Theorem mhmmulg 13755

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 6041	. . . . . 6 ⊢ (𝑛 = 0 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(0 · 𝑋)))
2		oveq1 6025	. . . . . 6 ⊢ (𝑛 = 0 → (𝑛 × (𝐹‘𝑋)) = (0 × (𝐹‘𝑋)))
3	1, 2	eqeq12d 2246	. . . . 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 6041	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(𝑚 · 𝑋)))
6		oveq1 6025	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑛 × (𝐹‘𝑋)) = (𝑚 × (𝐹‘𝑋)))
7	5, 6	eqeq12d 2246	. . . . 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 6041	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘((𝑚 + 1) · 𝑋)))
10		oveq1 6025	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 × (𝐹‘𝑋)) = ((𝑚 + 1) × (𝐹‘𝑋)))
11	9, 10	eqeq12d 2246	. . . . 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 6041	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝐹‘(𝑛 · 𝑋)) = (𝐹‘(𝑁 · 𝑋)))
14		oveq1 6025	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 × (𝐹‘𝑋)) = (𝑁 × (𝐹‘𝑋)))
15	13, 14	eqeq12d 2246	. . . . 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 2231	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
18		eqid 2231	. . . . . . 7 ⊢ (0g‘𝐻) = (0g‘𝐻)
19	17, 18	mhm0 13556	. . . . . 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 13717	. . . . . . 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 5643	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(0 · 𝑋)) = (𝐹‘(0g‘𝐺)))
26		eqid 2231	. . . . . . . 8 ⊢ (Base‘𝐻) = (Base‘𝐻)
27	21, 26	mhmf 13553	. . . . . . 7 ⊢ (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:𝐵⟶(Base‘𝐻))
28	27	ffvelcdmda 5782	. . . . . 6 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ (Base‘𝐻))
29		mhmmulg.t	. . . . . . 7 ⊢ × = (.g‘𝐻)
30	26, 18, 29	mulg0 13717	. . . . . 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 2274	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(0 · 𝑋)) = (0 × (𝐹‘𝑋)))
33		oveq1 6025	. . . . . . 7 ⊢ ((𝐹‘(𝑚 · 𝑋)) = (𝑚 × (𝐹‘𝑋)) → ((𝐹‘(𝑚 · 𝑋))(+g‘𝐻)(𝐹‘𝑋)) = ((𝑚 × (𝐹‘𝑋))(+g‘𝐻)(𝐹‘𝑋)))
34		mhmrcl1 13551	. . . . . . . . . . . 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 529	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → 𝑋 ∈ 𝐵)
38		eqid 2231	. . . . . . . . . . . 12 ⊢ (+g‘𝐺) = (+g‘𝐺)
39	21, 22, 38	mulgnn0p1 13725	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ 𝑚 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑚 + 1) · 𝑋) = ((𝑚 · 𝑋)(+g‘𝐺)𝑋))
40	35, 36, 37, 39	syl3anc 1273	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) · 𝑋) = ((𝑚 · 𝑋)(+g‘𝐺)𝑋))
41	40	fveq2d 5643	. . . . . . . . 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 529	. . . . . . . . . . . 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 13730	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Mnd ∧ 𝑚 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
47	43, 44, 45, 46	syl3anc 1273	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑚 ∈ ℕ0) ∧ 𝑋 ∈ 𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
48	47	an32s 570	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → (𝑚 · 𝑋) ∈ 𝐵)
49		eqid 2231	. . . . . . . . . . 11 ⊢ (+g‘𝐻) = (+g‘𝐻)
50	21, 38, 49	mhmlin 13555	. . . . . . . . . 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 1273	. . . . . . . . 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 2264	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → (𝐹‘((𝑚 + 1) · 𝑋)) = ((𝐹‘(𝑚 · 𝑋))(+g‘𝐻)(𝐹‘𝑋)))
53		mhmrcl2 13552	. . . . . . . . . 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 13725	. . . . . . . . 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 1273	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) × (𝐹‘𝑋)) = ((𝑚 × (𝐹‘𝑋))(+g‘𝐻)(𝐹‘𝑋)))
58	52, 57	eqeq12d 2246	. . . . . . 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 9594	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹‘𝑋))))
63	62	3impib 1227	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹‘𝑋)))
64	63	3com12 1233	1 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ‘cfv 5326  (class class class)co 6018  0cc0 8032  1c1 8033   + caddc 8035  ℕ₀cn0 9402  Basecbs 13087  +_gcplusg 13165  0_gc0g 13344  Mndcmnd 13504   MndHom cmhm 13545  ._gcmg 13711

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-map 6819  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-n0 9403  df-z 9480  df-uz 9756  df-seqfrec 10711  df-ndx 13090  df-slot 13091  df-base 13093  df-plusg 13178  df-0g 13346  df-mgm 13444  df-sgrp 13490  df-mnd 13505  df-mhm 13547  df-minusg 13592  df-mulg 13712

This theorem is referenced by:  ghmmulg  13848  lgseisenlem4  15808