Theorem frmdup1 18674

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 18669	. . 3 ⊢ (𝐼 ∈ 𝑋 → 𝑀 ∈ Mnd)
4	1, 3	syl 17	. 2 ⊢ (𝜑 → 𝑀 ∈ Mnd)
5		frmdup.g	. 2 ⊢ (𝜑 → 𝐺 ∈ Mnd)
6	5	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐼) → 𝐺 ∈ Mnd)
7		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐼) → 𝑥 ∈ Word 𝐼)
8		frmdup.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴:𝐼⟶𝐵)
9	8	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐼) → 𝐴:𝐼⟶𝐵)
10		wrdco 14720	. . . . . . 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 18649	. . . . . 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 7062	. . . 4 ⊢ (𝜑 → 𝐸:Word 𝐼⟶𝐵)
17		eqid 2736	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
18	2, 17	frmdbas 18662	. . . . . 6 ⊢ (𝐼 ∈ 𝑋 → (Base‘𝑀) = Word 𝐼)
19	1, 18	syl 17	. . . . 5 ⊢ (𝜑 → (Base‘𝑀) = Word 𝐼)
20	19	feq2d 6654	. . . 4 ⊢ (𝜑 → (𝐸:(Base‘𝑀)⟶𝐵 ↔ 𝐸:Word 𝐼⟶𝐵))
21	16, 20	mpbird 256	. . 3 ⊢ (𝜑 → 𝐸:(Base‘𝑀)⟶𝐵)
22	2, 17	frmdelbas 18663	. . . . . . . . 9 ⊢ (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
23	22	ad2antrl 726	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑦 ∈ Word 𝐼)
24	2, 17	frmdelbas 18663	. . . . . . . . 9 ⊢ (𝑧 ∈ (Base‘𝑀) → 𝑧 ∈ Word 𝐼)
25	24	ad2antll 727	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ Word 𝐼)
26	8	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝐴:𝐼⟶𝐵)
27		ccatco 14724	. . . . . . . 8 ⊢ ((𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼 ∧ 𝐴:𝐼⟶𝐵) → (𝐴 ∘ (𝑦 ++ 𝑧)) = ((𝐴 ∘ 𝑦) ++ (𝐴 ∘ 𝑧)))
28	23, 25, 26, 27	syl3anc 1371	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴 ∘ (𝑦 ++ 𝑧)) = ((𝐴 ∘ 𝑦) ++ (𝐴 ∘ 𝑧)))
29	28	oveq2d 7373	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))) = (𝐺 Σg ((𝐴 ∘ 𝑦) ++ (𝐴 ∘ 𝑧))))
30	5	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝐺 ∈ Mnd)
31		wrdco 14720	. . . . . . . 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 14720	. . . . . . . 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 2736	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
36	12, 35	gsumccat 18651	. . . . . . 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 1371	. . . . . 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 2776	. . . . 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 2736	. . . . . . . . 9 ⊢ (+g‘𝑀) = (+g‘𝑀)
40	2, 17, 39	frmdadd 18665	. . . . . . . 8 ⊢ ((𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → (𝑦(+g‘𝑀)𝑧) = (𝑦 ++ 𝑧))
41	40	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦(+g‘𝑀)𝑧) = (𝑦 ++ 𝑧))
42	41	fveq2d 6846	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g‘𝑀)𝑧)) = (𝐸‘(𝑦 ++ 𝑧)))
43		ccatcl 14462	. . . . . . . 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 5814	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ++ 𝑧) → (𝐴 ∘ 𝑥) = (𝐴 ∘ (𝑦 ++ 𝑧)))
46	45	oveq2d 7373	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ++ 𝑧) → (𝐺 Σg (𝐴 ∘ 𝑥)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
47		ovex 7390	. . . . . . . 8 ⊢ (𝐺 Σg (𝐴 ∘ 𝑥)) ∈ V
48	46, 15, 47	fvmpt3i 6953	. . . . . . 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 2776	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g‘𝑀)𝑧)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
51		coeq2 5814	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 ∘ 𝑥) = (𝐴 ∘ 𝑦))
52	51	oveq2d 7373	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐺 Σg (𝐴 ∘ 𝑥)) = (𝐺 Σg (𝐴 ∘ 𝑦)))
53	52, 15, 47	fvmpt3i 6953	. . . . . . 7 ⊢ (𝑦 ∈ Word 𝐼 → (𝐸‘𝑦) = (𝐺 Σg (𝐴 ∘ 𝑦)))
54		coeq2 5814	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝐴 ∘ 𝑥) = (𝐴 ∘ 𝑧))
55	54	oveq2d 7373	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐺 Σg (𝐴 ∘ 𝑥)) = (𝐺 Σg (𝐴 ∘ 𝑧)))
56	55, 15, 47	fvmpt3i 6953	. . . . . . 7 ⊢ (𝑧 ∈ Word 𝐼 → (𝐸‘𝑧) = (𝐺 Σg (𝐴 ∘ 𝑧)))
57	53, 56	oveqan12d 7376	. . . . . 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 2786	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g‘𝑀)𝑧)) = ((𝐸‘𝑦)(+g‘𝐺)(𝐸‘𝑧)))
60	59	ralrimivva 3197	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+g‘𝑀)𝑧)) = ((𝐸‘𝑦)(+g‘𝐺)(𝐸‘𝑧)))
61		wrd0 14427	. . . 4 ⊢ ∅ ∈ Word 𝐼
62		coeq2 5814	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝐴 ∘ 𝑥) = (𝐴 ∘ ∅))
63		co02 6212	. . . . . . . 8 ⊢ (𝐴 ∘ ∅) = ∅
64	62, 63	eqtrdi 2792	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 ∘ 𝑥) = ∅)
65	64	oveq2d 7373	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐺 Σg (𝐴 ∘ 𝑥)) = (𝐺 Σg ∅))
66		eqid 2736	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
67	66	gsum0 18539	. . . . . 6 ⊢ (𝐺 Σg ∅) = (0g‘𝐺)
68	65, 67	eqtrdi 2792	. . . . 5 ⊢ (𝑥 = ∅ → (𝐺 Σg (𝐴 ∘ 𝑥)) = (0g‘𝐺))
69	68, 15, 47	fvmpt3i 6953	. . . 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 18670	. . 3 ⊢ ∅ = (0g‘𝑀)
73	17, 12, 39, 35, 72, 66	ismhm 18603	. 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 1344	1 ⊢ (𝜑 → 𝐸 ∈ (𝑀 MndHom 𝐺))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∅c0 4282   ↦ cmpt 5188   ∘ ccom 5637  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  Word cword 14402   ++ cconcat 14458  Basecbs 17083  +_gcplusg 17133  0_gc0g 17321   Σ_g cgsu 17322  Mndcmnd 18556   MndHom cmhm 18599  freeMndcfrmd 18657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-word 14403  df-concat 14459  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-frmd 18659

This theorem is referenced by:  frmdup3  18677