Metamath Proof Explorer


Theorem fpwwe2lem4

Description: Lemma for fpwwe2 . (Contributed by Mario Carneiro, 15-May-2015) (Revised by AV, 20-Jul-2024)

Ref Expression
Hypotheses fpwwe2.1 W = x r | x A r x × x r We x y x [˙ r -1 y / u]˙ u F r u × u = y
fpwwe2.2 φ A V
fpwwe2.3 φ x A r x × x r We x x F r A
Assertion fpwwe2lem4 φ X A R X × X R We X X F R A

Proof

Step Hyp Ref Expression
1 fpwwe2.1 W = x r | x A r x × x r We x y x [˙ r -1 y / u]˙ u F r u × u = y
2 fpwwe2.2 φ A V
3 fpwwe2.3 φ x A r x × x r We x x F r A
4 2 adantr φ X A R X × X R We X A V
5 simpr1 φ X A R X × X R We X X A
6 4 5 ssexd φ X A R X × X R We X X V
7 6 6 xpexd φ X A R X × X R We X X × X V
8 simpr2 φ X A R X × X R We X R X × X
9 7 8 ssexd φ X A R X × X R We X R V
10 6 9 jca φ X A R X × X R We X X V R V
11 sseq1 x = X x A X A
12 xpeq12 x = X x = X x × x = X × X
13 12 anidms x = X x × x = X × X
14 13 sseq2d x = X r x × x r X × X
15 weeq2 x = X r We x r We X
16 11 14 15 3anbi123d x = X x A r x × x r We x X A r X × X r We X
17 16 anbi2d x = X φ x A r x × x r We x φ X A r X × X r We X
18 oveq1 x = X x F r = X F r
19 18 eleq1d x = X x F r A X F r A
20 17 19 imbi12d x = X φ x A r x × x r We x x F r A φ X A r X × X r We X X F r A
21 sseq1 r = R r X × X R X × X
22 weeq1 r = R r We X R We X
23 21 22 3anbi23d r = R X A r X × X r We X X A R X × X R We X
24 23 anbi2d r = R φ X A r X × X r We X φ X A R X × X R We X
25 oveq2 r = R X F r = X F R
26 25 eleq1d r = R X F r A X F R A
27 24 26 imbi12d r = R φ X A r X × X r We X X F r A φ X A R X × X R We X X F R A
28 20 27 3 vtocl2g X V R V φ X A R X × X R We X X F R A
29 10 28 mpcom φ X A R X × X R We X X F R A