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Theorem ghomco 33349
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 6020 . . . . . . 7 ((𝑇:ran 𝐻⟶ran 𝐾𝑆:ran 𝐺⟶ran 𝐻) → (𝑇𝑆):ran 𝐺⟶ran 𝐾)
21ancoms 469 . . . . . 6 ((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) → (𝑇𝑆):ran 𝐺⟶ran 𝐾)
32ad2ant2r 782 . . . . 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 6318 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆:ran 𝐺⟶ran 𝐻𝑥 ∈ ran 𝐺) → (𝑆𝑥) ∈ ran 𝐻)
6 ffvelrn 6318 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆:ran 𝐺⟶ran 𝐻𝑦 ∈ ran 𝐺) → (𝑆𝑦) ∈ ran 𝐻)
75, 6anim12da 33165 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑆𝑥) ∈ ran 𝐻 ∧ (𝑆𝑦) ∈ ran 𝐻))
8 fveq2 6153 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑢 = (𝑆𝑥) → (𝑇𝑢) = (𝑇‘(𝑆𝑥)))
98oveq1d 6625 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = (𝑆𝑥) → ((𝑇𝑢)𝐾(𝑇𝑣)) = ((𝑇‘(𝑆𝑥))𝐾(𝑇𝑣)))
10 oveq1 6617 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑢 = (𝑆𝑥) → (𝑢𝐻𝑣) = ((𝑆𝑥)𝐻𝑣))
1110fveq2d 6157 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = (𝑆𝑥) → (𝑇‘(𝑢𝐻𝑣)) = (𝑇‘((𝑆𝑥)𝐻𝑣)))
129, 11eqeq12d 2636 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = (𝑆𝑥) → (((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣)) ↔ ((𝑇‘(𝑆𝑥))𝐾(𝑇𝑣)) = (𝑇‘((𝑆𝑥)𝐻𝑣))))
13 fveq2 6153 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = (𝑆𝑦) → (𝑇𝑣) = (𝑇‘(𝑆𝑦)))
1413oveq2d 6626 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = (𝑆𝑦) → ((𝑇‘(𝑆𝑥))𝐾(𝑇𝑣)) = ((𝑇‘(𝑆𝑥))𝐾(𝑇‘(𝑆𝑦))))
15 oveq2 6618 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = (𝑆𝑦) → ((𝑆𝑥)𝐻𝑣) = ((𝑆𝑥)𝐻(𝑆𝑦)))
1615fveq2d 6157 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = (𝑆𝑦) → (𝑇‘((𝑆𝑥)𝐻𝑣)) = (𝑇‘((𝑆𝑥)𝐻(𝑆𝑦))))
1714, 16eqeq12d 2636 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = (𝑆𝑦) → (((𝑇‘(𝑆𝑥))𝐾(𝑇𝑣)) = (𝑇‘((𝑆𝑥)𝐻𝑣)) ↔ ((𝑇‘(𝑆𝑥))𝐾(𝑇‘(𝑆𝑦))) = (𝑇‘((𝑆𝑥)𝐻(𝑆𝑦)))))
1812, 17rspc2va 3311 . . . . . . . . . . . . . . . . . . . . 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))))
197, 18sylan 488 . . . . . . . . . . . . . . . . . . . 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))))
2019an32s 845 . . . . . . . . . . . . . . . . . . 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))))
2120adantllr 754 . . . . . . . . . . . . . . . . . 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))))
2221adantllr 754 . . . . . . . . . . . . . . . . 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))))
23 fveq2 6153 . . . . . . . . . . . . . . . . 17 (((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → (𝑇‘((𝑆𝑥)𝐻(𝑆𝑦))) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
2422, 23sylan9eq 2675 . . . . . . . . . . . . . . . 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))))
2524anasss 678 . . . . . . . . . . . . . . 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))))
26 fvco3 6237 . . . . . . . . . . . . . . . . . . 19 ((𝑆:ran 𝐺⟶ran 𝐻𝑥 ∈ ran 𝐺) → ((𝑇𝑆)‘𝑥) = (𝑇‘(𝑆𝑥)))
2726ad2ant2r 782 . . . . . . . . . . . . . . . . . 18 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑇𝑆)‘𝑥) = (𝑇‘(𝑆𝑥)))
28 fvco3 6237 . . . . . . . . . . . . . . . . . . 19 ((𝑆:ran 𝐺⟶ran 𝐻𝑦 ∈ ran 𝐺) → ((𝑇𝑆)‘𝑦) = (𝑇‘(𝑆𝑦)))
2928ad2ant2rl 784 . . . . . . . . . . . . . . . . . 18 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑇𝑆)‘𝑦) = (𝑇‘(𝑆𝑦)))
3027, 29oveq12d 6628 . . . . . . . . . . . . . . . . 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))))
3130adantlr 750 . . . . . . . . . . . . . . . 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))))
3231ad2ant2r 782 . . . . . . . . . . . . . . 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))))
33 eqid 2621 . . . . . . . . . . . . . . . . . . . 20 ran 𝐺 = ran 𝐺
3433grpocl 27221 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺) → (𝑥𝐺𝑦) ∈ ran 𝐺)
35343expb 1263 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → (𝑥𝐺𝑦) ∈ ran 𝐺)
36 fvco3 6237 . . . . . . . . . . . . . . . . . . 19 ((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑥𝐺𝑦) ∈ ran 𝐺) → ((𝑇𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
3736adantlr 750 . . . . . . . . . . . . . . . . . 18 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥𝐺𝑦) ∈ ran 𝐺) → ((𝑇𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
3835, 37sylan2 491 . . . . . . . . . . . . . . . . 17 (((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝐺 ∈ GrpOp ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺))) → ((𝑇𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
3938anassrs 679 . . . . . . . . . . . . . . . 16 ((((𝑆:ran 𝐺⟶ran 𝐻𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → ((𝑇𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
4039ad2ant2r 782 . . . . . . . . . . . . . . 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))))
4125, 32, 403eqtr4d 2665 . . . . . . . . . . . . . 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)))
4241expr 642 . . . . . . . . . . . . 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))))
4342ralimdvva 2959 . . . . . . . . . . . 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))))
4443an32s 845 . . . . . . . . . . 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))))
4544ex 450 . . . . . . . . . 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)))))
4645com23 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)))))
4746anasss 678 . . . . . . . 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)))))
4847imp 445 . . . . . . 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))))
4948an32s 845 . . . . . 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))))
5049com12 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))))
51503ad2ant1 1080 . . . 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))))
524, 51jcad 555 . . 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)))))
53 eqid 2621 . . . . . 6 ran 𝐻 = ran 𝐻
5433, 53elghomOLD 33345 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝑆 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑆𝑥)𝐻(𝑆𝑦)) = (𝑆‘(𝑥𝐺𝑦)))))
55543adant3 1079 . . . 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)))))
56 eqid 2621 . . . . . 6 ran 𝐾 = ran 𝐾
5753, 56elghomOLD 33345 . . . . 5 ((𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → (𝑇 ∈ (𝐻 GrpOpHom 𝐾) ↔ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻𝑣 ∈ ran 𝐻((𝑇𝑢)𝐾(𝑇𝑣)) = (𝑇‘(𝑢𝐻𝑣)))))
58573adant1 1077 . . . 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)))))
5955, 58anbi12d 746 . . 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))))))
6033, 56elghomOLD 33345 . . . 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)))))
61603adant2 1078 . . 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)))))
6252, 59, 613imtr4d 283 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → ((𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾)) → (𝑇𝑆) ∈ (𝐺 GrpOpHom 𝐾)))
6362imp 445 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  ran crn 5080  ccom 5083  wf 5848  cfv 5852  (class class class)co 6610  GrpOpcgr 27210   GrpOpHom cghomOLD 33341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-grpo 27214  df-ghomOLD 33342
This theorem is referenced by: (None)
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