Step | Hyp | Ref
| Expression |
1 | | subcidcl.j |
. . . 4
β’ (π β π½ β (SubcatβπΆ)) |
2 | | eqid 2732 |
. . . . 5
β’
(Homf βπΆ) = (Homf βπΆ) |
3 | | eqid 2732 |
. . . . 5
β’
(IdβπΆ) =
(IdβπΆ) |
4 | | subccocl.o |
. . . . 5
β’ Β· =
(compβπΆ) |
5 | | subcrcl 17759 |
. . . . . 6
β’ (π½ β (SubcatβπΆ) β πΆ β Cat) |
6 | 1, 5 | syl 17 |
. . . . 5
β’ (π β πΆ β Cat) |
7 | | subcidcl.2 |
. . . . 5
β’ (π β π½ Fn (π Γ π)) |
8 | 2, 3, 4, 6, 7 | issubc2 17782 |
. . . 4
β’ (π β (π½ β (SubcatβπΆ) β (π½ βcat
(Homf βπΆ) β§ βπ₯ β π (((IdβπΆ)βπ₯) β (π₯π½π₯) β§ βπ¦ β π βπ§ β π βπ β (π₯π½π¦)βπ β (π¦π½π§)(π(β¨π₯, π¦β© Β· π§)π) β (π₯π½π§))))) |
9 | 1, 8 | mpbid 231 |
. . 3
β’ (π β (π½ βcat
(Homf βπΆ) β§ βπ₯ β π (((IdβπΆ)βπ₯) β (π₯π½π₯) β§ βπ¦ β π βπ§ β π βπ β (π₯π½π¦)βπ β (π¦π½π§)(π(β¨π₯, π¦β© Β· π§)π) β (π₯π½π§)))) |
10 | 9 | simprd 496 |
. 2
β’ (π β βπ₯ β π (((IdβπΆ)βπ₯) β (π₯π½π₯) β§ βπ¦ β π βπ§ β π βπ β (π₯π½π¦)βπ β (π¦π½π§)(π(β¨π₯, π¦β© Β· π§)π) β (π₯π½π§))) |
11 | | subcidcl.x |
. . 3
β’ (π β π β π) |
12 | | subccocl.y |
. . . . . 6
β’ (π β π β π) |
13 | 12 | adantr 481 |
. . . . 5
β’ ((π β§ π₯ = π) β π β π) |
14 | | subccocl.z |
. . . . . . 7
β’ (π β π β π) |
15 | 14 | ad2antrr 724 |
. . . . . 6
β’ (((π β§ π₯ = π) β§ π¦ = π) β π β π) |
16 | | subccocl.f |
. . . . . . . . 9
β’ (π β πΉ β (ππ½π)) |
17 | 16 | ad3antrrr 728 |
. . . . . . . 8
β’ ((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β πΉ β (ππ½π)) |
18 | | simpllr 774 |
. . . . . . . . 9
β’ ((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β π₯ = π) |
19 | | simplr 767 |
. . . . . . . . 9
β’ ((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β π¦ = π) |
20 | 18, 19 | oveq12d 7423 |
. . . . . . . 8
β’ ((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β (π₯π½π¦) = (ππ½π)) |
21 | 17, 20 | eleqtrrd 2836 |
. . . . . . 7
β’ ((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β πΉ β (π₯π½π¦)) |
22 | | subccocl.g |
. . . . . . . . . 10
β’ (π β πΊ β (ππ½π)) |
23 | 22 | ad4antr 730 |
. . . . . . . . 9
β’
(((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β§ π = πΉ) β πΊ β (ππ½π)) |
24 | | simpllr 774 |
. . . . . . . . . 10
β’
(((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β§ π = πΉ) β π¦ = π) |
25 | | simplr 767 |
. . . . . . . . . 10
β’
(((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β§ π = πΉ) β π§ = π) |
26 | 24, 25 | oveq12d 7423 |
. . . . . . . . 9
β’
(((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β§ π = πΉ) β (π¦π½π§) = (ππ½π)) |
27 | 23, 26 | eleqtrrd 2836 |
. . . . . . . 8
β’
(((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β§ π = πΉ) β πΊ β (π¦π½π§)) |
28 | | simp-5r 784 |
. . . . . . . . . . . 12
β’
((((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β§ π = πΉ) β§ π = πΊ) β π₯ = π) |
29 | | simp-4r 782 |
. . . . . . . . . . . 12
β’
((((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β§ π = πΉ) β§ π = πΊ) β π¦ = π) |
30 | 28, 29 | opeq12d 4880 |
. . . . . . . . . . 11
β’
((((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β§ π = πΉ) β§ π = πΊ) β β¨π₯, π¦β© = β¨π, πβ©) |
31 | | simpllr 774 |
. . . . . . . . . . 11
β’
((((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β§ π = πΉ) β§ π = πΊ) β π§ = π) |
32 | 30, 31 | oveq12d 7423 |
. . . . . . . . . 10
β’
((((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β§ π = πΉ) β§ π = πΊ) β (β¨π₯, π¦β© Β· π§) = (β¨π, πβ© Β· π)) |
33 | | simpr 485 |
. . . . . . . . . 10
β’
((((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β§ π = πΉ) β§ π = πΊ) β π = πΊ) |
34 | | simplr 767 |
. . . . . . . . . 10
β’
((((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β§ π = πΉ) β§ π = πΊ) β π = πΉ) |
35 | 32, 33, 34 | oveq123d 7426 |
. . . . . . . . 9
β’
((((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β§ π = πΉ) β§ π = πΊ) β (π(β¨π₯, π¦β© Β· π§)π) = (πΊ(β¨π, πβ© Β· π)πΉ)) |
36 | 28, 31 | oveq12d 7423 |
. . . . . . . . 9
β’
((((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β§ π = πΉ) β§ π = πΊ) β (π₯π½π§) = (ππ½π)) |
37 | 35, 36 | eleq12d 2827 |
. . . . . . . 8
β’
((((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β§ π = πΉ) β§ π = πΊ) β ((π(β¨π₯, π¦β© Β· π§)π) β (π₯π½π§) β (πΊ(β¨π, πβ© Β· π)πΉ) β (ππ½π))) |
38 | 27, 37 | rspcdv 3604 |
. . . . . . 7
β’
(((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β§ π = πΉ) β (βπ β (π¦π½π§)(π(β¨π₯, π¦β© Β· π§)π) β (π₯π½π§) β (πΊ(β¨π, πβ© Β· π)πΉ) β (ππ½π))) |
39 | 21, 38 | rspcimdv 3602 |
. . . . . 6
β’ ((((π β§ π₯ = π) β§ π¦ = π) β§ π§ = π) β (βπ β (π₯π½π¦)βπ β (π¦π½π§)(π(β¨π₯, π¦β© Β· π§)π) β (π₯π½π§) β (πΊ(β¨π, πβ© Β· π)πΉ) β (ππ½π))) |
40 | 15, 39 | rspcimdv 3602 |
. . . . 5
β’ (((π β§ π₯ = π) β§ π¦ = π) β (βπ§ β π βπ β (π₯π½π¦)βπ β (π¦π½π§)(π(β¨π₯, π¦β© Β· π§)π) β (π₯π½π§) β (πΊ(β¨π, πβ© Β· π)πΉ) β (ππ½π))) |
41 | 13, 40 | rspcimdv 3602 |
. . . 4
β’ ((π β§ π₯ = π) β (βπ¦ β π βπ§ β π βπ β (π₯π½π¦)βπ β (π¦π½π§)(π(β¨π₯, π¦β© Β· π§)π) β (π₯π½π§) β (πΊ(β¨π, πβ© Β· π)πΉ) β (ππ½π))) |
42 | 41 | adantld 491 |
. . 3
β’ ((π β§ π₯ = π) β ((((IdβπΆ)βπ₯) β (π₯π½π₯) β§ βπ¦ β π βπ§ β π βπ β (π₯π½π¦)βπ β (π¦π½π§)(π(β¨π₯, π¦β© Β· π§)π) β (π₯π½π§)) β (πΊ(β¨π, πβ© Β· π)πΉ) β (ππ½π))) |
43 | 11, 42 | rspcimdv 3602 |
. 2
β’ (π β (βπ₯ β π (((IdβπΆ)βπ₯) β (π₯π½π₯) β§ βπ¦ β π βπ§ β π βπ β (π₯π½π¦)βπ β (π¦π½π§)(π(β¨π₯, π¦β© Β· π§)π) β (π₯π½π§)) β (πΊ(β¨π, πβ© Β· π)πΉ) β (ππ½π))) |
44 | 10, 43 | mpd 15 |
1
β’ (π β (πΊ(β¨π, πβ© Β· π)πΉ) β (ππ½π)) |