Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ghomco Structured version   Visualization version   GIF version

Theorem ghomco 37605
Description: The composition of two group homomorphisms is a group homomorphism. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
Assertion
Ref Expression
ghomco (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) ∧ (𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾))) → (𝑇𝑆) ∈ (𝐺 GrpOpHom 𝐾))

Proof of Theorem ghomco
Dummy variables 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fco 6744 . . . . . . 7 ((𝑇:ran 𝐻⟶ran 𝐾𝑆:ran 𝐺⟶ran 𝐻) → (𝑇𝑆):ran 𝐺⟶ran 𝐾)
21ancoms 457 . . . . . 6 ((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) → (𝑇𝑆):ran 𝐺⟶ran 𝐾)
32ad2ant2r 745 . . . . 5 (((𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦))) ∧ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣)))) → (𝑇𝑆):ran 𝐺⟶ran 𝐾)
43a1i 11 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → (((𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦))) ∧ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣)))) → (𝑇𝑆):ran 𝐺⟶ran 𝐾))
5 ffvelcdm 7087 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆:ran 𝐺⟶ran 𝐻𝑥 ∈ ran 𝐺) → (𝑆𝑥) ∈ ran 𝐻)
6 ffvelcdm 7087 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆:ran 𝐺⟶ran 𝐻𝑦 ∈ ran 𝐺) → (𝑆𝑦) ∈ ran 𝐻)
75, 6anim12dan 617 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑆𝑥) ∈ ran 𝐻 ∧ (𝑆𝑦) ∈ ran 𝐻))
8 fveq2 6893 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑢 = (𝑆𝑥) → (𝑇𝑢) = (𝑇‘(𝑆𝑥)))
98oveq1d 7431 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = (𝑆𝑥) → ((𝑇𝑢)𝐾(𝑇𝑣)) = ((𝑇‘(𝑆𝑥))𝐾(𝑇𝑣)))
10 fvoveq1 7439 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = (𝑆𝑥) → (𝑇‘(𝑢𝐻𝑣)) = (𝑇‘((𝑆𝑥)𝐻𝑣)))
119, 10eqeq12d 2742 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = (𝑆𝑥) → (((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣)) ↔ ((𝑇‘(𝑆𝑥))𝐾(𝑇𝑣)) = (𝑇‘((𝑆𝑥)𝐻𝑣))))
12 fveq2 6893 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = (𝑆𝑦) → (𝑇𝑣) = (𝑇‘(𝑆𝑦)))
1312oveq2d 7432 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = (𝑆𝑦) → ((𝑇‘(𝑆𝑥))𝐾(𝑇𝑣)) = ((𝑇‘(𝑆𝑥))𝐾(𝑇‘(𝑆𝑦))))
14 oveq2 7424 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = (𝑆𝑦) → ((𝑆𝑥)𝐻𝑣) = ((𝑆𝑥)𝐻(𝑆𝑦)))
1514fveq2d 6897 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = (𝑆𝑦) → (𝑇‘((𝑆𝑥)𝐻𝑣)) = (𝑇‘((𝑆𝑥)𝐻(𝑆𝑦))))
1613, 15eqeq12d 2742 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = (𝑆𝑦) → (((𝑇‘(𝑆𝑥))𝐾(𝑇𝑣)) = (𝑇‘((𝑆𝑥)𝐻𝑣)) ↔ ((𝑇‘(𝑆𝑥))𝐾(𝑇‘(𝑆𝑦))) = (𝑇‘((𝑆𝑥)𝐻(𝑆𝑦)))))
1711, 16rspc2va 3619 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑆𝑥) ∈ ran 𝐻 ∧ (𝑆𝑦) ∈ ran 𝐻) ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣))) → ((𝑇‘(𝑆𝑥))𝐾(𝑇‘(𝑆𝑦))) = (𝑇‘((𝑆𝑥)𝐻(𝑆𝑦))))
187, 17sylan 578 . . . . . . . . . . . . . . . . . . . 20 (((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣))) → ((𝑇‘(𝑆𝑥))𝐾(𝑇‘(𝑆𝑦))) = (𝑇‘((𝑆𝑥)𝐻(𝑆𝑦))))
1918an32s 650 . . . . . . . . . . . . . . . . . . 19 (((𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑇‘(𝑆𝑥))𝐾(𝑇‘(𝑆𝑦))) = (𝑇‘((𝑆𝑥)𝐻(𝑆𝑦))))
2019adantllr 717 . . . . . . . . . . . . . . . . . 18 ((((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑇‘(𝑆𝑥))𝐾(𝑇‘(𝑆𝑦))) = (𝑇‘((𝑆𝑥)𝐻(𝑆𝑦))))
2120adantllr 717 . . . . . . . . . . . . . . . . 17 (((((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑇‘(𝑆𝑥))𝐾(𝑇‘(𝑆𝑦))) = (𝑇‘((𝑆𝑥)𝐻(𝑆𝑦))))
22 fveq2 6893 . . . . . . . . . . . . . . . . 17 (((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → (𝑇‘((𝑆𝑥)𝐻(𝑆𝑦))) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
2321, 22sylan9eq 2786 . . . . . . . . . . . . . . . 16 ((((((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) ∧ ((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦))) → ((𝑇‘(𝑆𝑥))𝐾(𝑇‘(𝑆𝑦))) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
2423anasss 465 . . . . . . . . . . . . . . 15 (((((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ ((𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺) ∧ ((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦)))) → ((𝑇‘(𝑆𝑥))𝐾(𝑇‘(𝑆𝑦))) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
25 fvco3 6993 . . . . . . . . . . . . . . . . . . 19 ((𝑆:ran 𝐺⟶ran 𝐻𝑥 ∈ ran 𝐺) → ((𝑇𝑆)‘𝑥) = (𝑇‘(𝑆𝑥)))
2625ad2ant2r 745 . . . . . . . . . . . . . . . . . 18 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑇𝑆)‘𝑥) = (𝑇‘(𝑆𝑥)))
27 fvco3 6993 . . . . . . . . . . . . . . . . . . 19 ((𝑆:ran 𝐺⟶ran 𝐻𝑦 ∈ ran 𝐺) → ((𝑇𝑆)‘𝑦) = (𝑇‘(𝑆𝑦)))
2827ad2ant2rl 747 . . . . . . . . . . . . . . . . . 18 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑇𝑆)‘𝑦) = (𝑇‘(𝑆𝑦)))
2926, 28oveq12d 7434 . . . . . . . . . . . . . . . . 17 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → (((𝑇𝑆)‘𝑥)𝐾((𝑇𝑆)‘𝑦)) = ((𝑇‘(𝑆𝑥))𝐾(𝑇‘(𝑆𝑦))))
3029adantlr 713 . . . . . . . . . . . . . . . 16 ((((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → (((𝑇𝑆)‘𝑥)𝐾((𝑇𝑆)‘𝑦)) = ((𝑇‘(𝑆𝑥))𝐾(𝑇‘(𝑆𝑦))))
3130ad2ant2r 745 . . . . . . . . . . . . . . 15 (((((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ ((𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺) ∧ ((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦)))) → (((𝑇𝑆)‘𝑥)𝐾((𝑇𝑆)‘𝑦)) = ((𝑇‘(𝑆𝑥))𝐾(𝑇‘(𝑆𝑦))))
32 eqid 2726 . . . . . . . . . . . . . . . . . . . 20 ran 𝐺 = ran 𝐺
3332grpocl 30430 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺) → (𝑥𝐺𝑦) ∈ ran 𝐺)
34333expb 1117 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → (𝑥𝐺𝑦) ∈ ran 𝐺)
35 fvco3 6993 . . . . . . . . . . . . . . . . . . 19 ((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑥𝐺𝑦) ∈ ran 𝐺) → ((𝑇𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
3635adantlr 713 . . . . . . . . . . . . . . . . . 18 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥𝐺𝑦) ∈ ran 𝐺) → ((𝑇𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
3734, 36sylan2 591 . . . . . . . . . . . . . . . . 17 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝐺 ∈ GrpOp ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺))) → ((𝑇𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
3837anassrs 466 . . . . . . . . . . . . . . . 16 ((((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑇𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
3938ad2ant2r 745 . . . . . . . . . . . . . . 15 (((((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ ((𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺) ∧ ((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦)))) → ((𝑇𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
4024, 31, 393eqtr4d 2776 . . . . . . . . . . . . . 14 (((((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ ((𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺) ∧ ((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦)))) → (((𝑇𝑆)‘𝑥)𝐾((𝑇𝑆)‘𝑦)) = ((𝑇𝑆)‘(𝑥𝐺𝑦)))
4140expr 455 . . . . . . . . . . . . 13 (((((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → (((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → (((𝑇𝑆)‘𝑥)𝐾((𝑇𝑆)‘𝑦)) = ((𝑇𝑆)‘(𝑥𝐺𝑦))))
4241ralimdvva 3195 . . . . . . . . . . . 12 ((((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣))) → (∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(((𝑇𝑆)‘𝑥)𝐾((𝑇𝑆)‘𝑦)) = ((𝑇𝑆)‘(𝑥𝐺𝑦))))
4342an32s 650 . . . . . . . . . . 11 ((((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ 𝐺 ∈ GrpOp) → (∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(((𝑇𝑆)‘𝑥)𝐾((𝑇𝑆)‘𝑦)) = ((𝑇𝑆)‘(𝑥𝐺𝑦))))
4443ex 411 . . . . . . . . . 10 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣))) → (𝐺 ∈ GrpOp → (∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(((𝑇𝑆)‘𝑥)𝐾((𝑇𝑆)‘𝑦)) = ((𝑇𝑆)‘(𝑥𝐺𝑦)))))
4544com23 86 . . . . . . . . 9 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣))) → (∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → (𝐺 ∈ GrpOp → ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(((𝑇𝑆)‘𝑥)𝐾((𝑇𝑆)‘𝑦)) = ((𝑇𝑆)‘(𝑥𝐺𝑦)))))
4645anasss 465 . . . . . . . 8 ((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣)))) → (∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → (𝐺 ∈ GrpOp → ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(((𝑇𝑆)‘𝑥)𝐾((𝑇𝑆)‘𝑦)) = ((𝑇𝑆)‘(𝑥𝐺𝑦)))))
4746imp 405 . . . . . . 7 (((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣)))) ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦))) → (𝐺 ∈ GrpOp → ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(((𝑇𝑆)‘𝑥)𝐾((𝑇𝑆)‘𝑦)) = ((𝑇𝑆)‘(𝑥𝐺𝑦))))
4847an32s 650 . . . . . 6 (((𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦))) ∧ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣)))) → (𝐺 ∈ GrpOp → ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(((𝑇𝑆)‘𝑥)𝐾((𝑇𝑆)‘𝑦)) = ((𝑇𝑆)‘(𝑥𝐺𝑦))))
4948com12 32 . . . . 5 (𝐺 ∈ GrpOp → (((𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦))) ∧ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣)))) → ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(((𝑇𝑆)‘𝑥)𝐾((𝑇𝑆)‘𝑦)) = ((𝑇𝑆)‘(𝑥𝐺𝑦))))
50493ad2ant1 1130 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → (((𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦))) ∧ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣)))) → ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(((𝑇𝑆)‘𝑥)𝐾((𝑇𝑆)‘𝑦)) = ((𝑇𝑆)‘(𝑥𝐺𝑦))))
514, 50jcad 511 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → (((𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦))) ∧ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣)))) → ((𝑇𝑆):ran 𝐺⟶ran 𝐾 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(((𝑇𝑆)‘𝑥)𝐾((𝑇𝑆)‘𝑦)) = ((𝑇𝑆)‘(𝑥𝐺𝑦)))))
52 eqid 2726 . . . . . 6 ran 𝐻 = ran 𝐻
5332, 52elghomOLD 37601 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝑆 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦)))))
54533adant3 1129 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → (𝑆 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦)))))
55 eqid 2726 . . . . . 6 ran 𝐾 = ran 𝐾
5652, 55elghomOLD 37601 . . . . 5 ((𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → (𝑇 ∈ (𝐻 GrpOpHom 𝐾) ↔ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣)))))
57563adant1 1127 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → (𝑇 ∈ (𝐻 GrpOpHom 𝐾) ↔ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣)))))
5854, 57anbi12d 630 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → ((𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾)) ↔ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦))) ∧ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣))))))
5932, 55elghomOLD 37601 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → ((𝑇𝑆) ∈ (𝐺 GrpOpHom 𝐾) ↔ ((𝑇𝑆):ran 𝐺⟶ran 𝐾 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(((𝑇𝑆)‘𝑥)𝐾((𝑇𝑆)‘𝑦)) = ((𝑇𝑆)‘(𝑥𝐺𝑦)))))
60593adant2 1128 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → ((𝑇𝑆) ∈ (𝐺 GrpOpHom 𝐾) ↔ ((𝑇𝑆):ran 𝐺⟶ran 𝐾 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(((𝑇𝑆)‘𝑥)𝐾((𝑇𝑆)‘𝑦)) = ((𝑇𝑆)‘(𝑥𝐺𝑦)))))
6151, 58, 603imtr4d 293 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → ((𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾)) → (𝑇𝑆) ∈ (𝐺 GrpOpHom 𝐾)))
6261imp 405 1 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) ∧ (𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾))) → (𝑇𝑆) ∈ (𝐺 GrpOpHom 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  ran crn 5675  ccom 5678  wf 6542  cfv 6546  (class class class)co 7416  GrpOpcgr 30419   GrpOpHom cghomOLD 37597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-grpo 30423  df-ghomOLD 37598
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator