Theorem ghomco 33690

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))) → (𝑇 ∘ 𝑆) ∈ (𝐺 GrpOpHom 𝐾))

Proof of Theorem ghomco

Dummy variables 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fco 6058	. . . . . . 7 ⊢ ((𝑇:ran 𝐻⟶ran 𝐾 ∧ 𝑆:ran 𝐺⟶ran 𝐻) → (𝑇 ∘ 𝑆):ran 𝐺⟶ran 𝐾)
2	1	ancoms 469	. . . . . 6 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) → (𝑇 ∘ 𝑆):ran 𝐺⟶ran 𝐾)
3	2	ad2ant2r 783	. . . . 5 ⊢ (((𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦))) ∧ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣)))) → (𝑇 ∘ 𝑆):ran 𝐺⟶ran 𝐾)
4	3	a1i 11	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → (((𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦))) ∧ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣)))) → (𝑇 ∘ 𝑆):ran 𝐺⟶ran 𝐾))
5		ffvelrn 6357	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑥 ∈ ran 𝐺) → (𝑆‘𝑥) ∈ ran 𝐻)
6		ffvelrn 6357	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑦 ∈ ran 𝐺) → (𝑆‘𝑦) ∈ ran 𝐻)
7	5, 6	anim12da 33506	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → ((𝑆‘𝑥) ∈ ran 𝐻 ∧ (𝑆‘𝑦) ∈ ran 𝐻))
8		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = (𝑆‘𝑥) → (𝑇‘𝑢) = (𝑇‘(𝑆‘𝑥)))
9	8	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = (𝑆‘𝑥) → ((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘𝑣)))
10		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = (𝑆‘𝑥) → (𝑢𝐻𝑣) = ((𝑆‘𝑥)𝐻𝑣))
11	10	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = (𝑆‘𝑥) → (𝑇‘(𝑢𝐻𝑣)) = (𝑇‘((𝑆‘𝑥)𝐻𝑣)))
12	9, 11	eqeq12d 2637	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = (𝑆‘𝑥) → (((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣)) ↔ ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘𝑣)) = (𝑇‘((𝑆‘𝑥)𝐻𝑣))))
13		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = (𝑆‘𝑦) → (𝑇‘𝑣) = (𝑇‘(𝑆‘𝑦)))
14	13	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (𝑆‘𝑦) → ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘𝑣)) = ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘(𝑆‘𝑦))))
15		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = (𝑆‘𝑦) → ((𝑆‘𝑥)𝐻𝑣) = ((𝑆‘𝑥)𝐻(𝑆‘𝑦)))
16	15	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (𝑆‘𝑦) → (𝑇‘((𝑆‘𝑥)𝐻𝑣)) = (𝑇‘((𝑆‘𝑥)𝐻(𝑆‘𝑦))))
17	14, 16	eqeq12d 2637	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = (𝑆‘𝑦) → (((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘𝑣)) = (𝑇‘((𝑆‘𝑥)𝐻𝑣)) ↔ ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘(𝑆‘𝑦))) = (𝑇‘((𝑆‘𝑥)𝐻(𝑆‘𝑦)))))
18	12, 17	rspc2va 3323	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑆‘𝑥) ∈ ran 𝐻 ∧ (𝑆‘𝑦) ∈ ran 𝐻) ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣))) → ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘(𝑆‘𝑦))) = (𝑇‘((𝑆‘𝑥)𝐻(𝑆‘𝑦))))
19	7, 18	sylan 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣))) → ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘(𝑆‘𝑦))) = (𝑇‘((𝑆‘𝑥)𝐻(𝑆‘𝑦))))
20	19	an32s 846	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘(𝑆‘𝑦))) = (𝑇‘((𝑆‘𝑥)𝐻(𝑆‘𝑦))))
21	20	adantllr 755	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘(𝑆‘𝑦))) = (𝑇‘((𝑆‘𝑥)𝐻(𝑆‘𝑦))))
22	21	adantllr 755	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘(𝑆‘𝑦))) = (𝑇‘((𝑆‘𝑥)𝐻(𝑆‘𝑦))))
23		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → (𝑇‘((𝑆‘𝑥)𝐻(𝑆‘𝑦))) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
24	22, 23	sylan9eq 2676	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) ∧ ((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦))) → ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘(𝑆‘𝑦))) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
25	24	anasss 679	. . . . . . . . . . . . . . 15 ⊢ (((((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ ((𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺) ∧ ((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦)))) → ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘(𝑆‘𝑦))) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
26		fvco3 6275	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑥 ∈ ran 𝐺) → ((𝑇 ∘ 𝑆)‘𝑥) = (𝑇‘(𝑆‘𝑥)))
27	26	ad2ant2r 783	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → ((𝑇 ∘ 𝑆)‘𝑥) = (𝑇‘(𝑆‘𝑥)))
28		fvco3 6275	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑦 ∈ ran 𝐺) → ((𝑇 ∘ 𝑆)‘𝑦) = (𝑇‘(𝑆‘𝑦)))
29	28	ad2ant2rl 785	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → ((𝑇 ∘ 𝑆)‘𝑦) = (𝑇‘(𝑆‘𝑦)))
30	27, 29	oveq12d 6668	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → (((𝑇 ∘ 𝑆)‘𝑥)𝐾((𝑇 ∘ 𝑆)‘𝑦)) = ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘(𝑆‘𝑦))))
31	30	adantlr 751	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → (((𝑇 ∘ 𝑆)‘𝑥)𝐾((𝑇 ∘ 𝑆)‘𝑦)) = ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘(𝑆‘𝑦))))
32	31	ad2ant2r 783	. . . . . . . . . . . . . . 15 ⊢ (((((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ ((𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺) ∧ ((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦)))) → (((𝑇 ∘ 𝑆)‘𝑥)𝐾((𝑇 ∘ 𝑆)‘𝑦)) = ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘(𝑆‘𝑦))))
33		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 ⊢ ran 𝐺 = ran 𝐺
34	33	grpocl 27354	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺) → (𝑥𝐺𝑦) ∈ ran 𝐺)
35	34	3expb 1266	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → (𝑥𝐺𝑦) ∈ ran 𝐺)
36		fvco3 6275	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑥𝐺𝑦) ∈ ran 𝐺) → ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
37	36	adantlr 751	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥𝐺𝑦) ∈ ran 𝐺) → ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
38	35, 37	sylan2 491	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝐺 ∈ GrpOp ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺))) → ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
39	38	anassrs 680	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
40	39	ad2ant2r 783	. . . . . . . . . . . . . . 15 ⊢ (((((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ ((𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺) ∧ ((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦)))) → ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
41	25, 32, 40	3eqtr4d 2666	. . . . . . . . . . . . . 14 ⊢ (((((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ ((𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺) ∧ ((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦)))) → (((𝑇 ∘ 𝑆)‘𝑥)𝐾((𝑇 ∘ 𝑆)‘𝑦)) = ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦)))
42	41	expr 643	. . . . . . . . . . . . 13 ⊢ (((((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → (((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → (((𝑇 ∘ 𝑆)‘𝑥)𝐾((𝑇 ∘ 𝑆)‘𝑦)) = ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦))))
43	42	ralimdvva 2964	. . . . . . . . . . . 12 ⊢ ((((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣))) → (∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(((𝑇 ∘ 𝑆)‘𝑥)𝐾((𝑇 ∘ 𝑆)‘𝑦)) = ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦))))
44	43	an32s 846	. . . . . . . . . . 11 ⊢ ((((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣))) ∧ 𝐺 ∈ GrpOp) → (∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(((𝑇 ∘ 𝑆)‘𝑥)𝐾((𝑇 ∘ 𝑆)‘𝑦)) = ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦))))
45	44	ex 450	. . . . . . . . . 10 ⊢ (((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣))) → (𝐺 ∈ GrpOp → (∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(((𝑇 ∘ 𝑆)‘𝑥)𝐾((𝑇 ∘ 𝑆)‘𝑦)) = ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦)))))
46	45	com23 86	. . . . . . . . 9 ⊢ (((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣))) → (∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → (𝐺 ∈ GrpOp → ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(((𝑇 ∘ 𝑆)‘𝑥)𝐾((𝑇 ∘ 𝑆)‘𝑦)) = ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦)))))
47	46	anasss 679	. . . . . . . 8 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣)))) → (∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → (𝐺 ∈ GrpOp → ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(((𝑇 ∘ 𝑆)‘𝑥)𝐾((𝑇 ∘ 𝑆)‘𝑦)) = ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦)))))
48	47	imp 445	. . . . . . 7 ⊢ (((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣)))) ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦))) → (𝐺 ∈ GrpOp → ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(((𝑇 ∘ 𝑆)‘𝑥)𝐾((𝑇 ∘ 𝑆)‘𝑦)) = ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦))))
49	48	an32s 846	. . . . . 6 ⊢ (((𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦))) ∧ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣)))) → (𝐺 ∈ GrpOp → ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(((𝑇 ∘ 𝑆)‘𝑥)𝐾((𝑇 ∘ 𝑆)‘𝑦)) = ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦))))
50	49	com12 32	. . . . 5 ⊢ (𝐺 ∈ GrpOp → (((𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦))) ∧ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣)))) → ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(((𝑇 ∘ 𝑆)‘𝑥)𝐾((𝑇 ∘ 𝑆)‘𝑦)) = ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦))))
51	50	3ad2ant1 1082	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → (((𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦))) ∧ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣)))) → ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(((𝑇 ∘ 𝑆)‘𝑥)𝐾((𝑇 ∘ 𝑆)‘𝑦)) = ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦))))
52	4, 51	jcad 555	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → (((𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦))) ∧ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣)))) → ((𝑇 ∘ 𝑆):ran 𝐺⟶ran 𝐾 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(((𝑇 ∘ 𝑆)‘𝑥)𝐾((𝑇 ∘ 𝑆)‘𝑦)) = ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦)))))
53		eqid 2622	. . . . . 6 ⊢ ran 𝐻 = ran 𝐻
54	33, 53	elghomOLD 33686	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝑆 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦)))))
55	54	3adant3 1081	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → (𝑆 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦)))))
56		eqid 2622	. . . . . 6 ⊢ ran 𝐾 = ran 𝐾
57	53, 56	elghomOLD 33686	. . . . 5 ⊢ ((𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → (𝑇 ∈ (𝐻 GrpOpHom 𝐾) ↔ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣)))))
58	57	3adant1 1079	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → (𝑇 ∈ (𝐻 GrpOpHom 𝐾) ↔ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣)))))
59	55, 58	anbi12d 747	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → ((𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾)) ↔ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦))) ∧ (𝑇:ran 𝐻⟶ran 𝐾 ∧ ∀𝑢 ∈ ran 𝐻∀𝑣 ∈ ran 𝐻((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣))))))
60	33, 56	elghomOLD 33686	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → ((𝑇 ∘ 𝑆) ∈ (𝐺 GrpOpHom 𝐾) ↔ ((𝑇 ∘ 𝑆):ran 𝐺⟶ran 𝐾 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(((𝑇 ∘ 𝑆)‘𝑥)𝐾((𝑇 ∘ 𝑆)‘𝑦)) = ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦)))))
61	60	3adant2 1080	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → ((𝑇 ∘ 𝑆) ∈ (𝐺 GrpOpHom 𝐾) ↔ ((𝑇 ∘ 𝑆):ran 𝐺⟶ran 𝐾 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(((𝑇 ∘ 𝑆)‘𝑥)𝐾((𝑇 ∘ 𝑆)‘𝑦)) = ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦)))))
62	52, 59, 61	3imtr4d 283	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → ((𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾)) → (𝑇 ∘ 𝑆) ∈ (𝐺 GrpOpHom 𝐾)))
63	62	imp 445	1 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) ∧ (𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾))) → (𝑇 ∘ 𝑆) ∈ (𝐺 GrpOpHom 𝐾))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ran crn 5115   ∘ ccom 5118  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  GrpOpcgr 27343   GrpOpHom cghomOLD 33682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-grpo 27347  df-ghomOLD 33683

This theorem is referenced by: (None)