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Theorem ghomco 34701
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 6399 . . . . . . 7 ((𝑇:ran 𝐻⟶ran 𝐾𝑆:ran 𝐺⟶ran 𝐻) → (𝑇𝑆):ran 𝐺⟶ran 𝐾)
21ancoms 459 . . . . . 6 ((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) → (𝑇𝑆):ran 𝐺⟶ran 𝐾)
32ad2ant2r 743 . . . . 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 6714 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆:ran 𝐺⟶ran 𝐻𝑥 ∈ ran 𝐺) → (𝑆𝑥) ∈ ran 𝐻)
6 ffvelrn 6714 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆:ran 𝐺⟶ran 𝐻𝑦 ∈ ran 𝐺) → (𝑆𝑦) ∈ ran 𝐻)
75, 6anim12da 34519 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑆𝑥) ∈ ran 𝐻 ∧ (𝑆𝑦) ∈ ran 𝐻))
8 fveq2 6538 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑢 = (𝑆𝑥) → (𝑇𝑢) = (𝑇‘(𝑆𝑥)))
98oveq1d 7031 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = (𝑆𝑥) → ((𝑇𝑢)𝐾(𝑇𝑣)) = ((𝑇‘(𝑆𝑥))𝐾(𝑇𝑣)))
10 fvoveq1 7039 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = (𝑆𝑥) → (𝑇‘(𝑢𝐻𝑣)) = (𝑇‘((𝑆𝑥)𝐻𝑣)))
119, 10eqeq12d 2810 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = (𝑆𝑥) → (((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣)) ↔ ((𝑇‘(𝑆𝑥))𝐾(𝑇𝑣)) = (𝑇‘((𝑆𝑥)𝐻𝑣))))
12 fveq2 6538 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = (𝑆𝑦) → (𝑇𝑣) = (𝑇‘(𝑆𝑦)))
1312oveq2d 7032 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = (𝑆𝑦) → ((𝑇‘(𝑆𝑥))𝐾(𝑇𝑣)) = ((𝑇‘(𝑆𝑥))𝐾(𝑇‘(𝑆𝑦))))
14 oveq2 7024 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = (𝑆𝑦) → ((𝑆𝑥)𝐻𝑣) = ((𝑆𝑥)𝐻(𝑆𝑦)))
1514fveq2d 6542 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = (𝑆𝑦) → (𝑇‘((𝑆𝑥)𝐻𝑣)) = (𝑇‘((𝑆𝑥)𝐻(𝑆𝑦))))
1613, 15eqeq12d 2810 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = (𝑆𝑦) → (((𝑇‘(𝑆𝑥))𝐾(𝑇𝑣)) = (𝑇‘((𝑆𝑥)𝐻𝑣)) ↔ ((𝑇‘(𝑆𝑥))𝐾(𝑇‘(𝑆𝑦))) = (𝑇‘((𝑆𝑥)𝐻(𝑆𝑦)))))
1711, 16rspc2va 3573 . . . . . . . . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . . . . . . . . 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 648 . . . . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . . 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 6538 . . . . . . . . . . . . . . . . 17 (((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → (𝑇‘((𝑆𝑥)𝐻(𝑆𝑦))) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
2321, 22sylan9eq 2851 . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . 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 6627 . . . . . . . . . . . . . . . . . . 19 ((𝑆:ran 𝐺⟶ran 𝐻𝑥 ∈ ran 𝐺) → ((𝑇𝑆)‘𝑥) = (𝑇‘(𝑆𝑥)))
2625ad2ant2r 743 . . . . . . . . . . . . . . . . . 18 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑇𝑆)‘𝑥) = (𝑇‘(𝑆𝑥)))
27 fvco3 6627 . . . . . . . . . . . . . . . . . . 19 ((𝑆:ran 𝐺⟶ran 𝐻𝑦 ∈ ran 𝐺) → ((𝑇𝑆)‘𝑦) = (𝑇‘(𝑆𝑦)))
2827ad2ant2rl 745 . . . . . . . . . . . . . . . . . 18 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑇𝑆)‘𝑦) = (𝑇‘(𝑆𝑦)))
2926, 28oveq12d 7034 . . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . . 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 743 . . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . . . . . . . 20 ran 𝐺 = ran 𝐺
3332grpocl 27968 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺) → (𝑥𝐺𝑦) ∈ ran 𝐺)
34333expb 1113 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → (𝑥𝐺𝑦) ∈ ran 𝐺)
35 fvco3 6627 . . . . . . . . . . . . . . . . . . 19 ((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑥𝐺𝑦) ∈ ran 𝐺) → ((𝑇𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
3635adantlr 711 . . . . . . . . . . . . . . . . . 18 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥𝐺𝑦) ∈ ran 𝐺) → ((𝑇𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
3734, 36sylan2 592 . . . . . . . . . . . . . . . . 17 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝐺 ∈ GrpOp ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺))) → ((𝑇𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
3837anassrs 468 . . . . . . . . . . . . . . . 16 ((((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑇𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
3938ad2ant2r 743 . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . 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 648 . . . . . . . . . . 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 413 . . . . . . . . . 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 467 . . . . . . . 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 407 . . . . . . 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 648 . . . . . 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 1126 . . . 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 513 . . 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 2795 . . . . . 6 ran 𝐻 = ran 𝐻
5332, 52elghomOLD 34697 . . . . 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 1125 . . . 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 2795 . . . . . 6 ran 𝐾 = ran 𝐾
5652, 55elghomOLD 34697 . . . . 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 1123 . . . 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 34697 . . . 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 1124 . . 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 295 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → ((𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾)) → (𝑇𝑆) ∈ (𝐺 GrpOpHom 𝐾)))
6261imp 407 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 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wral 3105  ran crn 5444  ccom 5447  wf 6221  cfv 6225  (class class class)co 7016  GrpOpcgr 27957   GrpOpHom cghomOLD 34693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-grpo 27961  df-ghomOLD 34694
This theorem is referenced by: (None)
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