Theorem ghomco 38574

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 6734	. . . . . . 7 ⊢ ((𝑇:ran 𝐻⟶ran 𝐾 ∧ 𝑆:ran 𝐺⟶ran 𝐻) → (𝑇 ∘ 𝑆):ran 𝐺⟶ran 𝐾)
2	1	ancoms 464	. . . . . 6 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) → (𝑇 ∘ 𝑆):ran 𝐺⟶ran 𝐾)
3	2	ad2ant2r 760	. . . . 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		ffvelcdm 7080	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑥 ∈ ran 𝐺) → (𝑆‘𝑥) ∈ ran 𝐻)
6		ffvelcdm 7080	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑦 ∈ ran 𝐺) → (𝑆‘𝑦) ∈ ran 𝐻)
7	5, 6	anim12dan 631	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → ((𝑆‘𝑥) ∈ ran 𝐻 ∧ (𝑆‘𝑦) ∈ ran 𝐻))
8		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = (𝑆‘𝑥) → (𝑇‘𝑢) = (𝑇‘(𝑆‘𝑥)))
9	8	oveq1d 7431	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = (𝑆‘𝑥) → ((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘𝑣)))
10		fvoveq1 7439	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = (𝑆‘𝑥) → (𝑇‘(𝑢𝐻𝑣)) = (𝑇‘((𝑆‘𝑥)𝐻𝑣)))
11	9, 10	eqeq12d 2782	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = (𝑆‘𝑥) → (((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣)) ↔ ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘𝑣)) = (𝑇‘((𝑆‘𝑥)𝐻𝑣))))
12		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = (𝑆‘𝑦) → (𝑇‘𝑣) = (𝑇‘(𝑆‘𝑦)))
13	12	oveq2d 7432	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (𝑆‘𝑦) → ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘𝑣)) = ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘(𝑆‘𝑦))))
14		oveq2 7424	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = (𝑆‘𝑦) → ((𝑆‘𝑥)𝐻𝑣) = ((𝑆‘𝑥)𝐻(𝑆‘𝑦)))
15	14	fveq2d 6889	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (𝑆‘𝑦) → (𝑇‘((𝑆‘𝑥)𝐻𝑣)) = (𝑇‘((𝑆‘𝑥)𝐻(𝑆‘𝑦))))
16	13, 15	eqeq12d 2782	. . . . . . . . . . . . . . . . . . . . . 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)))))
17	11, 16	rspc2va 3596	. . . . . . . . . . . . . . . . . . . . 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))))
18	7, 17	sylan 592	. . . . . . . . . . . . . . . . . . . 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))))
19	18	an32s 665	. . . . . . . . . . . . . . . . . . 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))))
20	19	adantllr 732	. . . . . . . . . . . . . . . . . 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))))
21	20	adantllr 732	. . . . . . . . . . . . . . . . 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))))
22		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → (𝑇‘((𝑆‘𝑥)𝐻(𝑆‘𝑦))) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
23	21, 22	sylan9eq 2821	. . . . . . . . . . . . . . . 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))))
24	23	anasss 472	. . . . . . . . . . . . . . 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))))
25		fvco3 6985	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑥 ∈ ran 𝐺) → ((𝑇 ∘ 𝑆)‘𝑥) = (𝑇‘(𝑆‘𝑥)))
26	25	ad2ant2r 760	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → ((𝑇 ∘ 𝑆)‘𝑥) = (𝑇‘(𝑆‘𝑥)))
27		fvco3 6985	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑦 ∈ ran 𝐺) → ((𝑇 ∘ 𝑆)‘𝑦) = (𝑇‘(𝑆‘𝑦)))
28	27	ad2ant2rl 762	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → ((𝑇 ∘ 𝑆)‘𝑦) = (𝑇‘(𝑆‘𝑦)))
29	26, 28	oveq12d 7434	. . . . . . . . . . . . . . . . 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))))
30	29	adantlr 728	. . . . . . . . . . . . . . . 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))))
31	30	ad2ant2r 760	. . . . . . . . . . . . . . 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))))
32		eqid 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ ran 𝐺 = ran 𝐺
33	32	grpocl 30867	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺) → (𝑥𝐺𝑦) ∈ ran 𝐺)
34	33	3expb 1138	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → (𝑥𝐺𝑦) ∈ ran 𝐺)
35		fvco3 6985	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑥𝐺𝑦) ∈ ran 𝐺) → ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
36	35	adantlr 728	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥𝐺𝑦) ∈ ran 𝐺) → ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
37	34, 36	sylan2 605	. . . . . . . . . . . . . . . . 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))))
38	37	anassrs 473	. . . . . . . . . . . . . . . 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))))
39	38	ad2ant2r 760	. . . . . . . . . . . . . . 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))))
40	24, 31, 39	3eqtr4d 2811	. . . . . . . . . . . . . 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)))
41	40	expr 462	. . . . . . . . . . . . 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))))
42	41	ralimdvva 3215	. . . . . . . . . . . 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))))
43	42	an32s 665	. . . . . . . . . . 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))))
44	43	ex 418	. . . . . . . . . 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)))))
45	44	com23 87	. . . . . . . . 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)))))
46	45	anasss 472	. . . . . . . 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)))))
47	46	imp 412	. . . . . . 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))))
48	47	an32s 665	. . . . . 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))))
49	48	com12 33	. . . . 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))))
50	49	3ad2ant1 1151	. . . 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))))
51	4, 50	jcad 522	. . 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)))))
52		eqid 2766	. . . . . 6 ⊢ ran 𝐻 = ran 𝐻
53	32, 52	elghomOLD 38570	. . . . 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)))))
54	53	3adant3 1150	. . . 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)))))
55		eqid 2766	. . . . . 6 ⊢ ran 𝐾 = ran 𝐾
56	52, 55	elghomOLD 38570	. . . . 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)))))
57	56	3adant1 1148	. . . 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)))))
58	54, 57	anbi12d 644	. . 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))))))
59	32, 55	elghomOLD 38570	. . . 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)))))
60	59	3adant2 1149	. . 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)))))
61	51, 58, 60	3imtr4d 297	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → ((𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾)) → (𝑇 ∘ 𝑆) ∈ (𝐺 GrpOpHom 𝐾)))
62	61	imp 412	1 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) ∧ (𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾))) → (𝑇 ∘ 𝑆) ∈ (𝐺 GrpOpHom 𝐾))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ran crn 5665   ∘ ccom 5668  ⟶wf 6536  ‘cfv 6540  (class class class)co 7416  GrpOpcgr 30856   GrpOpHom cghomOLD 38566

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5241  ax-sep 5260  ax-nul 5272  ax-pow 5339  ax-pr 5407  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4876  df-iun 4961  df-br 5113  df-opab 5177  df-mpt 5196  df-id 5559  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-grpo 30860  df-ghomOLD 38567

This theorem is used by: (None)