Theorem ghomco 38402

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 6720	. . . . . . 7 ⊢ ((𝑇:ran 𝐻⟶ran 𝐾 ∧ 𝑆:ran 𝐺⟶ran 𝐻) → (𝑇 ∘ 𝑆):ran 𝐺⟶ran 𝐾)
2	1	ancoms 463	. . . . . 6 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) → (𝑇 ∘ 𝑆):ran 𝐺⟶ran 𝐾)
3	2	ad2ant2r 759	. . . . 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 7066	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑥 ∈ ran 𝐺) → (𝑆‘𝑥) ∈ ran 𝐻)
6		ffvelcdm 7066	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑦 ∈ ran 𝐺) → (𝑆‘𝑦) ∈ ran 𝐻)
7	5, 6	anim12dan 630	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → ((𝑆‘𝑥) ∈ ran 𝐻 ∧ (𝑆‘𝑦) ∈ ran 𝐻))
8		fveq2 6871	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = (𝑆‘𝑥) → (𝑇‘𝑢) = (𝑇‘(𝑆‘𝑥)))
9	8	oveq1d 7415	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = (𝑆‘𝑥) → ((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘𝑣)))
10		fvoveq1 7423	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = (𝑆‘𝑥) → (𝑇‘(𝑢𝐻𝑣)) = (𝑇‘((𝑆‘𝑥)𝐻𝑣)))
11	9, 10	eqeq12d 2781	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = (𝑆‘𝑥) → (((𝑇‘𝑢)𝐾(𝑇‘𝑣)) = (𝑇‘(𝑢𝐻𝑣)) ↔ ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘𝑣)) = (𝑇‘((𝑆‘𝑥)𝐻𝑣))))
12		fveq2 6871	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = (𝑆‘𝑦) → (𝑇‘𝑣) = (𝑇‘(𝑆‘𝑦)))
13	12	oveq2d 7416	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (𝑆‘𝑦) → ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘𝑣)) = ((𝑇‘(𝑆‘𝑥))𝐾(𝑇‘(𝑆‘𝑦))))
14		oveq2 7408	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = (𝑆‘𝑦) → ((𝑆‘𝑥)𝐻𝑣) = ((𝑆‘𝑥)𝐻(𝑆‘𝑦)))
15	14	fveq2d 6875	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (𝑆‘𝑦) → (𝑇‘((𝑆‘𝑥)𝐻𝑣)) = (𝑇‘((𝑆‘𝑥)𝐻(𝑆‘𝑦))))
16	13, 15	eqeq12d 2781	. . . . . . . . . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . . . . . . . . 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 664	. . . . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . 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 6871	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆‘𝑥)𝐻(𝑆‘𝑦)) = (𝑆‘(𝑥𝐺𝑦)) → (𝑇‘((𝑆‘𝑥)𝐻(𝑆‘𝑦))) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
23	21, 22	sylan9eq 2820	. . . . . . . . . . . . . . . 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 471	. . . . . . . . . . . . . . 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 6971	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑥 ∈ ran 𝐺) → ((𝑇 ∘ 𝑆)‘𝑥) = (𝑇‘(𝑆‘𝑥)))
26	25	ad2ant2r 759	. . . . . . . . . . . . . . . . . 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 6971	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑦 ∈ ran 𝐺) → ((𝑇 ∘ 𝑆)‘𝑦) = (𝑇‘(𝑆‘𝑦)))
28	27	ad2ant2rl 761	. . . . . . . . . . . . . . . . . 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 7418	. . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . 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 759	. . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . . 20 ⊢ ran 𝐺 = ran 𝐺
33	32	grpocl 30761	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺) → (𝑥𝐺𝑦) ∈ ran 𝐺)
34	33	3expb 1136	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → (𝑥𝐺𝑦) ∈ ran 𝐺)
35		fvco3 6971	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆:ran 𝐺⟶ran 𝐻 ∧ (𝑥𝐺𝑦) ∈ ran 𝐺) → ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
36	35	adantlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆:ran 𝐺⟶ran 𝐻 ∧ 𝑇:ran 𝐻⟶ran 𝐾) ∧ (𝑥𝐺𝑦) ∈ ran 𝐺) → ((𝑇 ∘ 𝑆)‘(𝑥𝐺𝑦)) = (𝑇‘(𝑆‘(𝑥𝐺𝑦))))
37	34, 36	sylan2 604	. . . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . . 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 759	. . . . . . . . . . . . . . 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 2810	. . . . . . . . . . . . . 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 461	. . . . . . . . . . . . 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 3212	. . . . . . . . . . . 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 664	. . . . . . . . . . 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 417	. . . . . . . . . 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 471	. . . . . . . 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 411	. . . . . . 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 664	. . . . . 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 1149	. . . 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 521	. . 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 2765	. . . . . 6 ⊢ ran 𝐻 = ran 𝐻
53	32, 52	elghomOLD 38398	. . . . 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 1148	. . . 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 2765	. . . . . 6 ⊢ ran 𝐾 = ran 𝐾
56	52, 55	elghomOLD 38398	. . . . 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 1146	. . . 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 643	. . 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 38398	. . . 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 1147	. . 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 411	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 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ran crn 5653   ∘ ccom 5656  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  GrpOpcgr 30750   GrpOpHom cghomOLD 38394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-grpo 30754  df-ghomOLD 38395

This theorem is referenced by: (None)