Theorem ghomco 32860

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 5971	. . . . . . 7 ⊢ ((𝑇:ran 𝐻⟶ran 𝐾 ∧ 𝑆:ran 𝐺⟶ran 𝐻) → (𝑇 ∘ 𝑆):ran 𝐺⟶ran 𝐾)
2	1	ancoms 468	. . . . . 6 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) → (𝑇 ∘ 𝑆):ran 𝐺⟶ran 𝐾)
3	2	ad2ant2r 779	. . . . 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 6265	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑥 ∈ ran 𝐺) → (𝑆‘𝑥) ∈ ran 𝐻)
6		ffvelrn 6265	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑦 ∈ ran 𝐺) → (𝑆‘𝑦) ∈ ran 𝐻)
7	5, 6	anim12da 32676	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → ((𝑆‘𝑥) ∈ ran 𝐻 ∧ (𝑆‘𝑦) ∈ ran 𝐻))
8		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = (𝑆‘𝑥) → (𝑇‘𝑢) = (𝑇‘(𝑆‘𝑥)))
9	8	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = (𝑆‘𝑥) → ((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘𝑣)))
10		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = (𝑆‘𝑥) → (𝑢𝐻𝑣) = ((𝑆‘𝑥)𝐻𝑣))
11	10	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = (𝑆‘𝑥) → (𝑇‘(𝑢𝐻𝑣)) = (𝑇‘((𝑆‘𝑥)𝐻𝑣)))
12	9, 11	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = (𝑆‘𝑥) → (((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣)) ↔ ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘𝑣)) = (𝑇‘((𝑆‘𝑥)𝐻𝑣))))
13		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = (𝑆‘𝑦) → (𝑇‘𝑣) = (𝑇‘(𝑆‘𝑦)))
14	13	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (𝑆‘𝑦) → ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘𝑣)) = ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘(𝑆‘𝑦))))
15		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = (𝑆‘𝑦) → ((𝑆‘𝑥)𝐻𝑣) = ((𝑆‘𝑥)𝐻(𝑆‘𝑦)))
16	15	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (𝑆‘𝑦) → (𝑇‘((𝑆‘𝑥)𝐻𝑣)) = (𝑇‘((𝑆‘𝑥)𝐻(𝑆‘𝑦))))
17	14, 16	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . . 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 3294	. . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . 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 842	. . . . . . . . . . . . . . . . . . 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 751	. . . . . . . . . . . . . . . . . 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 751	. . . . . . . . . . . . . . . . 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 6103	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → (𝑇‘((𝑆‘𝑥)𝐻(𝑆‘𝑦))) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
24	22, 23	sylan9eq 2664	. . . . . . . . . . . . . . . 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 677	. . . . . . . . . . . . . . 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 6185	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑥 ∈ ran 𝐺) → ((𝑇 ∘ 𝑆)‘𝑥) = (𝑇‘(𝑆‘𝑥)))
27	26	ad2ant2r 779	. . . . . . . . . . . . . . . . . 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 6185	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑦 ∈ ran 𝐺) → ((𝑇 ∘ 𝑆)‘𝑦) = (𝑇‘(𝑆‘𝑦)))
29	28	ad2ant2rl 781	. . . . . . . . . . . . . . . . . 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 6567	. . . . . . . . . . . . . . . . 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 747	. . . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . 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 2610	. . . . . . . . . . . . . . . . . . . 20 ⊢ ran 𝐺 = ran 𝐺
34	33	grpocl 26738	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺) → (𝑥𝐺𝑦) ∈ ran 𝐺)
35	34	3expb 1258	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → (𝑥𝐺𝑦) ∈ ran 𝐺)
36		fvco3 6185	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑥𝐺𝑦) ∈ ran 𝐺) → ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
37	36	adantlr 747	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥𝐺𝑦) ∈ ran 𝐺) → ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
38	35, 37	sylan2 490	. . . . . . . . . . . . . . . . 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 678	. . . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . 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 2654	. . . . . . . . . . . . . 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 641	. . . . . . . . . . . . 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 2947	. . . . . . . . . . . 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 842	. . . . . . . . . . 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 449	. . . . . . . . . 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 84	. . . . . . . . 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 677	. . . . . . . 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 444	. . . . . . 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 842	. . . . . 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 1075	. . . 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 554	. . 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 2610	. . . . . 6 ⊢ ran 𝐻 = ran 𝐻
54	33, 53	elghomOLD 32856	. . . . 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 1074	. . . 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 2610	. . . . . 6 ⊢ ran 𝐾 = ran 𝐾
57	53, 56	elghomOLD 32856	. . . . 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 1072	. . . 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 743	. . 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 32856	. . . 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 1073	. . 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 282	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) → ((𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾)) → (𝑇 ∘ 𝑆) ∈ (𝐺 GrpOpHom 𝐾)))
63	62	imp 444	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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ran crn 5039   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  GrpOpcgr 26727   GrpOpHom cghomOLD 32852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-grpo 26731  df-ghomOLD 32853

This theorem is referenced by: (None)