Metamath Proof Explorer


Theorem pwsle

Description: Ordering in a structure power. (Contributed by Mario Carneiro, 16-Aug-2015)

Ref Expression
Hypotheses pwsle.y Y = R 𝑠 I
pwsle.v B = Base Y
pwsle.o O = R
pwsle.l ˙ = Y
Assertion pwsle R V I W ˙ = r O B × B

Proof

Step Hyp Ref Expression
1 pwsle.y Y = R 𝑠 I
2 pwsle.v B = Base Y
3 pwsle.o O = R
4 pwsle.l ˙ = Y
5 vex f V
6 vex g V
7 5 6 prss f B g B f g B
8 eqid Scalar R = Scalar R
9 1 8 pwsval R V I W Y = Scalar R 𝑠 I × R
10 9 fveq2d R V I W Base Y = Base Scalar R 𝑠 I × R
11 2 10 eqtrid R V I W B = Base Scalar R 𝑠 I × R
12 11 sseq2d R V I W f g B f g Base Scalar R 𝑠 I × R
13 7 12 syl5bb R V I W f B g B f g Base Scalar R 𝑠 I × R
14 13 anbi1d R V I W f B g B x I f x I × R x g x f g Base Scalar R 𝑠 I × R x I f x I × R x g x
15 fvconst2g R V x I I × R x = R
16 15 ad4ant14 R V I W f B g B x I I × R x = R
17 16 fveq2d R V I W f B g B x I I × R x = R
18 17 3 eqtr4di R V I W f B g B x I I × R x = O
19 18 breqd R V I W f B g B x I f x I × R x g x f x O g x
20 19 ralbidva R V I W f B g B x I f x I × R x g x x I f x O g x
21 eqid Base R = Base R
22 simpll R V I W f B g B R V
23 simplr R V I W f B g B I W
24 simprl R V I W f B g B f B
25 1 21 2 22 23 24 pwselbas R V I W f B g B f : I Base R
26 25 ffnd R V I W f B g B f Fn I
27 simprr R V I W f B g B g B
28 1 21 2 22 23 27 pwselbas R V I W f B g B g : I Base R
29 28 ffnd R V I W f B g B g Fn I
30 inidm I I = I
31 eqidd R V I W f B g B x I f x = f x
32 eqidd R V I W f B g B x I g x = g x
33 26 29 24 27 30 31 32 ofrfvalg R V I W f B g B f O f g x I f x O g x
34 20 33 bitr4d R V I W f B g B x I f x I × R x g x f O f g
35 34 pm5.32da R V I W f B g B x I f x I × R x g x f B g B f O f g
36 brinxp2 f r O B × B g f B g B f O f g
37 35 36 bitr4di R V I W f B g B x I f x I × R x g x f r O B × B g
38 14 37 bitr3d R V I W f g Base Scalar R 𝑠 I × R x I f x I × R x g x f r O B × B g
39 38 opabbidv R V I W f g | f g Base Scalar R 𝑠 I × R x I f x I × R x g x = f g | f r O B × B g
40 9 fveq2d R V I W Y = Scalar R 𝑠 I × R
41 4 40 eqtrid R V I W ˙ = Scalar R 𝑠 I × R
42 eqid Scalar R 𝑠 I × R = Scalar R 𝑠 I × R
43 fvexd R V I W Scalar R V
44 simpr R V I W I W
45 snex R V
46 xpexg I W R V I × R V
47 44 45 46 sylancl R V I W I × R V
48 eqid Base Scalar R 𝑠 I × R = Base Scalar R 𝑠 I × R
49 snnzg R V R
50 49 adantr R V I W R
51 dmxp R dom I × R = I
52 50 51 syl R V I W dom I × R = I
53 eqid Scalar R 𝑠 I × R = Scalar R 𝑠 I × R
54 42 43 47 48 52 53 prdsle R V I W Scalar R 𝑠 I × R = f g | f g Base Scalar R 𝑠 I × R x I f x I × R x g x
55 41 54 eqtrd R V I W ˙ = f g | f g Base Scalar R 𝑠 I × R x I f x I × R x g x
56 relinxp Rel r O B × B
57 56 a1i R V I W Rel r O B × B
58 dfrel4v Rel r O B × B r O B × B = f g | f r O B × B g
59 57 58 sylib R V I W r O B × B = f g | f r O B × B g
60 39 55 59 3eqtr4d R V I W ˙ = r O B × B