Theorem frmdup1 18769

Description: Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 27-Sep-2015.)

Hypotheses

Ref	Expression
frmdup.m	⊢ 𝑀 = (freeMnd‘𝐼)
frmdup.b	⊢ 𝐵 = (Base‘𝐺)
frmdup.e	⊢ 𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥)))
frmdup.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
frmdup.i	⊢ (𝜑 → 𝐼 ∈ 𝑋)
frmdup.a	⊢ (𝜑 → 𝐴:𝐼⟶𝐵)

Assertion

Ref	Expression
frmdup1	⊢ (𝜑 → 𝐸 ∈ (𝑀 MndHom 𝐺))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐺   𝜑,𝑥   𝑥,𝐼

Allowed substitution hints:   𝐸(𝑥)   𝑀(𝑥)   𝑋(𝑥)

Proof of Theorem frmdup1

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

Step	Hyp	Ref	Expression
1		frmdup.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑋)
2		frmdup.m	. . . 4 ⊢ 𝑀 = (freeMnd‘𝐼)
3	2	frmdmnd 18764	. . 3 ⊢ (𝐼 ∈ 𝑋 → 𝑀 ∈ Mnd)
4	1, 3	syl 17	. 2 ⊢ (𝜑 → 𝑀 ∈ Mnd)
5		frmdup.g	. 2 ⊢ (𝜑 → 𝐺 ∈ Mnd)
6	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐼) → 𝐺 ∈ Mnd)
7		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐼) → 𝑥 ∈ Word 𝐼)
8		frmdup.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴:𝐼⟶𝐵)
9	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐼) → 𝐴:𝐼⟶𝐵)
10		wrdco 14735	. . . . . . 7 ⊢ ((𝑥 ∈ Word 𝐼 ∧ 𝐴:𝐼⟶𝐵) → (𝐴 ∘ 𝑥) ∈ Word 𝐵)
11	7, 9, 10	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐼) → (𝐴 ∘ 𝑥) ∈ Word 𝐵)
12		frmdup.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
13	12	gsumwcl 18744	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∘ 𝑥) ∈ Word 𝐵) → (𝐺 Σg (𝐴 ∘ 𝑥)) ∈ 𝐵)
14	6, 11, 13	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐼) → (𝐺 Σg (𝐴 ∘ 𝑥)) ∈ 𝐵)
15		frmdup.e	. . . . 5 ⊢ 𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥)))
16	14, 15	fmptd 7047	. . . 4 ⊢ (𝜑 → 𝐸:Word 𝐼⟶𝐵)
17		eqid 2731	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
18	2, 17	frmdbas 18757	. . . . . 6 ⊢ (𝐼 ∈ 𝑋 → (Base‘𝑀) = Word 𝐼)
19	1, 18	syl 17	. . . . 5 ⊢ (𝜑 → (Base‘𝑀) = Word 𝐼)
20	19	feq2d 6635	. . . 4 ⊢ (𝜑 → (𝐸:(Base‘𝑀)⟶𝐵 ↔ 𝐸:Word 𝐼⟶𝐵))
21	16, 20	mpbird 257	. . 3 ⊢ (𝜑 → 𝐸:(Base‘𝑀)⟶𝐵)
22	2, 17	frmdelbas 18758	. . . . . . . . 9 ⊢ (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
23	22	ad2antrl 728	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑦 ∈ Word 𝐼)
24	2, 17	frmdelbas 18758	. . . . . . . . 9 ⊢ (𝑧 ∈ (Base‘𝑀) → 𝑧 ∈ Word 𝐼)
25	24	ad2antll 729	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ Word 𝐼)
26	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝐴:𝐼⟶𝐵)
27		ccatco 14739	. . . . . . . 8 ⊢ ((𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼 ∧ 𝐴:𝐼⟶𝐵) → (𝐴 ∘ (𝑦 ++ 𝑧)) = ((𝐴 ∘ 𝑦) ++ (𝐴 ∘ 𝑧)))
28	23, 25, 26, 27	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴 ∘ (𝑦 ++ 𝑧)) = ((𝐴 ∘ 𝑦) ++ (𝐴 ∘ 𝑧)))
29	28	oveq2d 7362	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))) = (𝐺 Σg ((𝐴 ∘ 𝑦) ++ (𝐴 ∘ 𝑧))))
30	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝐺 ∈ Mnd)
31		wrdco 14735	. . . . . . . 8 ⊢ ((𝑦 ∈ Word 𝐼 ∧ 𝐴:𝐼⟶𝐵) → (𝐴 ∘ 𝑦) ∈ Word 𝐵)
32	23, 26, 31	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴 ∘ 𝑦) ∈ Word 𝐵)
33		wrdco 14735	. . . . . . . 8 ⊢ ((𝑧 ∈ Word 𝐼 ∧ 𝐴:𝐼⟶𝐵) → (𝐴 ∘ 𝑧) ∈ Word 𝐵)
34	25, 26, 33	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴 ∘ 𝑧) ∈ Word 𝐵)
35		eqid 2731	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
36	12, 35	gsumccat 18746	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∘ 𝑦) ∈ Word 𝐵 ∧ (𝐴 ∘ 𝑧) ∈ Word 𝐵) → (𝐺 Σg ((𝐴 ∘ 𝑦) ++ (𝐴 ∘ 𝑧))) = ((𝐺 Σg (𝐴 ∘ 𝑦))(+g‘𝐺)(𝐺 Σg (𝐴 ∘ 𝑧))))
37	30, 32, 34, 36	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg ((𝐴 ∘ 𝑦) ++ (𝐴 ∘ 𝑧))) = ((𝐺 Σg (𝐴 ∘ 𝑦))(+g‘𝐺)(𝐺 Σg (𝐴 ∘ 𝑧))))
38	29, 37	eqtrd 2766	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))) = ((𝐺 Σg (𝐴 ∘ 𝑦))(+g‘𝐺)(𝐺 Σg (𝐴 ∘ 𝑧))))
39		eqid 2731	. . . . . . . . 9 ⊢ (+g‘𝑀) = (+g‘𝑀)
40	2, 17, 39	frmdadd 18760	. . . . . . . 8 ⊢ ((𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → (𝑦(+g‘𝑀)𝑧) = (𝑦 ++ 𝑧))
41	40	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦(+g‘𝑀)𝑧) = (𝑦 ++ 𝑧))
42	41	fveq2d 6826	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g‘𝑀)𝑧)) = (𝐸‘(𝑦 ++ 𝑧)))
43		ccatcl 14478	. . . . . . . 8 ⊢ ((𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
44	23, 25, 43	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
45		coeq2 5798	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ++ 𝑧) → (𝐴 ∘ 𝑥) = (𝐴 ∘ (𝑦 ++ 𝑧)))
46	45	oveq2d 7362	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ++ 𝑧) → (𝐺 Σg (𝐴 ∘ 𝑥)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
47		ovex 7379	. . . . . . . 8 ⊢ (𝐺 Σg (𝐴 ∘ 𝑥)) ∈ V
48	46, 15, 47	fvmpt3i 6934	. . . . . . 7 ⊢ ((𝑦 ++ 𝑧) ∈ Word 𝐼 → (𝐸‘(𝑦 ++ 𝑧)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
49	44, 48	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦 ++ 𝑧)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
50	42, 49	eqtrd 2766	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g‘𝑀)𝑧)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
51		coeq2 5798	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 ∘ 𝑥) = (𝐴 ∘ 𝑦))
52	51	oveq2d 7362	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐺 Σg (𝐴 ∘ 𝑥)) = (𝐺 Σg (𝐴 ∘ 𝑦)))
53	52, 15, 47	fvmpt3i 6934	. . . . . . 7 ⊢ (𝑦 ∈ Word 𝐼 → (𝐸‘𝑦) = (𝐺 Σg (𝐴 ∘ 𝑦)))
54		coeq2 5798	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝐴 ∘ 𝑥) = (𝐴 ∘ 𝑧))
55	54	oveq2d 7362	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐺 Σg (𝐴 ∘ 𝑥)) = (𝐺 Σg (𝐴 ∘ 𝑧)))
56	55, 15, 47	fvmpt3i 6934	. . . . . . 7 ⊢ (𝑧 ∈ Word 𝐼 → (𝐸‘𝑧) = (𝐺 Σg (𝐴 ∘ 𝑧)))
57	53, 56	oveqan12d 7365	. . . . . 6 ⊢ ((𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼) → ((𝐸‘𝑦)(+g‘𝐺)(𝐸‘𝑧)) = ((𝐺 Σg (𝐴 ∘ 𝑦))(+g‘𝐺)(𝐺 Σg (𝐴 ∘ 𝑧))))
58	23, 25, 57	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝐸‘𝑦)(+g‘𝐺)(𝐸‘𝑧)) = ((𝐺 Σg (𝐴 ∘ 𝑦))(+g‘𝐺)(𝐺 Σg (𝐴 ∘ 𝑧))))
59	38, 50, 58	3eqtr4d 2776	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g‘𝑀)𝑧)) = ((𝐸‘𝑦)(+g‘𝐺)(𝐸‘𝑧)))
60	59	ralrimivva 3175	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+g‘𝑀)𝑧)) = ((𝐸‘𝑦)(+g‘𝐺)(𝐸‘𝑧)))
61		wrd0 14443	. . . 4 ⊢ ∅ ∈ Word 𝐼
62		coeq2 5798	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝐴 ∘ 𝑥) = (𝐴 ∘ ∅))
63		co02 6208	. . . . . . . 8 ⊢ (𝐴 ∘ ∅) = ∅
64	62, 63	eqtrdi 2782	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 ∘ 𝑥) = ∅)
65	64	oveq2d 7362	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐺 Σg (𝐴 ∘ 𝑥)) = (𝐺 Σg ∅))
66		eqid 2731	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
67	66	gsum0 18589	. . . . . 6 ⊢ (𝐺 Σg ∅) = (0g‘𝐺)
68	65, 67	eqtrdi 2782	. . . . 5 ⊢ (𝑥 = ∅ → (𝐺 Σg (𝐴 ∘ 𝑥)) = (0g‘𝐺))
69	68, 15, 47	fvmpt3i 6934	. . . 4 ⊢ (∅ ∈ Word 𝐼 → (𝐸‘∅) = (0g‘𝐺))
70	61, 69	mp1i 13	. . 3 ⊢ (𝜑 → (𝐸‘∅) = (0g‘𝐺))
71	21, 60, 70	3jca 1128	. 2 ⊢ (𝜑 → (𝐸:(Base‘𝑀)⟶𝐵 ∧ ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+g‘𝑀)𝑧)) = ((𝐸‘𝑦)(+g‘𝐺)(𝐸‘𝑧)) ∧ (𝐸‘∅) = (0g‘𝐺)))
72	2	frmd0 18765	. . 3 ⊢ ∅ = (0g‘𝑀)
73	17, 12, 39, 35, 72, 66	ismhm 18690	. 2 ⊢ (𝐸 ∈ (𝑀 MndHom 𝐺) ↔ ((𝑀 ∈ Mnd ∧ 𝐺 ∈ Mnd) ∧ (𝐸:(Base‘𝑀)⟶𝐵 ∧ ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+g‘𝑀)𝑧)) = ((𝐸‘𝑦)(+g‘𝐺)(𝐸‘𝑧)) ∧ (𝐸‘∅) = (0g‘𝐺))))
74	4, 5, 71, 73	syl21anbrc 1345	1 ⊢ (𝜑 → 𝐸 ∈ (𝑀 MndHom 𝐺))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∅c0 4283   ↦ cmpt 5172   ∘ ccom 5620  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  Word cword 14417   ++ cconcat 14474  Basecbs 17117  +_gcplusg 17158  0_gc0g 17340   Σ_g cgsu 17341  Mndcmnd 18639   MndHom cmhm 18686  freeMndcfrmd 18752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-card 9829  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-2 12185  df-n0 12379  df-z 12466  df-uz 12730  df-fz 13405  df-fzo 13552  df-seq 13906  df-hash 14235  df-word 14418  df-concat 14475  df-struct 17055  df-sets 17072  df-slot 17090  df-ndx 17102  df-base 17118  df-ress 17139  df-plusg 17171  df-0g 17342  df-gsum 17343  df-mgm 18545  df-sgrp 18624  df-mnd 18640  df-mhm 18688  df-submnd 18689  df-frmd 18754

This theorem is referenced by:  frmdup3  18772