Theorem mhmmulg 13029

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 5900	. . . . . 6 ⊢ (𝑛 = 0 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(0 · 𝑋)))
2		oveq1 5884	. . . . . 6 ⊢ (𝑛 = 0 → (𝑛 × (𝐹‘𝑋)) = (0 × (𝐹‘𝑋)))
3	1, 2	eqeq12d 2192	. . . . 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 5900	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(𝑚 · 𝑋)))
6		oveq1 5884	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑛 × (𝐹‘𝑋)) = (𝑚 × (𝐹‘𝑋)))
7	5, 6	eqeq12d 2192	. . . . 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 5900	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘((𝑚 + 1) · 𝑋)))
10		oveq1 5884	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 × (𝐹‘𝑋)) = ((𝑚 + 1) × (𝐹‘𝑋)))
11	9, 10	eqeq12d 2192	. . . . 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 5900	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(𝑁 · 𝑋)))
14		oveq1 5884	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 × (𝐹‘𝑋)) = (𝑁 × (𝐹‘𝑋)))
15	13, 14	eqeq12d 2192	. . . . 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 2177	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
18		eqid 2177	. . . . . . 7 ⊢ (0g‘𝐻) = (0g‘𝐻)
19	17, 18	mhm0 12864	. . . . . 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 12993	. . . . . . 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 5521	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(0 · 𝑋)) = (𝐹‘(0g‘𝐺)))
26		eqid 2177	. . . . . . . 8 ⊢ (Base‘𝐻) = (Base‘𝐻)
27	21, 26	mhmf 12861	. . . . . . 7 ⊢ (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:𝐵⟶(Base‘𝐻))
28	27	ffvelcdmda 5653	. . . . . 6 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ (Base‘𝐻))
29		mhmmulg.t	. . . . . . 7 ⊢ × = (.g‘𝐻)
30	26, 18, 29	mulg0 12993	. . . . . 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 2220	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(0 · 𝑋)) = (0 × (𝐹‘𝑋)))
33		oveq1 5884	. . . . . . 7 ⊢ ((𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹‘𝑋)) → ((𝐹‘(𝑚 · 𝑋))(+g‘𝐻)(𝐹‘𝑋)) = ((𝑚 × (𝐹‘𝑋))(+g‘𝐻)(𝐹‘𝑋)))
34		mhmrcl1 12859	. . . . . . . . . . . 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 2177	. . . . . . . . . . . 12 ⊢ (+g‘𝐺) = (+g‘𝐺)
39	21, 22, 38	mulgnn0p1 12999	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ 𝑚 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑚 + 1) · 𝑋) = ((𝑚 · 𝑋)(+g‘𝐺)𝑋))
40	35, 36, 37, 39	syl3anc 1238	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) = ((𝑚 · 𝑋)(+g‘𝐺)𝑋))
41	40	fveq2d 5521	. . . . . . . . 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 13004	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Mnd ∧ 𝑚 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
47	43, 44, 45, 46	syl3anc 1238	. . . . . . . . . . 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 2177	. . . . . . . . . . 11 ⊢ (+g‘𝐻) = (+g‘𝐻)
50	21, 38, 49	mhmlin 12863	. . . . . . . . . 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 1238	. . . . . . . . 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 2210	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g‘𝐻)(𝐹‘𝑋)))
53		mhmrcl2 12860	. . . . . . . . . 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 12999	. . . . . . . . 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 1238	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) × (𝐹‘𝑋)) = ((𝑚 × (𝐹‘𝑋))(+g‘𝐻)(𝐹‘𝑋)))
58	52, 57	eqeq12d 2192	. . . . . . 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 9369	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹‘𝑋))))
63	62	3impib 1201	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹‘𝑋)))
64	63	3com12 1207	1 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ‘cfv 5218  (class class class)co 5877  0cc0 7813  1c1 7814   + caddc 7816  ℕ₀cn0 9178  Basecbs 12464  +_gcplusg 12538  0_gc0g 12710  Mndcmnd 12822   MndHom cmhm 12854  ._gcmg 12988

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-map 6652  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-2 8980  df-n0 9179  df-z 9256  df-uz 9531  df-seqfrec 10448  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-mhm 12856  df-minusg 12886  df-mulg 12989

This theorem is referenced by: (None)