ ((ocβπΎ)βπ) β€ π)) |
31 | 30 | 3ad2ant1 1133 |
. . . . . 6
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β (((ocβπΎ)βπ) β π΄ β§ Β¬ ((ocβπΎ)βπ) β€ π)) |
32 | | simp2 1137 |
. . . . . 6
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β (π β π΄ β§ Β¬ π β€ π)) |
33 | | eqid 2731 |
. . . . . . 7
β’
(β©π
β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π) = (β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π) |
34 | 2, 3, 4, 9, 33 | ltrniotacl 39148 |
. . . . . 6
β’ (((πΎ β HL β§ π β π») β§ (((ocβπΎ)βπ) β π΄ β§ Β¬ ((ocβπΎ)βπ) β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β (β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π) β ((LTrnβπΎ)βπ)) |
35 | 1, 31, 32, 34 | syl3anc 1371 |
. . . . 5
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β (β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π) β ((LTrnβπΎ)βπ)) |
36 | | eqid 2731 |
. . . . . 6
β’ (π β ((TEndoβπΎ)βπ), π‘ β ((TEndoβπΎ)βπ) β¦ (β β ((LTrnβπΎ)βπ) β¦ ((π ββ) β (π‘ββ)))) = (π β ((TEndoβπΎ)βπ), π‘ β ((TEndoβπΎ)βπ) β¦ (β β ((LTrnβπΎ)βπ) β¦ ((π ββ) β (π‘ββ)))) |
37 | 9, 36 | tendospdi2 39591 |
. . . . 5
β’
(((2nd βπ) β ((TEndoβπΎ)βπ) β§ (2nd βπ) β ((TEndoβπΎ)βπ) β§ (β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π) β ((LTrnβπΎ)βπ)) β (((2nd βπ)(π β ((TEndoβπΎ)βπ), π‘ β ((TEndoβπΎ)βπ) β¦ (β β ((LTrnβπΎ)βπ) β¦ ((π ββ) β (π‘ββ))))(2nd βπ))β(β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π)) = (((2nd βπ)β(β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π)) β ((2nd βπ)β(β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π)))) |
38 | 26, 28, 35, 37 | syl3anc 1371 |
. . . 4
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β (((2nd βπ)(π β ((TEndoβπΎ)βπ), π‘ β ((TEndoβπΎ)βπ) β¦ (β β ((LTrnβπΎ)βπ) β¦ ((π ββ) β (π‘ββ))))(2nd βπ))β(β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π)) = (((2nd βπ)β(β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π)) β ((2nd βπ)β(β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π)))) |
39 | 4, 9, 10, 6, 20, 36, 22 | dvhfplusr 39653 |
. . . . . . 7
β’ ((πΎ β HL β§ π β π») β
(+gβ(Scalarβπ)) = (π β ((TEndoβπΎ)βπ), π‘ β ((TEndoβπΎ)βπ) β¦ (β β ((LTrnβπΎ)βπ) β¦ ((π ββ) β (π‘ββ))))) |
40 | 39 | 3ad2ant1 1133 |
. . . . . 6
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β
(+gβ(Scalarβπ)) = (π β ((TEndoβπΎ)βπ), π‘ β ((TEndoβπΎ)βπ) β¦ (β β ((LTrnβπΎ)βπ) β¦ ((π ββ) β (π‘ββ))))) |
41 | 40 | oveqd 7394 |
. . . . 5
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β ((2nd βπ)(+gβ(Scalarβπ))(2nd βπ)) = ((2nd
βπ)(π β ((TEndoβπΎ)βπ), π‘ β ((TEndoβπΎ)βπ) β¦ (β β ((LTrnβπΎ)βπ) β¦ ((π ββ) β (π‘ββ))))(2nd βπ))) |
42 | 41 | fveq1d 6864 |
. . . 4
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β (((2nd βπ)(+gβ(Scalarβπ))(2nd βπ))β(β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π)) = (((2nd βπ)(π β ((TEndoβπΎ)βπ), π‘ β ((TEndoβπΎ)βπ) β¦ (β β ((LTrnβπΎ)βπ) β¦ ((π ββ) β (π‘ββ))))(2nd βπ))β(β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π))) |
43 | | eqid 2731 |
. . . . . . 7
β’
((ocβπΎ)βπ) = ((ocβπΎ)βπ) |
44 | 2, 3, 4, 43, 9, 5 | dicelval1sta 39756 |
. . . . . 6
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ π β (πΌβπ)) β (1st βπ) = ((2nd
βπ)β(β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π))) |
45 | 44 | 3adant3r 1181 |
. . . . 5
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β (1st βπ) = ((2nd
βπ)β(β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π))) |
46 | 2, 3, 4, 43, 9, 5 | dicelval1sta 39756 |
. . . . . 6
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ π β (πΌβπ)) β (1st βπ) = ((2nd
βπ)β(β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π))) |
47 | 46 | 3adant3l 1180 |
. . . . 5
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β (1st βπ) = ((2nd
βπ)β(β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π))) |
48 | 45, 47 | coeq12d 5840 |
. . . 4
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β ((1st βπ) β (1st
βπ)) =
(((2nd βπ)β(β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π)) β ((2nd βπ)β(β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π)))) |
49 | 38, 42, 48 | 3eqtr4rd 2782 |
. . 3
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β ((1st βπ) β (1st
βπ)) =
(((2nd βπ)(+gβ(Scalarβπ))(2nd βπ))β(β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π))) |
50 | 4, 9, 10, 36 | tendoplcl 39350 |
. . . . 5
β’ (((πΎ β HL β§ π β π») β§ (2nd βπ) β ((TEndoβπΎ)βπ) β§ (2nd βπ) β ((TEndoβπΎ)βπ)) β ((2nd βπ)(π β ((TEndoβπΎ)βπ), π‘ β ((TEndoβπΎ)βπ) β¦ (β β ((LTrnβπΎ)βπ) β¦ ((π ββ) β (π‘ββ))))(2nd βπ)) β ((TEndoβπΎ)βπ)) |
51 | 1, 26, 28, 50 | syl3anc 1371 |
. . . 4
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β ((2nd βπ)(π β ((TEndoβπΎ)βπ), π‘ β ((TEndoβπΎ)βπ) β¦ (β β ((LTrnβπΎ)βπ) β¦ ((π ββ) β (π‘ββ))))(2nd βπ)) β ((TEndoβπΎ)βπ)) |
52 | 41, 51 | eqeltrd 2832 |
. . 3
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β ((2nd βπ)(+gβ(Scalarβπ))(2nd βπ)) β ((TEndoβπΎ)βπ)) |
53 | | fvex 6875 |
. . . . . 6
β’
(1st βπ) β V |
54 | | fvex 6875 |
. . . . . 6
β’
(1st βπ) β V |
55 | 53, 54 | coex 7887 |
. . . . 5
β’
((1st βπ) β (1st βπ)) β V |
56 | | ovex 7410 |
. . . . 5
β’
((2nd βπ)(+gβ(Scalarβπ))(2nd βπ)) β V |
57 | 2, 3, 4, 43, 9, 10, 5, 55, 56 | dicopelval 39746 |
. . . 4
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β (β¨((1st
βπ) β
(1st βπ)),
((2nd βπ)(+gβ(Scalarβπ))(2nd βπ))β© β (πΌβπ) β (((1st βπ) β (1st
βπ)) =
(((2nd βπ)(+gβ(Scalarβπ))(2nd βπ))β(β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π)) β§ ((2nd βπ)(+gβ(Scalarβπ))(2nd βπ)) β ((TEndoβπΎ)βπ)))) |
58 | 57 | 3adant3 1132 |
. . 3
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β (β¨((1st
βπ) β
(1st βπ)),
((2nd βπ)(+gβ(Scalarβπ))(2nd βπ))β© β (πΌβπ) β (((1st βπ) β (1st
βπ)) =
(((2nd βπ)(+gβ(Scalarβπ))(2nd βπ))β(β©π β ((LTrnβπΎ)βπ)(πβ((ocβπΎ)βπ)) = π)) β§ ((2nd βπ)(+gβ(Scalarβπ))(2nd βπ)) β ((TEndoβπΎ)βπ)))) |
59 | 49, 52, 58 | mpbir2and 711 |
. 2
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β β¨((1st
βπ) β
(1st βπ)),
((2nd βπ)(+gβ(Scalarβπ))(2nd βπ))β© β (πΌβπ)) |
60 | 24, 59 | eqeltrd 2832 |
1
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β (πΌβπ) β§ π β (πΌβπ))) β (π + π) β (πΌβπ)) |