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Theorem ghomco 35329
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 6505 . . . . . . 7 ((𝑇:ran 𝐻⟶ran 𝐾𝑆:ran 𝐺⟶ran 𝐻) → (𝑇𝑆):ran 𝐺⟶ran 𝐾)
21ancoms 462 . . . . . 6 ((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) → (𝑇𝑆):ran 𝐺⟶ran 𝐾)
32ad2ant2r 746 . . . . 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 ffvelrn 6826 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆:ran 𝐺⟶ran 𝐻𝑥 ∈ ran 𝐺) → (𝑆𝑥) ∈ ran 𝐻)
6 ffvelrn 6826 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆:ran 𝐺⟶ran 𝐻𝑦 ∈ ran 𝐺) → (𝑆𝑦) ∈ ran 𝐻)
75, 6anim12dan 621 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑆𝑥) ∈ ran 𝐻 ∧ (𝑆𝑦) ∈ ran 𝐻))
8 fveq2 6645 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑢 = (𝑆𝑥) → (𝑇𝑢) = (𝑇‘(𝑆𝑥)))
98oveq1d 7150 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = (𝑆𝑥) → ((𝑇𝑢)𝐾(𝑇𝑣)) = ((𝑇‘(𝑆𝑥))𝐾(𝑇𝑣)))
10 fvoveq1 7158 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = (𝑆𝑥) → (𝑇‘(𝑢𝐻𝑣)) = (𝑇‘((𝑆𝑥)𝐻𝑣)))
119, 10eqeq12d 2814 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = (𝑆𝑥) → (((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣)) ↔ ((𝑇‘(𝑆𝑥))𝐾(𝑇𝑣)) = (𝑇‘((𝑆𝑥)𝐻𝑣))))
12 fveq2 6645 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = (𝑆𝑦) → (𝑇𝑣) = (𝑇‘(𝑆𝑦)))
1312oveq2d 7151 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = (𝑆𝑦) → ((𝑇‘(𝑆𝑥))𝐾(𝑇𝑣)) = ((𝑇‘(𝑆𝑥))𝐾(𝑇‘(𝑆𝑦))))
14 oveq2 7143 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = (𝑆𝑦) → ((𝑆𝑥)𝐻𝑣) = ((𝑆𝑥)𝐻(𝑆𝑦)))
1514fveq2d 6649 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = (𝑆𝑦) → (𝑇‘((𝑆𝑥)𝐻𝑣)) = (𝑇‘((𝑆𝑥)𝐻(𝑆𝑦))))
1613, 15eqeq12d 2814 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = (𝑆𝑦) → (((𝑇‘(𝑆𝑥))𝐾(𝑇𝑣)) = (𝑇‘((𝑆𝑥)𝐻𝑣)) ↔ ((𝑇‘(𝑆𝑥))𝐾(𝑇‘(𝑆𝑦))) = (𝑇‘((𝑆𝑥)𝐻(𝑆𝑦)))))
1711, 16rspc2va 3582 . . . . . . . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . . . . . . . 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 651 . . . . . . . . . . . . . . . . . . 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 718 . . . . . . . . . . . . . . . . . 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 718 . . . . . . . . . . . . . . . . 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 6645 . . . . . . . . . . . . . . . . 17 (((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → (𝑇‘((𝑆𝑥)𝐻(𝑆𝑦))) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
2321, 22sylan9eq 2853 . . . . . . . . . . . . . . . 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 470 . . . . . . . . . . . . . . 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 6737 . . . . . . . . . . . . . . . . . . 19 ((𝑆:ran 𝐺⟶ran 𝐻𝑥 ∈ ran 𝐺) → ((𝑇𝑆)‘𝑥) = (𝑇‘(𝑆𝑥)))
2625ad2ant2r 746 . . . . . . . . . . . . . . . . . 18 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑇𝑆)‘𝑥) = (𝑇‘(𝑆𝑥)))
27 fvco3 6737 . . . . . . . . . . . . . . . . . . 19 ((𝑆:ran 𝐺⟶ran 𝐻𝑦 ∈ ran 𝐺) → ((𝑇𝑆)‘𝑦) = (𝑇‘(𝑆𝑦)))
2827ad2ant2rl 748 . . . . . . . . . . . . . . . . . 18 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑇𝑆)‘𝑦) = (𝑇‘(𝑆𝑦)))
2926, 28oveq12d 7153 . . . . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . . . . 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 746 . . . . . . . . . . . . . . 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 2798 . . . . . . . . . . . . . . . . . . . 20 ran 𝐺 = ran 𝐺
3332grpocl 28283 . . . . . . . . . . . . . . . . . . 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 6737 . . . . . . . . . . . . . . . . . . 19 ((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑥𝐺𝑦) ∈ ran 𝐺) → ((𝑇𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
3635adantlr 714 . . . . . . . . . . . . . . . . . 18 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥𝐺𝑦) ∈ ran 𝐺) → ((𝑇𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
3734, 36sylan2 595 . . . . . . . . . . . . . . . . 17 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝐺 ∈ GrpOp ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺))) → ((𝑇𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
3837anassrs 471 . . . . . . . . . . . . . . . 16 ((((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑇𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
3938ad2ant2r 746 . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . 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 460 . . . . . . . . . . . . 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 3146 . . . . . . . . . . . 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 651 . . . . . . . . . . 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 416 . . . . . . . . . 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 470 . . . . . . . 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 410 . . . . . . 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 651 . . . . . 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 516 . . 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 2798 . . . . . 6 ran 𝐻 = ran 𝐻
5332, 52elghomOLD 35325 . . . . 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 2798 . . . . . 6 ran 𝐾 = ran 𝐾
5652, 55elghomOLD 35325 . . . . 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 633 . . 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 35325 . . . 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 297 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → ((𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾)) → (𝑇𝑆) ∈ (𝐺 GrpOpHom 𝐾)))
6261imp 410 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  wral 3106  ran crn 5520  ccom 5523  wf 6320  cfv 6324  (class class class)co 7135  GrpOpcgr 28272   GrpOpHom cghomOLD 35321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-grpo 28276  df-ghomOLD 35322
This theorem is referenced by: (None)
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