Theorem mhmmulg 13695

Description: A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)

Hypotheses

Ref	Expression
mhmmulg.b	⊢ 𝐵 = (Base‘𝐺)
mhmmulg.s	⊢ · = (.g‘𝐺)
mhmmulg.t	⊢ × = (.g‘𝐻)

Assertion

Ref	Expression
mhmmulg	⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹‘𝑋)))

Proof of Theorem mhmmulg

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvoveq1 6023	. . . . . 6 ⊢ (𝑛 = 0 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(0 · 𝑋)))
2		oveq1 6007	. . . . . 6 ⊢ (𝑛 = 0 → (𝑛 × (𝐹‘𝑋)) = (0 × (𝐹‘𝑋)))
3	1, 2	eqeq12d 2244	. . . . 5 ⊢ (𝑛 = 0 → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹‘𝑋)) ↔ (𝐹‘(0 · 𝑋)) = (0 × (𝐹‘𝑋))))
4	3	imbi2d 230	. . . 4 ⊢ (𝑛 = 0 → (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹‘𝑋))) ↔ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(0 · 𝑋)) = (0 × (𝐹‘𝑋)))))
5		fvoveq1 6023	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(𝑚 · 𝑋)))
6		oveq1 6007	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑛 × (𝐹‘𝑋)) = (𝑚 × (𝐹‘𝑋)))
7	5, 6	eqeq12d 2244	. . . . 5 ⊢ (𝑛 = 𝑚 → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹‘𝑋)) ↔ (𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹‘𝑋))))
8	7	imbi2d 230	. . . 4 ⊢ (𝑛 = 𝑚 → (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹‘𝑋))) ↔ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹‘𝑋)))))
9		fvoveq1 6023	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘((𝑚 + 1) · 𝑋)))
10		oveq1 6007	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 × (𝐹‘𝑋)) = ((𝑚 + 1) × (𝐹‘𝑋)))
11	9, 10	eqeq12d 2244	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹‘𝑋)) ↔ (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹‘𝑋))))
12	11	imbi2d 230	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹‘𝑋))) ↔ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹‘𝑋)))))
13		fvoveq1 6023	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(𝑁 · 𝑋)))
14		oveq1 6007	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 × (𝐹‘𝑋)) = (𝑁 × (𝐹‘𝑋)))
15	13, 14	eqeq12d 2244	. . . . 5 ⊢ (𝑛 = 𝑁 → ((𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹‘𝑋)) ↔ (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹‘𝑋))))
16	15	imbi2d 230	. . . 4 ⊢ (𝑛 = 𝑁 → (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑛 · 𝑋)) = (𝑛 × (𝐹‘𝑋))) ↔ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹‘𝑋)))))
17		eqid 2229	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
18		eqid 2229	. . . . . . 7 ⊢ (0g‘𝐻) = (0g‘𝐻)
19	17, 18	mhm0 13496	. . . . . 6 ⊢ (𝐹 ∈ (𝐺 MndHom 𝐻) → (𝐹‘(0g‘𝐺)) = (0g‘𝐻))
20	19	adantr 276	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(0g‘𝐺)) = (0g‘𝐻))
21		mhmmulg.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
22		mhmmulg.s	. . . . . . . 8 ⊢ · = (.g‘𝐺)
23	21, 17, 22	mulg0 13657	. . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = (0g‘𝐺))
24	23	adantl 277	. . . . . 6 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (0 · 𝑋) = (0g‘𝐺))
25	24	fveq2d 5630	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(0 · 𝑋)) = (𝐹‘(0g‘𝐺)))
26		eqid 2229	. . . . . . . 8 ⊢ (Base‘𝐻) = (Base‘𝐻)
27	21, 26	mhmf 13493	. . . . . . 7 ⊢ (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:𝐵⟶(Base‘𝐻))
28	27	ffvelcdmda 5769	. . . . . 6 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ (Base‘𝐻))
29		mhmmulg.t	. . . . . . 7 ⊢ × = (.g‘𝐻)
30	26, 18, 29	mulg0 13657	. . . . . 6 ⊢ ((𝐹‘𝑋) ∈ (Base‘𝐻) → (0 × (𝐹‘𝑋)) = (0g‘𝐻))
31	28, 30	syl 14	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (0 × (𝐹‘𝑋)) = (0g‘𝐻))
32	20, 25, 31	3eqtr4d 2272	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(0 · 𝑋)) = (0 × (𝐹‘𝑋)))
33		oveq1 6007	. . . . . . 7 ⊢ ((𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹‘𝑋)) → ((𝐹‘(𝑚 · 𝑋))(+g‘𝐻)(𝐹‘𝑋)) = ((𝑚 × (𝐹‘𝑋))(+g‘𝐻)(𝐹‘𝑋)))
34		mhmrcl1 13491	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐺 ∈ Mnd)
35	34	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → 𝐺 ∈ Mnd)
36		simpr 110	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
37		simplr 528	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → 𝑋 ∈ 𝐵)
38		eqid 2229	. . . . . . . . . . . 12 ⊢ (+g‘𝐺) = (+g‘𝐺)
39	21, 22, 38	mulgnn0p1 13665	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ 𝑚 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑚 + 1) · 𝑋) = ((𝑚 · 𝑋)(+g‘𝐺)𝑋))
40	35, 36, 37, 39	syl3anc 1271	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) = ((𝑚 · 𝑋)(+g‘𝐺)𝑋))
41	40	fveq2d 5630	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 + 1) · 𝑋)) = (𝐹‘((𝑚 · 𝑋)(+g‘𝐺)𝑋)))
42		simpll 527	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → 𝐹 ∈ (𝐺 MndHom 𝐻))
43	34	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋 ∈ 𝐵) → 𝐺 ∈ Mnd)
44		simplr 528	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋 ∈ 𝐵) → 𝑚 ∈ ℕ0)
45		simpr 110	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
46	21, 22	mulgnn0cl 13670	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Mnd ∧ 𝑚 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
47	43, 44, 45, 46	syl3anc 1271	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋 ∈ 𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
48	47	an32s 568	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → (𝑚 · 𝑋) ∈ 𝐵)
49		eqid 2229	. . . . . . . . . . 11 ⊢ (+g‘𝐻) = (+g‘𝐻)
50	21, 38, 49	mhmlin 13495	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ (𝑚 · 𝑋) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐹‘((𝑚 · 𝑋)(+g‘𝐺)𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g‘𝐻)(𝐹‘𝑋)))
51	42, 48, 37, 50	syl3anc 1271	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 · 𝑋)(+g‘𝐺)𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g‘𝐻)(𝐹‘𝑋)))
52	41, 51	eqtrd 2262	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g‘𝐻)(𝐹‘𝑋)))
53		mhmrcl2 13492	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐻 ∈ Mnd)
54	53	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → 𝐻 ∈ Mnd)
55	28	adantr 276	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘𝑋) ∈ (Base‘𝐻))
56	26, 29, 49	mulgnn0p1 13665	. . . . . . . . 9 ⊢ ((𝐻 ∈ Mnd ∧ 𝑚 ∈ ℕ0 ∧ (𝐹‘𝑋) ∈ (Base‘𝐻)) → ((𝑚 + 1) × (𝐹‘𝑋)) = ((𝑚 × (𝐹‘𝑋))(+g‘𝐻)(𝐹‘𝑋)))
57	54, 36, 55, 56	syl3anc 1271	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) × (𝐹‘𝑋)) = ((𝑚 × (𝐹‘𝑋))(+g‘𝐻)(𝐹‘𝑋)))
58	52, 57	eqeq12d 2244	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹‘𝑋)) ↔ ((𝐹‘(𝑚 · 𝑋))(+g‘𝐻)(𝐹‘𝑋)) = ((𝑚 × (𝐹‘𝑋))(+g‘𝐻)(𝐹‘𝑋))))
59	33, 58	imbitrrid 156	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹‘𝑋)) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹‘𝑋))))
60	59	expcom 116	. . . . 5 ⊢ (𝑚 ∈ ℕ0 → ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → ((𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹‘𝑋)) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹‘𝑋)))))
61	60	a2d 26	. . . 4 ⊢ (𝑚 ∈ ℕ0 → (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹‘𝑋))) → ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝑚 + 1) × (𝐹‘𝑋)))))
62	4, 8, 12, 16, 32, 61	nn0ind 9557	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹‘𝑋))))
63	62	3impib 1225	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹‘𝑋)))
64	63	3com12 1231	1 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ‘cfv 5317  (class class class)co 6000  0cc0 7995  1c1 7996   + caddc 7998  ℕ₀cn0 9365  Basecbs 13027  +_gcplusg 13105  0_gc0g 13284  Mndcmnd 13444   MndHom cmhm 13485  ._gcmg 13651

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-map 6795  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-2 9165  df-n0 9366  df-z 9443  df-uz 9719  df-seqfrec 10665  df-ndx 13030  df-slot 13031  df-base 13033  df-plusg 13118  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-mhm 13487  df-minusg 13532  df-mulg 13652

This theorem is referenced by:  ghmmulg  13788  lgseisenlem4  15746