Theorem mhmeql 13124

Description: The equalizer of two monoid homomorphisms is a submonoid. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)

Assertion

Ref	Expression
mhmeql	⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubMnd‘𝑆))

Proof of Theorem mhmeql

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

Step	Hyp	Ref	Expression
1		eqid 2196	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2196	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
3	1, 2	mhmf 13097	. . . . 5 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
4	3	adantr 276	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
5	4	ffnd 5408	. . 3 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
6	1, 2	mhmf 13097	. . . . 5 ⊢ (𝐺 ∈ (𝑆 MndHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
7	6	adantl 277	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
8	7	ffnd 5408	. . 3 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐺 Fn (Base‘𝑆))
9		fndmin 5669	. . 3 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) → dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
10	5, 8, 9	syl2anc 411	. 2 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
11		ssrab2 3268	. . . 4 ⊢ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ⊆ (Base‘𝑆)
12	11	a1i 9	. . 3 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ⊆ (Base‘𝑆))
13		fveq2 5558	. . . . 5 ⊢ (𝑧 = (0g‘𝑆) → (𝐹‘𝑧) = (𝐹‘(0g‘𝑆)))
14		fveq2 5558	. . . . 5 ⊢ (𝑧 = (0g‘𝑆) → (𝐺‘𝑧) = (𝐺‘(0g‘𝑆)))
15	13, 14	eqeq12d 2211	. . . 4 ⊢ (𝑧 = (0g‘𝑆) → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘(0g‘𝑆)) = (𝐺‘(0g‘𝑆))))
16		mhmrcl1 13095	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
17	16	adantr 276	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Mnd)
18		eqid 2196	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
19	1, 18	mndidcl 13071	. . . . 5 ⊢ (𝑆 ∈ Mnd → (0g‘𝑆) ∈ (Base‘𝑆))
20	17, 19	syl 14	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (0g‘𝑆) ∈ (Base‘𝑆))
21		eqid 2196	. . . . . . 7 ⊢ (0g‘𝑇) = (0g‘𝑇)
22	18, 21	mhm0 13100	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
23	22	adantr 276	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
24	18, 21	mhm0 13100	. . . . . 6 ⊢ (𝐺 ∈ (𝑆 MndHom 𝑇) → (𝐺‘(0g‘𝑆)) = (0g‘𝑇))
25	24	adantl 277	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐺‘(0g‘𝑆)) = (0g‘𝑇))
26	23, 25	eqtr4d 2232	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g‘𝑆)) = (𝐺‘(0g‘𝑆)))
27	15, 20, 26	elrabd 2922	. . 3 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (0g‘𝑆) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
28		fveq2 5558	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑥(+g‘𝑆)𝑦) → (𝐹‘𝑧) = (𝐹‘(𝑥(+g‘𝑆)𝑦)))
29		fveq2 5558	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑥(+g‘𝑆)𝑦) → (𝐺‘𝑧) = (𝐺‘(𝑥(+g‘𝑆)𝑦)))
30	28, 29	eqeq12d 2211	. . . . . . . . . 10 ⊢ (𝑧 = (𝑥(+g‘𝑆)𝑦) → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘(𝑥(+g‘𝑆)𝑦)) = (𝐺‘(𝑥(+g‘𝑆)𝑦))))
31	17	ad2antrr 488	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝑆 ∈ Mnd)
32		simplrl 535	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝑥 ∈ (Base‘𝑆))
33		simprl 529	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝑦 ∈ (Base‘𝑆))
34		eqid 2196	. . . . . . . . . . . 12 ⊢ (+g‘𝑆) = (+g‘𝑆)
35	1, 34	mndcl 13064	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
36	31, 32, 33, 35	syl3anc 1249	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
37		simplll 533	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝐹 ∈ (𝑆 MndHom 𝑇))
38		eqid 2196	. . . . . . . . . . . . 13 ⊢ (+g‘𝑇) = (+g‘𝑇)
39	1, 34, 38	mhmlin 13099	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
40	37, 32, 33, 39	syl3anc 1249	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
41		simpllr 534	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝐺 ∈ (𝑆 MndHom 𝑇))
42	1, 34, 38	mhmlin 13099	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐺‘(𝑥(+g‘𝑆)𝑦)) = ((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦)))
43	41, 32, 33, 42	syl3anc 1249	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐺‘(𝑥(+g‘𝑆)𝑦)) = ((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦)))
44		simplrr 536	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐹‘𝑥) = (𝐺‘𝑥))
45		simprr 531	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐹‘𝑦) = (𝐺‘𝑦))
46	44, 45	oveq12d 5940	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) = ((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦)))
47	43, 46	eqtr4d 2232	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐺‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
48	40, 47	eqtr4d 2232	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = (𝐺‘(𝑥(+g‘𝑆)𝑦)))
49	30, 36, 48	elrabd 2922	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
50	49	expr 375	. . . . . . . 8 ⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑦) = (𝐺‘𝑦) → (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))
51	50	ralrimiva 2570	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → ∀𝑦 ∈ (Base‘𝑆)((𝐹‘𝑦) = (𝐺‘𝑦) → (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))
52		fveq2 5558	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
53		fveq2 5558	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝐺‘𝑧) = (𝐺‘𝑦))
54	52, 53	eqeq12d 2211	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑦) = (𝐺‘𝑦)))
55	54	ralrab 2925	. . . . . . 7 ⊢ (∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ↔ ∀𝑦 ∈ (Base‘𝑆)((𝐹‘𝑦) = (𝐺‘𝑦) → (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))
56	51, 55	sylibr 134	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
57	56	expr 375	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹‘𝑥) = (𝐺‘𝑥) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))
58	57	ralrimiva 2570	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)((𝐹‘𝑥) = (𝐺‘𝑥) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))
59		fveq2 5558	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
60		fveq2 5558	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝐺‘𝑧) = (𝐺‘𝑥))
61	59, 60	eqeq12d 2211	. . . . 5 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
62	61	ralrab 2925	. . . 4 ⊢ (∀𝑥 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ↔ ∀𝑥 ∈ (Base‘𝑆)((𝐹‘𝑥) = (𝐺‘𝑥) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))
63	58, 62	sylibr 134	. . 3 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ∀𝑥 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
64	1, 18, 34	issubm 13104	. . . 4 ⊢ (𝑆 ∈ Mnd → ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ∈ (SubMnd‘𝑆) ↔ ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ⊆ (Base‘𝑆) ∧ (0g‘𝑆) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ∧ ∀𝑥 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})))
65	17, 64	syl 14	. . 3 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ∈ (SubMnd‘𝑆) ↔ ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ⊆ (Base‘𝑆) ∧ (0g‘𝑆) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ∧ ∀𝑥 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})))
66	12, 27, 63, 65	mpbir3and 1182	. 2 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ∈ (SubMnd‘𝑆))
67	10, 66	eqeltrd 2273	1 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubMnd‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  {crab 2479   ∩ cin 3156   ⊆ wss 3157  dom cdm 4663   Fn wfn 5253  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922  Basecbs 12678  +_gcplusg 12755  0_gc0g 12927  Mndcmnd 13057   MndHom cmhm 13089  SubMndcsubmnd 13090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-map 6709  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-mhm 13091  df-submnd 13092

This theorem is referenced by:  ghmeql  13397  rhmeql  13806